how to do math research as an undergraduate

Undergraduate Research

Where to start:.

A good starting point is the Harvard College Undergraduate Research and Fellowships page. The Office of Undergraduate Research and Fellowships administers research programs for Harvard College undergraduates. Check out the website . Another resource is OCS , the Harvard Office of Career Services. It offers help on preparing a CV or cover letters and gives advice on how to network, interview, etc. Their website is here . Other Sources that can provide additional information on Scholarships, awards, and other grants:

• Committee on General Scholarships: more …
• Office of International Programs: more …
• Student Employment Office: more …

Independent study in Mathematics

Students who would like to do some independent study or a reading class please read the pamphlet page . about Math 91r.

THE ANNUAL OCS SUMMER OPPORTUNITIES FAIR

The Office of Career Services hosts summer programs to help you begin your summer search. Programs are both Harvard affiliated and public or private sector and include internships, public service, funding, travel, and research (URAF staff will be there to answer your questions!). Check out the website.

Harvard-Amgen Scholars program in Biotechnology

Check out the Harvard-Amgen Scholars Program Learn about Harvard’s Amgen 10-week intensive summer research program, one of ten Amgen U.S. programs that support research in biotechnology. The Harvard program includes faculty projects in FAS science departments, SEAS, the Wyss Institute for Biologically-inspired Engineering, and the School of Medicine, open to rising juniors and seniors in biotechnology-related fields.

PRIMO program

The Program for research in Markets and Organizations (PRIMO) is a 10-week program for Harvard undergraduates who wish to work closely with Harvard Business School faculty on research projects.

Harvard Undergraduate Research Events

• Wednesday, October 10, 12:00-1: 20 PM – Fall Undergraduate Research Spotlight. Come and meet Harvard undergraduate peers who will showcase their research projects and share their experiences conducting research at Harvard and abroad, followed by reception and deserts. Event program and list of presentations can be found here: here (pizza and desserts while supplies last). Free for Harvard students. Cabot Library 1st floor Discovery Bar.
• Wednesday, October 17, 12:00-1: 00 PM – Undergraduate Science Research Workshop. Workshop facilitators Dr. Margaret A. Lynch, (Assoc. Director of Science #Education) and Dr. Anna Babakhanyan, (Undergraduate Research Advisor) will help Harvard students learn about science research landscape at Harvard. You will learn about what kind of research (basic science vs. clinical, various research areas) is available at Harvard, where you can conduct research, the types of undergraduate research appointments, how to find a lab that fits, interviewing and more. In addition, the workshop will provide strategies for students to prepare for the Annual HUROS Fair, see below. No registration is required for this event (pizza while supplies last). Free for all Harvard students. Cabot Library first floor Discover Bar. More.

Outside Programs

Caltech always announces two summer research opportunities available to continuing undergraduate students. Examples: WAVE Student-Faculty Programs The WAVE Fellows program provides support for talented undergraduates intent on pursuing a Ph.D. to conduct a 10-week summer research project at Caltech. And then there is the AMGEN Scholars program. See the website for more details.

Johns Hopkins Summer 2018 Opportunities

The Johns Hopkins University Center for Talented Youth (CTY) is seeking instructors and teaching assistants for our summer programs. CTY offers challenging academic programs for highly talented elementary, middle, and high school students from across the country and around the world. Positions are available at residential and day sites at colleges, universities, and schools on the East and West coasts, as well as internationally in Hong Kong. Website

Math REU list from AMS

Mellon Mays opportunities awareness

The Mellon Mays Undergraduate Fellowship Program ( MMUF ) selects ten students in their sophomore year to join a tightly-knit research community during junior and senior years to conduct independent research in close collaboration with a faculty mentor. Join us at this information session to find out more about the program. MMUF exists to counter the under-representation of minority groups on college and university faculties nationwide through activities designed to encourage the pursuit of the Ph.D. in the humanities and core sciences.

MIT Amgen and UROP

You may be familiar with the Amgen Scholars Program, a summer research program in science and biotechnology. The Massachusetts Institute of Technology is a participant in the Amgen-UROP Scholars Program for a ninth year. UROP is MIT’s Undergraduate Research Opportunities Program. The mission of the Amgen-UROP Scholars Program is to provide students with a strong science research experience that may be pivotal in their undergraduate career, cultivate a passion for science, encourage the pursuit of graduate studies in the sciences, and stimulate interest in research and scientific careers. MIT is delighted to invite undergraduate students from other colleges and universities to join our research enterprise. We value the knowledge, experience, and enthusiasm these young scholars will bring to our campus and appreciate this opportunity to build a relationship with your faculty and campus.

More REU's, not only math

The National Science Foundation Research Experiences for Undergraduates (REU) NSF funds a large number of research opportunities for undergraduate students through its REU Sites program. An REU Site consists of a group of ten or so undergraduates who work in the research programs of the host institution. Each student is associated with a specific research project, where he/she works closely with the faculty and other researchers. Students are granted stipends and, in many cases, assistance with housing and travel. Undergraduate students supported with NSF funds must be citizens or permanent residents of the United States or its possessions. An REU Site may be at either the US or foreign location. By using the web page , search for an REU Site, you may examine opportunities in the subject areas supported by various NSF units. Also, you may search by keywords to identify sites in particular research areas or with certain features, such as a particular location. Students must contact the individual sites for information and application materials. NSF does not have application materials and does not select student participants. A contact person and contact information are listed for each site.

Here is a link with more information about summer programs for undergraduates at NSA: NSA The most math-related one is DSP, but those students who are more interested in computer science could also look at, say, CES SP. They are all paid with benefits and housing is covered. Note that application deadlines are pretty early (usually mid-October). The application process will involve usually a few interviews and a trip down to DC.

NSF Graduate Research Fellowships

US citizens and permanent residents who are planning to enter graduate school in the fall of 2019 are eligible (as are those in the first two years of such a graduate program, or who are returning to graduate school after being out for two or more years). The program solicitation contains full details. Information about the NSF Graduate Research Fellowship Program (GRFP) is here . The GRFP supports outstanding graduate students in NSF-supported science, technology, engineering, and mathematics disciplines who are pursuing research-based Masters and doctoral degrees at accredited United States institutions. The program provides up to three years of graduate education support, including an annual, 000 stipend. Applications for Mathematical Sciences topics are due October 26, 2018.

Pathway to Science

summer research listings from pathways to science.

Perimeter Institute

Applications are now being accepted for Perimeter Institute’s Undergraduate Theoretical Physics Summer Program. The program consists of two parts:

• Fully-Funded Two Week Summer School (May 27 to June 7, 2019) Students are immersed in Perimeter’s dynamic research environment — attending courses on cutting-edge topics in physics, learning new techniques to solve interesting problems, working on group research projects, and potentially even publishing their work. All meals, accommodation, and transportation provided
• Paid Research Internship (May 1 to August 30, 2019, negotiable) Students will work on projects alongside Perimeter researchers. Students will have the opportunity to develop their research skills and absorb the rich variety of talks, conferences, and events at the Perimeter Institute. Applicants can apply for the two-week summer school or for both the summer school and the research internship. Summer school and internship positions will be awarded by February 28, 2019. Selected interns will be contacted with the research projects topics. All research interns must complete the two-week summer school.

Apply online at perimeterinstitute.ca/undergrad

Stanford resident counselors

Stanford Pre-Collegiate Institutes is hiring Residential Counselors for the summer to work with the following courses:

• Cryptography (grades 9-10)
• Knot Theory (grades 10-11)
• Logic and Problem Solving (grades 8-9)
• Number Theory (grades 9-11)
• Excursions in Probability (grades 8-9)
• Discrete Mathematics (grades 9-10)
• The Mathematics of Symmetry (grades 10-11)
• Mathematical Puzzles and Games (grades 8-9)

Stanford Pre-Collegiate Institutes offers three-week sessions for academically talented high school students during June and July. Interested candidates can learn more about our positions and apply by visiting our employment website .

Summer Research 2019 at Nebraska

We are now accepting applications for the University of Nebraska’s 2019 Summer Research Program, and we’d like to encourage your students to apply. Details.

Princeton University Math Club

Princeton University

Undergraduate Research

Research in mathematics and its allies takes a variety of forms: from the most abstract algebraic geometry to the most concrete problems in finance and everything in between. Accordingly, there are many ways to get involved in mathematical research. In general, as an underclassman, the best way to do so is to participate in an REU or other research program during the summer. After that, in your junior and senior years, Princeton provides a natural avenue to research: the junior papers (JP), one per semester during your junior year and, of course, the senior thesis, a year-long project undertaken in your final year. Depending on your background and level of interest in research, however, you might want to consider looking for research opportunities during the year, even in your first two years.

Getting Started [Show] Getting Started [Hide] How do you get started doing research? In general, the answer is to build background through relevant coursework first. Particularly in pure mathematics, it’s very difficult to jump in without having good preparation. Choosing courses that will prepare you for what you want to do, and help you figure out what you want to do for that matter, is important. Necessarily, the details will vary from person to person, but some advice applies across the board. First: choose carefully and plan ahead. Choose your courses thoughtfully. That means thinking about what you want to get out of each course, how your courses in any given semester fit together (in terms of workload, etc.), and what trajectory you’re trying to follow. A lot of this will be uncertain, especially at the start, but thinking about these issues will help remove that uncertainty. Second: consult your peers, advisers, and professors (not necessarily in that order). They have been where you are now, and they can help you get where you are trying to go. You will of course have to choose whom you ask, and have to combine multiple—often conflicting—sources of advice, but the people around you are an invaluable resource in finding your path. That being said, you should take advantage of opportunities to get involved in research early on: it’s often possible to find a good project to work on (at an REU, other research program, or at Princeton) even as a freshman. Early in your undergraduate years, you should be open to the possibilities; research experience, even if it isn’t in the area you ultimately want to pursue, is very useful, notably in helping you find your ultimate interests. As always, consult your academic advisers, professors, and friends (especially older friends) for advice.

Seminars, Lectures, and Colloquia [Show] Seminars, Lectures, and Colloquia [Hide] Attending talks is an important way to find areas of research that interest you. These come in at least three varieties: the Undergraduate Colloquium, which includes faculty, graduate, and undergraduate speakers; the Graduate Student Seminar (GSS), given by graduate students on their research for the benefit of their peers and undergrads; and the various department seminars. If you’re interested in the former two, join the Math Club Listserv and talk to LeeAnn Coleman in the department office to find out how to be added to the GSS mailing list. To get an idea of what goes on in the latter, just look at the seminar listings on the math website at the beginning of every week and see if anything looks interesting. Most seminars provide abstracts, and these will give you an idea about whether you will understand the talk. Look out especially for the department colloquia, because these are usually pitched at a non-specialist level and reasonably accessible—not to mention generally given by very good speakers. The colloquium speaker also sometimes gives a lunchtime talk the day of the colloquium, a practice unique to Princeton; these are also worth attending. As you attend talks, keep in mind that you will sometimes misgauge the difficulty of a talk and at times the speaker will not be very engaging. This will inevitably happen some of the time, but don’t let it discourage you. Try to get at least a little bit out of each talk you go to; an excellent mathematician once remarked to me that he was satisfied if he could come away from a talk with a single sentence of new knowledge: this is a bit extreme, but the general idea is important. In the end, you should attend seminars because you find their subject matter interesting (or intriguing). While they may help you get ideas for research, when you are just starting out, they will more likely point you to areas that are worth looking at. Moreover, attending seminars has long-term benefits for the aspiring researcher.

Junior Seminars [Show] Junior Seminars [Hide] Juniors in the department are required either to write a junior paper or participate in a junior seminar during both semesters. Junior seminars are a learning environment with which you are probably completely unfamiliar. First, it’s largely up to you how much to engage with the lectures and the course overall. For this reason, it’s especially important to pick a seminar whose subject matter interests you. This may be a challenge, as very few seminars are offered, but do your best. Actively following the speaker’s exposition and asking questions where appropriate—never be afraid to ask a question if something is unclear!—are great ways to stay focused on lectures while attending them. To keep up with the course overall, you will want to do the assigned readings, even those that are for talks other than your own. At some point during the semester, you will spend a couple of weeks learning a certain bit of the main topic of the seminar, often contained in a chapter of a textbook or a journal article, and then make a presentation at the seminar. It’s important to give a good talk: it’s your turn to teach your classmates the material. Whether this is your first talk, or you are a veteran lecturer, the best way to ensure a good talk is practice, practice, practice. Practice it at least twice before you give it in the seminar, ideally with another student or even the instructor (feel free to ask). Junior Seminars culminate in a final paper on some topic related to the theme of the seminar. The details vary from seminar to seminar, but, unlike the junior paper and senior thesis, which sometimes include original components, the paper will be purely expository and will generally represent a much more significant effort than a problem set. For advice on finding a topic, consult the section below. Your adviser in this context will be the seminar instructor. Also keep in mind that your paper will not be as elaborate as a senior thesis or junior paper.

What Type of Project is a Senior Thesis (or Junior Paper)? [Show] What Type of Project is a Senior Thesis (or Junior Paper)? [Hide] It’s probably a good idea to start off by saying a bit about what type of “project” a senior thesis and junior paper is. Even if you have participated in an REU or another summer math program for undergraduates, your biggest question might just be, “What type of project does one do with a faculty adviser?” Most theses and JP’s center on a particular important result, research area or program, or major conjecture. The thesis could give an exposition of a proof of a major result (perhaps extending it to slightly more cases), or of major partial results, or of recent results in a research area. It is also possible to undertake original research—though if this is what you want to do, you should ideally prepare yourself thoroughly for it during your first three years (see above). Junior papers are similar, though necessarily less involved, and the project can vary from a specific unsolved problem to an introductory exploration of an area unfamiliar to the student. Ultimately, a thesis or JP is an early step in the career of an aspiring researcher and, accordingly, it is foundational—you will not prove the Riemann hypothesis (probably), you might not end up proving anything, but you will gain valuable experience and learn mathematics that will continue to be useful to you later on. For specific examples, look up old theses in the library and, as always, talk to your older friends! Finally, remember that both a senior thesis and JP require significant writing. Mathematical writing is rather different from the writing you have probably done in other contexts. Various resources on the subject can be found on Terry Tao’s blog, here . In this guide, we’ll just point out that writing clear proofs and definitions makes the relevant concepts much clearer to you and can help you notice subtleties of—and subtle mistakes in—your arguments. One way to help yourself do this, and to find a style to start yourself off with, is to recall a textbook you read that was particularly written and model your writing on its style (not the content, though).

Finding an Adviser [Show] Finding an Adviser [Hide] Starting a thesis or JP requires two major steps: choosing an adviser and choosing a topic. In the overwhelming majority of cases, the former comes first. Choosing your adviser carefully is important. Your adviser’s style and the compatibility between the two of you will deeply influence the quality of your experience. Since you have virtually no time as an upperclassman to experiment with possible advisers—though this will likely happen accidentally anyway—you should consult older students and the faculty academic advisers to figure out which professors might be a good fit for you in terms of research interests and advising style. As always, you will have to be proactive to ensure your experience is all that it can be; ask your peers many questions: about the frequency and content of meetings, the expectations for an undergraduate project (too low? too high?), the level of preparation expected, and so on. Once you have an adviser, you will still need to find an effective way to work together. Sometimes, this will come naturally; that’s especially likely if your adviser often takes undergraduate students. Be that as it may, figure out how often meeting with your adviser is productive; once a week is standard, but some professors prefer biweekly meetings. Even if you have nothing to report, meeting with your adviser helps both of you stay in touch with the project and is an integral part of the research experience. You will also want to prepare for your meetings so as to get the most out of them. While meetings will be your primary interaction with your adviser, e-mails and other day-to-day interactions can be nearly as important. These generally take the form of questions and, here again, you will need to figure out how to make them work best for you. Experience is, for better or for worse, the only real way to learn how to do this. Finally, whoever your adviser is, you will benefit from making friends with their graduate students and postdocs (short for “postdoctoral fellows,” researchers who have recently obtained Ph.D.’s); they can serve as secondary advisers who can help you on a day-to-day basis—and share their experiences with early career research.

Finding a Project [Show] Finding a Project [Hide] Choosing a project is just as important as choosing an adviser. Working on a problem that fascinates and excites you will make your research more enjoyable and rewarding: it is also the best way to ensure you stay motivated throughout the project. Your adviser will likely suggest at least one project at your first meeting, or over e-mail beforehand. Starting the discussion early, whether by e-mail or in person, is helpful; doing so will allow you to go through a few possibilities before committing to one. To decide if an idea is one you want to pursue, you should read about it online, in books, and in the technical literature (for specific places to look, consult your adviser). After you’ve had a chance to look into it, don’t hesitate to voice your doubts about a project if it seems over your head or not interesting to you! All this being said, you should choose a problem early in the term—certainly within the first two or three meetings with your adviser. The easiest way to achieve this is to approach potential advisers well before term starts and to be pro-active about finding a project in advance. While it’s hard to give general advice about choosing a problem, there are a few things to keep in mind. First, choose a project in an area of active research. This will have many benefits, most notably perhaps that you will have access to a wide selection of papers that can help you as you work on your problem. Second, as mentioned above, choosing a project is to a large extent a judgment call: you should choose a topic you find interesting and beautiful if at all possible. Finally, keep in mind that in research, especially early on, things rarely go as planned: balance perseverance with flexibility, and don’t be afraid to change course if necessary, particularly as you figure out what you really want to do with your research.

Advice on the Research Process [Show] Advice on the Research Process [Hide] Read, read, read. And Google. Take time to read about things related to your research. Google—especially Google Scholar—is an excellent way to find material useful to you, often better than arXiv or mathscinet, two popular online technical libraries. Being a Princeton undergraduate means getting access to most journal articles you’ll need, and you should definitely take advantage of the convenience! Google can help answer queries both big and small, from “Find a readable introduction to algebraic K-theory” (huge) to “What does convex cocompact mean?” (tiny). In fact, Googling is almost always the fastest way to resolve any confusion you have as you struggle to understand an idea or term. Ultimately, you may need to read in a more structured way—by focusing on a book or a specific collection of journal articles, for instance—but unstructured reading can help prepare you for that, and help give you a general idea of how the area you’re working in, well, works. Daydreaming (about research) is good for you. If you have a specific problem you need to solve, whether it’s finishing a mostly finished proof, solving a tricky problem, or understand a particularly hard part of a paper you’re reading, it’s frequently useful to daydream about it at random quiet moments during the way: while you walk to class, shower, wait for a friend to meet you for dinner, or whatever. You’ll likely find, as countless mathematicians, scientists, and humanists before you have, that you have more idea during this unstructured time than during scheduled blocks set aside for research (though they have their place—see below). You need structure, too. Regularly setting aside time to work on research is essential. Progress in research is nearly always incremental and non-linear: it takes time and patience. A thesis or a junior paper is a fundamentally long-term project, and to do your best work you will need to treat it that way. To keep yourself productive during structured time, it helps to focus on concrete tasks (e.g., “read this paper”, “work on this part of this proof”, “understand this technique I need”) and to work on answering specific questions.

Other Useful Resources

Terence Tao on time management

Ravi Vakil’s advice for graduate students (some of which is applicable to undergraduates)

Acknowledgements

Many thanks to John Pardon ’11 and Max Rabinovich ’13 for contributing this article.

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Department of Mathematics

Undergraduate math research.

Performing undergraduate research is one of the very best ways to make the most out of your college experience. It will empower you to take the tools you learn in the classroom and apply them to real-world problems, all while building invaluable mentoring relationships with faculty in a professional setting.

For more than 25 years, the mathematics department has been proud to partner with the National Science Foundation to host Research Experiences for Undergraduates (REU). Remember to explore the additional funding opportunities from OSU and the College of Science which can help you get paid while doing the research you love!

Research Experiences for Undergraduates

Learn more about the reu program here , summer undergraduate research experience, learn more about the sure science program and how to apply , undergraduate research, scholarship and the arts (ursa).

The URSA Engage program provides funding for undergraduate students to work alongside mathematics faculty on mathematical research projects. URSA Engage is designed to establish mentoring relationships for undergraduates early in their academic programs at OSU and provide students opportunities to pursue research under the guidance of an OSU faculty member.

Learn more about URSA Engage and how to apply 

Going above and beyond in undergraduate research.

Mathematics and writing senior awarded Department of Energy fellowship

Prestigious research internship opens new possibilities for double-major science student

Using mathematical biology to explore everything from ecological phenomenon to cancer cells

URSA Engage funds undergraduates to work on mathematics research

Related stories, across the department, explore related stories.

Mathematics graduate thrives with simple philosophy: ‘Why not?’

Classroom puzzles to cosmic insights: Students and professor demystify mathematical theorem

A ‘limitless degree’: Physics senior pushes the boundaries of his field

Mathematics senior finds effective teaching strategies on her path to a graduate degree

Quick Links +

Undergraduate research.

Undergraduate Research is an great opportunity to get more involved in the Math Department while working directly with faculty to expand the bounds of existing knowledge. There are many benefits to conducting undergraduate research, including the opportunity to:

• Explore an area of interest more deeply
• Learn first-hand about research to determine if you would like to pursue advanced study after your bachelor's degree
• Gain experience that is often highly valued by graduate school admissions committees
• Present your findings at the UC Davis Undergraduate Research Conference or other symposia, and possibly co-author a published paper
• Build relationships with faculty, which can lead to personalized letters of recommendation

MAT 099/199: Research Credit

Students completing undergraduate research (MAT 99/199) will receive lower/upper division credit toward graduation requirements (180 unit requirement) but will not receive credit toward their major. Every 1 unit of credit corresponds to 3 hours of work a week, or 30 hours of work per quarter.

MAT 099 : Undergraduates students who have 83 units or less completed (lower division credit)

MAT 199 : Undergraduate students who have 84 units or more completed (upper division credit)

Academic Year Research Projects

Each quarter, a list of advertised research projects (along with a link to apply to these projects) can be found on the Quarterly Research Projects webpage .

Undergrad Research Conference 2021

The following are presentations held during the annual conference: 

Research in Ramsey Theory and Automatic Theorem Proving

One-Dimensional range restrcited C^2 Interpolation Algorithm

Approximating K-Means using an ADMM Approach

Recovering Individual Based Model Outcomes on Spatiotemporally Coarsened Data

Summer Research Projects

We have research projects happening each summer. For a list of summer projects (and to apply, if they are taking new students), see the Summer Research webpage .

Undergraduate Research Advisor

The undergraduate research advisor is a faculty member who has agreed to help students with all details related to a Research Experience for Undergraduates (REU). Primary responsibilities include helping students on finding a mentor, selecting students for available fellowships, and advertising other REU programs and fellowships outside of UC Davis.

Contact him for help!

Prof. Bob Guy! [email protected]

Undergraduate Senior Thesis in Mathematics

The Department of Mathematics at UC Davis offers undergraduate students the opportunity to participate in original mathematical research that will culminate in a senior undergraduate thesis. A student taking advantage of this opportunity would work under the guidance of a faculty mentor, pursuing original research.

Eligibility

To be eligible to write a senior thesis, advanced undergraduate students must

• Have a GPA of 3.5 or above in upper level courses in Mathematics OR be in the Honors Program .
• Find an appropriate research mentor willing to supervise their research project. Research supervisors can be faculty from the Mathematics Department or faculty from the Graduate Group in Applied Mathematics (GGAM) .
• Students that do not meet the GPA requirement may also be permitted to write a senior thesis. However, they must be nominated by a faculty member who is willing and able to supervise them in a research project, and the nomination must be approved by Undergraduate Research Committee.

Requirements for completion of an undergraduate thesis

• Students must complete two full quarters of research prior to graduation.
• While pursuing their research, students must complete MAT 199 (Special Studies for Advanced Undergraduates) and/or MAT 194 (Undergraduate Thesis) for a total of at least 6 units of credit over two quarters.
• Students must complete a thesis of sufficient quality and substance. Both the research supervisor and the Undergraduate Program Committee Chair (UPC) will evaluate a student's thesis.
• The UPC must receive a draft of the thesis no later than the beginning of the eighth week of the quarter in which the student plans to graduate. The UPC must also receive the supervisor's evaluation of the thesis at this time. The final copy of the thesis must be submitted to the department by the end of finals week.

Students who have a cumulative GPA that exceeds a College of Letters & Science requirement, complete at least 6 units of credits of MAT 199 and/or 194 over two quarters, and complete theses may be considered for graduation with High Honors or Highest Honors. All students completing theses will be recognized at the June Graduation Reception and in the Fall Department Newsletter.

Undergraduate Senior Thesis - Archived Submissions

Research programs at uc davis.

UC LEADS is a comprehensive two-year program designed to prepare economically or educationally disadvantaged students for success in doctoral degree programs in science, engineering, technology, or mathematics. Students are assigned individual faculty mentors, who guide them in academic year and summer research experiences. Scholars receive stipends and housing for their summer research and participate in extensive academic enrichment activities, including presenting their research at the annual UC LEADS Symposium.

MURPPS is a UC Davis undergraduate mentoring program designed to increase the number of disadvantaged students who pursue graduate studies in the physical and mathematical sciences by offering students the chance to work with professors on research projects relevant to their major. The goal of MURPPS is to help create a diverse post-graduate population in the Physical and Mathematical Sciences. Students are paid a quarterly stipend. MURPPS also runs seminars which introduce students to faculty and research projects, offers academic guidance, and access to the tutoring at the Academic Assistance and Tutoring Centers .

The Mentorships for Undergraduate Research in Agriculture, Letters, and Science (MURALS) is a two-quarter program that encourages students to further their education beyond the baccalaureate degree by providing opportunities to participate in academic research with a faculty mentor. MURALS welcomes students from all academic disciplines. For application information, please go to their "How to Apply" webpage.

Online Research Search Engines

The following are just a few suggested links that may prove helpful when you begin your research endeavors at UC Davis. Each site — or search engine — offers a wide variety of research aides including published articles and reviews, mathematical documents for specific phrases, or membership listings.

• Undergraduate Research Opportunities at UC Davis A comprehensive site relative to undergraduate research related to UC Davis, both on and off campus. Includes a link to the Undergraduate Research Center.
• eScholarship The publication listings for the entire University of California. This includes all research topics, but also mathematics.
• Front for the Mathematics ArXiv A front end to assist in searching the mathematics articles archive maintained by Cornell University.
• MathSciNet American Mathematical Society's listing of math reviews on the web.
• American Mathematical Society
• Society for Industrial and Applied Mathematics
• Combined Membership List A searchable database of the combined membership listings of the American Mathematical Society (AMS), Mathematical Association of America (MAA), Society for Industrial and Applied Mathematics (SIAM), American Mathematical Association of Two-Year Colleges (AMATYC), and the Association for Women in Mathematics (AWM).

PRIMES: How to Succeed in Mathematical Research

Professor Pavel Etingof, the Chief Research Advisor of PRIMES, gives his advice.

A recipe for succeeding in mathematical research would have been a laughable oxymoron, as this is a quintessentially creative endeavor. Yet, I'd like to offer a few tips for those who are taking their first steps along this fascinating path.

1. Be stubborn and at the same time flexible. In mathematical research, unlike olympiads, solving a problem takes weeks and months rather than hours, and there is no instant gratification. Yet, you don't want to rack your brains for too long without progress if you get stuck. Ask for help, or switch to another problem!

2. Be knowledge-seeking. If you did not make progress on your problem but learned something instead, then you did in fact make progress. Besides being intellectually rewarding by itself, at the end of the day learning always pays off in practical terms as well -- it helps you obtain better results. In fact, look for directions of study that will force you to learn something! The more you learn to enjoy the process of doing mathematics (rather than the result), the better mathematician you will be.

3. Mathematical research is an intrinsically social activity. Discuss a lot, seek help/advice/feedback from others, rather than be stuck for a long time.

4. Split the problem into small, bite-size steps, or ask your mentor to do so for you. You want to have something doable on your agenda at all times.

5. Consider examples. Look for the simplest example that captures the phenomenon ( Gelfand's principle). Also tracing through the proof with the simplest nontrivial example is a great way of checking a proof. It often uncovers subtle errors that are harder to see in the more general context, or ways to drastically simplify the proof, by identifying parts which are a "red herring."

6. Have several questions to think about so that you can switch from one to the other.

7. Use the Internet ( Wikipedia is usually good for math, even though you have to be careful with it). Also Google (or Google Scholar ) keyword search is often helpful. But you have to know the right keywords, and it sometimes takes some thinking to come up with them! Also, a good source is MathOverflow , where you may ask a question and professional mathematicians will answer it online. But make sure that your question is well stated, according to the rules of MathOverflow!

8. Use analogies. There are many mathematical problems but much fewer methods for solving them. So a method used in one problem may also work in another, analogous one.

9. Do computer tests, look for patterns in data, make conjectures. The On-Line Encyclopedia of Integer Sequences is often useful. It has an advanced tool called Superseeker which seeks patterns.

10. Confirm your results and proofs by computer calculations whenever possible, to avoid mistakes.

11. Keep good notes of what you are doing (preferably in LaTeX ) at all times. Good bookkeeping is a big part of doing math!

12. Try to write clearly and concisely, in logical sequence. A mathematical text should be highly structured. A good way of writing, at least for beginners, is to make sure that each piece of text is a definition, proposition, proof, remark, example, question, conjecture, etc., so that the text is split into small pieces and there is little (if any) loose text that does not fall into one of these categories. About each piece of text it should be clear what its status is. Text should be proofread and edited several times after it's written.

13. Try to understand statements and proofs of the results that you use as well as you can. Not only is it more honest and reliable, but this will also give you more power in handling the actual problem you deal with.

14. Be motivated and guided by beauty and harmony. It is the most important motivation in mathematics. If you have a proof but don't like it, if it seems ugly, it is much more likely that it is actually wrong. And even if it's correct, it probably will become much simpler or more powerful, and you'll learn something if you try to understand things better so you can write a better proof. It is worth trying to understand better the things you already understand to some extent, rather than jumping forward to entirely new things. Although this seemingly slows down the process, you will surely make up for it and be rewarded further down the road.

15. Listen to your heart. As in all important things in life, what you want and what you dream about is the most essential. Try to find your own voice. The main point of mathematical research is for you to enjoy it!

With questions, contact PRIMES Program Coordinator André Dixon at

• Undergraduate Research

Undergraduate Research programs are a great opportunity for undergraduates to build research experience, connect with faculty and researchers, and (sometimes) even earn some money. Undergraduate Research programs can take a variety of formats. Some are informal arrangements with a professor where you work independently on a problem but with guidance from the professor. Other programs are more formal, such as the numerous summer REU programs funded by the National Science Foundation.

These programs are typically an 8-10 week residential program with other students from various universities where you work together on a problem.

Summer REU programs typically involve paid travel expenses and a summer stipend and are very competitive to get admitted to. If you are interested in finding out more about Undergraduate Research opportunities at Purdue, or how to apply to summer REU programs, contact Jon Peterson at [email protected] .

Summer is traditionally a time to kick back and take a break from studies, but not so for several mathematics students who are in residence in the Mathematics Department during summers.

With support provided by Purdue alumni Andy Zoltners, Joel Spira, as well as the National Science Foundation and other funding, undergraduate math students engage in research projects under the guidance of mathematics faculty members.

• Purdue REU Opportunities

Summer REU Opportunities

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Research Opportunities for Math Undergraduates

Undergraduate mathematics research is an excellent way to connect with faculty, researchers, and existing projects, and to be hands-on with emerging possibilities and challenges within the field. As you build skills like critical thinking and problem-solving, you'll be developing your professional identity. 

Explore research opportunities

National reu listings.

Several organizations maintain lists of REU opportunities across the country:

• National Science Foundation (NSF)
• American Mathematical Society
• Mathematics Project in Minnesota
• U of M Office of Undergraduate Research

Math REUs at the U of M 

• Summer REU program in Combinatorics

Undergraduate Mathematics Office 115 Vincent Hall

[email protected] 612-625-4848

[email protected] Schedule an appointment

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Undergraduate Research in Mathematics

The  UGA Mathematics Undergraduate Research Program   (URP) is committed to supporting undergraduates with an interest in research. We have a number of faculty and graduate students who regularly engage in mentoring undergraduates, and a variety of options through which students may pursue their mathematical interests. 

Getting Started  

• Our standard entry-point for undergraduate researchers is the  UGA Mathematics Directed Reading Program (DRP) , which pairs undergraduates with graduate student mentors to research topics of their mutual interest. The DRP is a beneficial collaboration for both the undergraduate and their mentor. 

Research Opportunities

• After delving into new topics with graduate student DRP mentors, some undergraduates may seek deeper study of a particular field with a faculty or graduate student mentor. Other students may already have specialized interests and seek faculty mentorship right away, or strike up a research collaboration organically. Faculty-guided URP projects can be undertaken in partnership with the  UGA Center for Undergraduate Research Opportunities  (CURO), and/or satisfy  Experiential Learning  requirements, and/or count for course credit in an Undergraduate Research Group  or other independent research project (with some limitations regarding satisfying major/minor requirements).

Explore Your Interests

• The UGA Mathematics URP is very flexible. We are here to support undergraduate curiosity, innovation and creativity in mathematics. Whether you want to find new approaches to applied problems, or explore the universe of pure mathematics, please reach out to us at  [email protected] .

Undergraduate research is a student success activity offered by the Department of Mathematics.

Resources for Undergraduates

UNDERGRADUATE RESEARCH PROGRAM The UGA Mathematics Undergraduate Research Program (URP) has many faculty and postdocs interested to mentor  undergraduates.

DIRECTED READING PROGRAM The UGA Mathematics Directed Reading Program (DRP) pairs undergraduates with mathematics graduate student mentors to research topics of their mutual interest.

E XPERIENTIAL LEARNING Many UGA Mathematics faculty members engage in  Experiential Learning  projects with undergraduates.

UNDERGRADUATE RESEARCH GROUPS The UGA Department of Mathematics sponsors a number of undergraduate research groups (listed below with faculty mentors):

Knot Theory Summer REU ( Akram Alishahi , Melissa Zhang )

Applied Mathematics: Control and Optimization ( Weiwei Hu )

Mathematical Physics ( Jimmy Dilles ,  Gary Iliev )

Applied Mathematics and Scientific Computing ( Seulip Lee ,  Lin Mu )

Topology ( Akram Alishahi , Trenton Schirmer ,  Melissa Zhang )

Mathematics of Card Shuffling FYO ( Leonard Chastkofsky )

Mathematics of Trading ( Qing Zhang )

SUMMER UNDERGRADUATE MATHEMATICS RESEARCH (SUMR) CONFERENCE The UGA Department of Mathematics' annual conference to highlight undergraduate research projects, as well as the work of our graduate student and faculty research mentors.

CURO The UGA Center for Undergraduate Research Opportunities (CURO) offers University of Georgia undergraduates the opportunity to engage in faculty-mentored research regardless of discipline, major or GPA – even students in their first year.

MAA Undergraduate Research Resources The Mathematical Association of America  (MAA) has information about undergraduate research opportunities, as well as advice on preparing posters, presentations and papers, and other useful resources.

National Science Foundation REU The NSF Research Experiences for Undergraduates (REU) program supports active research participation by undergraduate students in any of the areas of research funded by the National Science Foundation. 

American Mathematical Society REU Here you will find information about the AMS R esearch Experience for Undergraduates Summer Programs.

Whether you want to find new approaches to applied problems or explore the universe of pure mathematics, please reach out to:

[email protected]

Fill out this DRP/URP form   to get started today!

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Department of Mathematics

Undergraduate research, summer undergraduate research at yale (sumry).

The SUMRY program is a ten-­week undergraduate research program run by the mathematics department at Yale University, usually between early June and early August. In a recent year , there were 15-20 funded positions for undergraduates to investigate open research problems in the mathematical sciences. Students work either individually or in small groups, directed by faculty members, post­doctoral fellows, and graduate students. The work pursued in this program will give participants an idea of what research in mathematics is like.

Directed Reading Program

The Directed Reading Program pairs undergraduate students with graduate student mentors to read and work through a mathematics text over the course of one semester. The pairs meet once each week for one hour, with the undergraduates expected to do about 4 hours of independent reading per week. At the end of the semester, undergraduates either give a talk to their peers or prepare a short exposition of some of the material from the semester. Undergraduates are expected to have a high level of mathematical maturity and eagerness to learn the topic.

Math 470 is an individual studies course, it can be taken for graduation credit (but not applied toward undergraduate math major requirements). By default, it can be taken only once, though under exceptional circumstances, the DUS may permit it to be taken twice. Interested students must submit a proposal to math.dus@yale.edu at least three days before the end of add / drop period, with the name of their adviser, and details about the proposed study (both its content and the structure of the course). Typically, the class will require weekly meetings with the adviser, it will have some assignments along the way (that are to be written up or presented to the adviser), and it will terminate with a final paper or project. Please note that university rules do not allow independent study on topics that are taught in existing courses (there can be a bit of overlap, but you cannot do independent study to learn Math 370, for example). 

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Undergraduate math research

I believe this is the right place to ask this, so I was wondering if anyone could give me advice on research at the undergraduate level.

I was recently accepted into the McNair Scholars program . It is a preparatory program for students who want to go on to graduate school. I am expected to submit a research topic proposal in the middle of the spring semester and study it during the summer with a mentor.

Since I am currently in the B.S. Mathematics program and I want to get my Masters later. I figured that while my topic can be in any area, it should be in math since it is my main interest as well.

I am a junior at the moment and taking: One-Dimensional Real Analysis, Intro to Numerical Methods, and Abstract Algebra. I frequently search MathWorld and Wikipedia for topics that interest me, although I don't consider myself a brilliant student or particularly strong. I have begun speaking with professors about their research also.

I have not met any other students doing undergraduate math research and my current feeling is that many or all the problems in math are far beyond my ability to research them. This may seem a little defeatist but it seems mathematics is progressively becoming more specialized. I know that there are many areas emerging in Applied mathematics but they seem to be using much higher mathematics as well.

My current interest is Abstract Algebra and Game Theory and I have been considering if there are possibilities to apply the former to the latter.

So my questions are: 1) Are my beliefs about the possibilities of undergraduate research unfounded? 2) Where can I find online math journals? 3) How can I go about finding what has been explored in areas of interest. Should I search through Wikipedia and MathWorld bibliographies and or look in the library for research?

Thanks I hope someone can help to clarify and guide me.

• soft-question
• 4 $\begingroup$ All of the papers published in the journal Involve have a significant student contribution. You can read the abstracts at their website (involvemath.org). I am not saying this is the place to go to find a research topic (or that you should attempt to find a topic on your own via the internet). But I know from experience that it can be fun and inspirational to see that your peers are researching and publishing in a variety of areas. $\endgroup$ –  user4977 Nov 13, 2010 at 2:25
• 7 $\begingroup$ Note to the advice-givers: there is also a user Metahominid at the math.SE site who has asked a few questions about basic abstract algebra and real analysis. Reading over them gives some information (far from definitive, of course) about the student and his current level, which might result in more personalized advice. $\endgroup$ –  Pete L. Clark Nov 13, 2010 at 19:26
• 3 $\begingroup$ That is me. They are basic because I am in the class. I am not trying to give the impression I can do any significant research. There have been almost no math McNair students, and I suspect this because of the difficulty. The point of the program from what I have been told so far is to look into a topic of interest, and pursue it. There is only the expectation to submit, and not that it be good or profound. Mainly it is to prepare students for the research experience. The only other math McNair students have done math education, I would prefer not to. Thank you to everyone. $\endgroup$ –  user7504 Nov 14, 2010 at 0:22
• 5 $\begingroup$ Congratulations for making it into the program. There are already tons of answers, so I won't add much to them, except to say first that undergraduate research can be extremely valuable for your development (to help you appreciate the difference between solving exercises and tackling real questions, even if someone already knows the answer btw). Ans second, a point that has already been made but cannot be stressed enough, that a successful UG research experience requires a good mentor , so find her or him first, and then worry about the topic. $\endgroup$ –  Thierry Zell Nov 16, 2010 at 18:14
• $\begingroup$ I want to say good luck! I sympathize with the challenge. I'm trying to catch up on research-level math while no longer being in school, and if I could give my younger, enrolled self one piece of advice it would be to seek out anyone who studies what you want to study and take advantage of the fact that they're enthusiastic nerds who generally want to share their fascination with the subject. $\endgroup$ –  Malice Vidrine Aug 27, 2013 at 1:02

16 Answers 16

Since you are a student who's already interested in going on to graduate school and is specifically asking about finding a topic to study at your undergraduate level program at McNair, please disregard the negative nattering nabobs whose answers and comments suggest that undergraduates have no place or business in trying to perform research, whether it's research as defined for all scientists or the "research experience" that is put together for undergraduates and for advanced high-school students. Undergraduates can definitely perform research, or even benefit from going through a structured and well-administered "research experience".

I agree with Peter Shor about finding a mentor, or multiple mentors, as soon as possible. There's no reason you have to be limited to getting advice from just one professor or teacher.

I agree with Ben Webster, specifically about speaking with professors in order to get a reasonable idea about the level of work that would be needed for you to perform useful research at an undergraduate level. A few other suggestions come to mind:

if you are at an institution that offers Masters and Ph.D. level degrees in mathematics, then your institution's library should have multiple research journals in hard-copy . I have found that it is much easier to go to the stacks in the library and browse through one or two year's worth of Tables of Contents and Abstracts in one journal in an afternoon or evening. This will familiarize you with the types of research papers being published currently, and make you aware of what "quanta" of research is enough to be a single research article.

make sure to attend Seminars, Colloquia, and (if your school's graduate students have one) any graduate research seminar courses that you can find time for. This will allow you to become more familiar with various subtopics within the topics of your interests, and to see what the current areas of interest are for local and visiting faculty members.

Colloquia are great as they often start by including a brief history of the topic by an expert in that field.

Seminars are great because they allow students to see the social aspect of math, including the give-and-take and the critical comments and requests for more detail and explanation, even by tenured faculty who don't follow a speaker's thought processes.

Graduate student seminar presentations are great because a student observes how graduate students can falter during presentations, how they are quizzed/coached/criticized/mentored/assisted by faculty during their presentations.

I'll admit that I'm not sure attending dissertation defenses would be of any serious benefit to the undergraduate student, other than observing the interaction level (animosity level?) between faculty and graduate students.

absolutely make sure to schedule some time to meet with mathematics professors who specialize in the fields of your interest, and communicate your desire to do research while you are an undergraduate, and communicate your desire to go on to graduate studies in mathematics.

look on the internet and search for undergraduate opportunities for research in mathematics. I guarantee you will find quite a number of web sites that can give you more information. MIT has an undergraduate research opportunity program that many of their students take advantage of. Your institution may have professors who can speak with you and give you advice.

Also, make sure to speak with more than one professor, and do not take any single person's advice as being the final word. Mathematicians are human beings too, and subject to the foibles and inclinations and disinclinations that all human beings have. If you run into disgruntled and critical individuals, do not let that dissuade you from going on into mathematics or decrease your desires. If you run into overly optimistic individuals who praise you too much and are too eager to take you on to do "scut work" computer programming, thank them for their time and let them know you'll come back to speak with them after you've spoken with other professors and weighed your options. Don't turn anyone down immediately. Always be polite in speaking with professors and teachers. Ask them how they chose their topics for their degrees, and you'll learn a lot.

• 29 $\begingroup$ I'm sorry, but this seems totally unrealistic. Anyone who, as an undergraduate taking real analysis and abstract algebra, can follow research-level seminars, either A) should expect the Fields Medal within a few years, or B) doesn't exist. Most of the WORDS in most seminars would be literally meaningless! For a student who already seems discouraged about their ability, this is a recipe for crushing disappointment. My advice: don't worry so much about research as an undergraduate. Find a problem you are interested in, open or not, and a professor at your school you feel you can talk to. $\endgroup$ –  Tom Church Nov 12, 2010 at 13:34
• 29 $\begingroup$ @Tom-Church, much like pre-med students who can volunteer and observe at hospitals without being able to participate (or understand all of the details) at the level of medical doctors, it is possible for students to attend research-level seminars to get an idea of the type of topics which are being discussed and are at the fore-front of research currently. How is the student going to find a problem to be interested in without at a minimum glancing at the field, reading abstracts, and seeing if the topics tickle his/her fancy? This isn't unrealistic. I, and many others, published as undergrads. $\endgroup$ –  sleepless in beantown Nov 12, 2010 at 14:06
• 26 $\begingroup$ I would encourage enthusiastic undergraduates to attend colloquia (meant for general audiences of mathematicians), not necessarily just to follow research trends, but also to watch interactions between mathematicians. Eavesdropping on mathematicians at tea can be interesting too. The point is that mathematics is intensely social, and it's good to see what mathematics and mathematicians are like away from books, articles, and lectures. If OP maintains a healthy balance between "I've got a lot to learn" and "this looks like fun, and something I want to do", where's the harm in it? $\endgroup$ –  Todd Trimble ♦ Nov 12, 2010 at 16:36
• $\begingroup$ Colloquia are great as they often start by including a brief history of the topic by an expert in that field. Seminars are great because they allow students to see the social aspect of math, including the give-and-take and the critical comments and requests for more detail and explanation, even by tenured faculty who don't follow a speaker's thought processes. Graduate student seminar presentations are great because a student observes how graduate students can falter during presentations, how they are quizzed/coached/criticized/mentored by faculty during their presentations. $\endgroup$ –  sleepless in beantown Nov 13, 2010 at 0:20
• 9 $\begingroup$ sleepless: your advice is the essence of good common sense, and I am sure that the OP will benefit from it if they take it at heart. When I was an undergrad, we attended talks made for us, but the faculty would attend too and the questions at the end could get serious (this is where I heard the term "gauge theory" for the first time). We were also encouraged to attend regular seminars with no expectations that we would understand anything , but, as one person said, for the music . You get the music first, then you add in the lyrics. $\endgroup$ –  Thierry Zell Nov 16, 2010 at 18:29

As an undergraduate in the US with some research experience, let me offer my take on the situation.

1) I think it's important not to have a finished product (that is, a piece of original research) as the end goal. This past summer I did some research through MIT's SPUR program, and this is their definition of success:

Significant progress, relative to one's own background and experience, in developing interests, satisfaction, skill, and ideas, rather than getting the complete solution to a problem.

I think this is a really nice sentiment. The goal is not for you to start making serious contributions to mathematics but to prepare you in several ways for a graduate experience. (If you're curious, I blogged a little about my research here and for several posts afterwards. I did not prove a new result, but I learned a lot and thought it was a valuable experience.)

2) It depends on what your institution has subscriptions to. Click around on scholar.google.com to see what you can access without paying for. Many institutions, for example, have access to JSTOR or SpringerLink .

3) Find someone who knows the subject and ask them to mentor you. Or, find a mentor and ask them for a subject. This is hard to do without guidance.

Let me also give you some advice you didn't ask for. If you are seriously planning on graduate studies, I think you should expand your mathematical worldview as much as possible beforehand. The easiest way to do that, in my opinion, is to read math blogs. I recommend starting with Terence Tao's and Tim Gowers' and working from there, and I also recommend John Baez's This Week's Finds (actually, read the rest of his stuff too). Math blogs are a valuable source of insight into mathematics and how mathematicians work, and these three are particularly interesting and well-written. Terence Tao's blog also contains his career advice, which is worth a read.

• 1 $\begingroup$ @Qiaochu-Yuan, excellent answer,+1. Also, the mathematics department may have its own internal library room/area, with hardcopy issues of journals that may not be in the main library or in the Barker engineering library. I still find that quickly skimming through the table of contents and abstracts is a good way to get an overview of what's going on in a particular field, and then being able to drill down into a topic by reading an article if the abstract catches my interest. The blogs you pointed out are excellent, particularly for the career advice. $\endgroup$ –  sleepless in beantown Nov 13, 2010 at 0:36
• 1 $\begingroup$ Yes, very good. The "problem-solving" model is invidious, as is the "no-previous-experience-required" idea. Awareness (not "originality", not fussiness over small, long-done details) is perhaps the most important goal for a beginning grad student. All the sadder that some REU's and other potentially encouraging, uplifting experiences give people the idea that they're "already done" in terms of knowledge, awareness, etc. $\endgroup$ –  paul garrett Aug 27, 2013 at 1:18
• $\begingroup$ The link to SpringerLink is broken; the site can now be found at link.springer.com . The link to MIT's SPUR program is also broken. The current website does not appear to contain the mentioned quote, but that can be seen in a snapshot preserved at the Wayback Machine. $\endgroup$ –  The Amplitwist May 3, 2023 at 19:13

There are quite a few undergrads who have done significant research in mathematics at your level. Even if you don't end up with a published paper (you shouldn't expect this, although it probably happens more often than you might expect), you will gain significant experience into what doing original research in mathematics means. However, I think expecting to find a problem to work on yourself, rather than have a mentor suggest one (or several) to you, is absolutely unrealistic . Maybe it's realistic for other fields, but in my opinion not mathematics.

Find a mentor who is willing to suggest a problem that you can tackle at your level, and (hopefully) give you ideas during the summer if you get stuck. Finding the right problems to work on is a major component of doing mathematical research, and sometimes the hardest one. If you find it on your own (rather than a mentor suggesting it), your mentor is likely not going to have any good ideas of how to attack it, it may end up a harder problem than is realistic for you to solve, and your mentor will be less motivated to help you. So my advice is to start looking for mentors now.

• $\begingroup$ Pity if the faculty in the environment are less able to think new thoughts than the student... though I suppose it happens all too often, given various dynamics. But, "srsly", one doesn't want a mentor to feel superior-to, in any case, indeed, for sure. $\endgroup$ –  paul garrett Aug 27, 2013 at 1:15

So on the one hand I have a very strong cultural bias against undergraduate research programs. I don't think trying to emphasize originality is a good idea. I think it would be much better to give people problems to work on that have already been solved and so you know lead to good and interesting mathematics. By forcing people to work on "new" questions I think you are often forcing them to work on bad math.

On the other hand, just because I think it would be better for people to do other sorts of programs, REU-style programs are what exist and they seem to work reasonably well for a lot of people. Furthermore, they're certainly valuable as an alternative to classroom learning. Real math research is not like what happens at most REUs, but it's also not like what happens in a classroom, so doing an REU is still going to help you get closer to understanding the scope of what a graduate student does.

So yes it's certainly possible and somewhat valuable for undergraduates to try to do "research," but you shouldn't expect that research to be the same sort of research that mathematicians are really doing.

• 3 $\begingroup$ What's "bad math"? $\endgroup$ –  Igor Belegradek Nov 13, 2010 at 3:25
• 13 $\begingroup$ Bad math may mean problems which are open because they seem uninteresting, or not clearly connected to anything. Working on odd, tedious but open problems may allow a student to do original work, but might be worse than a course for learning or for stimulating interest. $\endgroup$ –  Douglas Zare Nov 13, 2010 at 4:00
• 3 $\begingroup$ Of course, I solved this issue during my REU experience by working on a problem that had been solved 40 years previously (and was actually easy, when viewed correctly), though neither my mentor or I knew this. $\endgroup$ –  Ben Webster ♦ Nov 13, 2010 at 7:17
• 2 $\begingroup$ Several points: "bad" math includes artificial problems, and problems posed as though they were mysteries, when they are not. Also, is it always about "problems"? This, too, is corruptive. I'd not tell people to work to understand something by_prescribed_means , but to understand it however they can. And that surely many important things are already understood (by hard-working, able people), but everyone has to get themselves caught up to the present. The "standard" that genuinely worthwhile new contributions be made by people who don't know anything is both a vicious fiction... $\endgroup$ –  paul garrett Aug 27, 2013 at 1:12
• 1 $\begingroup$ @PaulGarrett: I think some of your points are overstated. It is true that most REU's address questions that don't really matter to mathematics, but there are exceptions. For instance, your colleague Vic Reiner has a good record of getting interesting research out of undergrads. My own approach to running REU's is based on the observation that many serious research papers have two parts: a reduction via sophisticated techniques to a concrete question of combinatorics, linear algebra, or calculus, and then an ad hoc treatment of that concrete question. Undergraduates can solve these concrete... $\endgroup$ –  Michael Zieve Aug 30, 2013 at 14:44

Your beliefs are somewhat unfounded; lots of people (for example, me) do research in various fora as undergrads, and in fact, the NSF is pushing undergrad research quite hard nowadays.

On the other hand, it's not something that's easy to do on your own. While you may get some other reasonable suggestions from people, there's an obvious first step here, which is talking to a professor (possibly several). Decide on the mentor, and then have them help you prepare the research proposal. As an undergraduate, trying to go out and find articles on your own without any direction is like searching for a needle in a haystack. You might find something cool, but I wouldn't recommend it as a first approach.

• 21 $\begingroup$ You say "lots of people (for example, me) do research in various fora as undergrads..." Interesting, but extremely unusual in my experience. To my knowledge, almost no undergrads at my university ( or any subsequent places I went to as a graduate and teacher ) did ( or would have been capable of ) any proper research at all (apart from small little "projects" and "investigations"). Still, that was back in the nineties in England, so maybe it doesn't apply here. The NSF sounds totally crazy to me, but I admit I'm not qualified to judge. $\endgroup$ –  Zen Harper Nov 12, 2010 at 10:07
• 3 $\begingroup$ In my (limited but recent) experience, undergrad research is encouraged much more in the US than the UK, and thought of as more achievable. In the UK (my own undergrad was Cambridge, early 2000’s) we weren’t for the most part encouraged to think of research as something accessible to us at all as undergraduates, like Zen Harper says. In the US (my own grad school, and what I've seen at other schools) it's widely encouraged (though not ubiquitous or essential); and it turns out that undergrads, with good mentoring, can often reach some non-trivial original work. (ct’d) $\endgroup$ –  Peter LeFanu Lumsdaine Nov 12, 2010 at 17:30
• 4 $\begingroup$ …so I think there may be a bit of “if you don’t believe you can do it, then you can’t”: in the UK, we don’t have expectations that it’s achievable, and we don't have much of a model or experience of how to do it, without which our expectations of impossiblity are pretty much correct! (I don’t mean to disparage the teachers I had in the UK, by the way: they were excellent, and encouraged us in many useful directions; undergraduate research just wasn’t one of them.) $\endgroup$ –  Peter LeFanu Lumsdaine Nov 12, 2010 at 17:36
• 4 $\begingroup$ Felipe- I (obviously) don't think this advice is over-optimistic at all. Maybe the McNair Scholars program is too optimistic, but that's an issue you should take up with them. I'm not sure "over-optimistic" is really the right characterization of the problem in MO advice; I still think its more of an issue of getting advice from people who don't really understand your situation, as Zen as demonstrated. There are so many details necessary for getting good advice on these professional issues that "Go talk to someone who actually knows you" is essentially always the right advice. $\endgroup$ –  Ben Webster ♦ Nov 12, 2010 at 19:48
• 8 $\begingroup$ Qiaochu: while I hope you are enjoying Cambridge, I suggest you try the sentence "the structure of the UK system means that students are all at fairly similar levels of mathematical maturity and knowledge as they progress" out on some people and see what their reaction is ;-) Let me just say that Cambridge is not a representative sample of UK tertiary maths education... $\endgroup$ –  Yemon Choi Nov 13, 2010 at 0:23

" I am a junior at the moment and taking: One Dimensional Real Analysis, Intro to Numerical Methods, and Abstract Algebra ".

Based on this information, I think it is a complete waste of your time even to consider research seriously at this stage; you need several more years of study as a minimum. Right now, you are still learning the basic language of mathematics. It's similar to, say, a student who wants to begin reading classical German literature, but only knows 100 words -- premature, to say the least. The maths you know right now is probably less than 1% of what you will need. Even after my Ph.D., I feel that my knowledge is very limited in comparison to most good researchers.

But do you really mean "research", i.e. new, original, nontrivial and interesting, and publishable in a good quality journal, i.e. one which your professors would publish in?

Or do you mean a kind of "investigation" or "project" instead? These are not required or expected to contain anything new or original. This would be highly worthwhile -- but only for your personal interest and satisfaction.

The question is, what do you expect to get out of it? If you're at a good university, their lecture courses should already provide you with all you need.

Please don't take offense, and apologies if I've formed the wrong impression, but it sounds to me ( from your statement "I have begun speaking with professors about their research also" ) like you might be the kind of student that irritates professors, always bugging them and asking them questions about their own research, but lacking the knowledge to understand the answers. ( But it's not your fault you lack knowledge - that's what you're at university to learn! ) As an analogy, imagine a ten-year-old, knowing nothing more than how to add fractions, constantly harrassing you to teach them about calculus; my response (unless I were in a very good mood that day) would be: " go back to school and stop bothering me, for at least another 3 years! " Unless you're an exceptionally good, enthusiastic student, or your professors are far more patient than me, that might be what they're thinking also, but are too polite to tell you.

But just my opinion, don't take my word for it; why don't you ask them directly if that's what they're thinking?!

• 13 $\begingroup$ +1: Why was this answer down-voted? Are you suggesting that juniors are ready to do research? $\endgroup$ –  Douglas S. Stones Nov 12, 2010 at 12:38
• 30 $\begingroup$ I think it is possible to find a suitable project for a math-interested student at any level. For example, I would be happy to discuss calculus with any ten-year-old who was interested enough to learn about it; it would be an excuse to talk about graphs and rates of change and the concept of limits and the effect of minute changes. One can explain a part of these ideas even to someone with little background. Similarly, one can find an interesting suitable project for an undergraduate. The surreal numbers, for example, would be an attractive topic at the boundary of algebra and game theory. $\endgroup$ –  Joel David Hamkins Nov 12, 2010 at 14:20
• 40 $\begingroup$ The down vote might have been because the same thoughts could have been conveyed in a much more polite manner (just a guess). $\endgroup$ –  BCnrd Nov 12, 2010 at 14:20
• 17 $\begingroup$ Agree with BCnrd. "like you might be the kind of student that irritates professors, always bugging them and asking them questions about their own research, but lacking the knowledge to understand the answers" - isn't that reading a bit much into it? OP said "research at the undergraduate level", which I take to mean investigations into subjects he finds attractive, not publishing in the Annals as an undergraduate. Also, "their lecture courses should already provide you with all you need" - are books and papers all that pros need? One-on-one conversation is something we all benefit from. $\endgroup$ –  Todd Trimble ♦ Nov 12, 2010 at 14:56
• 28 $\begingroup$ -1: This answer latches on to one fact about the student while ignoring another big one: the OP is in a structured program, not just doing this on a lark. You can, of course, doubt whether that program will produce very high quality research; I think most of us do. But that's not really the point of such programs. They're mainly aimed at grad school preparation/promotion. Frankly, students need to do something over the summer, and they may as well have a experience that shows them mathematics from another angle than just the classroom. $\endgroup$ –  Ben Webster ♦ Nov 12, 2010 at 17:46

Metahominid,

I am an undergraduate myself who was in a situation two years ago similar to the one you're in right now. As a sophomore (I'm a senior now), I wanted to do some sort of research but I hadn't taken that many courses. I was just taking real analysis and abstract algebra at the time. However, I asked around the math department for research opportunities algebra and game theory (yes, even my interests were similar!) and a professor recommended another professor to me who had just what I was looking for: an approach to combinatorial game theory using algebra (and a little bit of geometry). I have been working on this topic with him since the summer after my sophomore year. Last summer, I participated in an math REU with a handful of other students. Research in the REU was more independent, with the students doing pretty much all of the work while the mentors served more as helpful sounding boards than co-researchers. Here are the differences I have found between the two research experiences:

Research with professor

Since the problems I am working on are of direct interest to my professor as well, the topics I have to learn in order to even begin to approach the research tend to be more advanced, so I end up learning a lot of interesting theory (my research has led me into combinatorial commutative algebra and local cohomology)

Again, since the professor is working on this with me, I spend a great deal of time talking with him, bouncing ideas back and forth, and this creates a very strong mentor-student relationship that I feel is very beneficial. My mentor gives me advice not just on how to do math, but also on applying to graduate schools, writing good papers and abstracts, giving talks and presentations, etc.

As the students were expected to work independently of the professor, I was thrown into the deep end basically. What entailed were weeks of intensive reading and thinking. Although the mentor was there to help me make sure I was sane by letting me bounce ideas off him, I was still responsible for all of the original thinking and problem solving. The benefits of this are absurdly great: my problem solving skills have improved greatly and I find it far easier to follow lectures and do homework problems now. In fact, math classes feel like nothing now that I have done some research on my own.

We were free to work with other students, and I did collaborate with a few students, which I think is an invaluable experience. By talking to these students every day, I learned different ways of thinking about things, and different approaches to solving problems. Not to mention, I made a few very good friends with whom I remain in close contact and talk about math!

I got to do some nontrivial original work (although I wouldn't say the problems I solved were important) and wrote some publishable material. This needn't happen, though. The point is to get the experience.

The bottom line is this: if you're eager to do research, ask around for professors who are looking for motivated undergraduate students. Make sure they know that you are motivated and willing to learn and work hard. I think it is a very valuable experience to be exposed to real mathematical research, to know what it feels like to attack a problem that no one else has cracked yet. It's completely different from coursework. I never considered myself particularly talented at mathematics, but over the years I have realized that being good at doing math is much more about practice and experience rather than some "natural" talent. I also highly recommend one of the NSF sponsored REUs, as the mentors are usually very skilled at picking problems that are at an appropriate level for undergraduate students. I hope this helped.

The following might be more appropriate as a comment to sleepless in beantown's answer instead of an answer, but for some reason I am not able to comment.

The following website of the AMS contains numerous links to Research Experience for Undergraduates (REU) programs

http://www.ams.org/programs/students/undergrad/emp-reu

If I understand correctly, these REU programs are funded by the NSF to offer undergraduate students (not necessarily from the institution that is hosting the program) an opportunity to do research, under supervison, over the summer. (Since there was some debate what "research" should mean, I add that here "research" means, or at least can mean [and not only rarely], something that in the end is published in well-established mathematical research journals.)

Also, there is a somewhat recently founded journal Involve specifically dedicated to "showcasing and encouraging high quality mathematical research involving students (at all levels)"

http://pjm.math.berkeley.edu/involve/about/journal/about.html

• 1 $\begingroup$ Hi, unknown (you can register and add a name or nickname also), you couldn't add a comment because you don't have 50 reputation points yet. Thanks for the comment/reply. I hadn't head about the AMS REU program web site. $\endgroup$ –  sleepless in beantown Nov 12, 2010 at 14:19

The web site on the McNair Graduate Opportunity program gives no indication that mathematics students were in mind, and a quick survey of some past students' topics showed none in mathematics. I do not think the program was designed for mathematics students or the way mathematics research is done, even undergraduate mathematics research.

Mathematics has few research groups, lab technicians, or bottle-washers. We usually do mathematics with 1-2 people involved. In other areas, you can learn to run some tests, and collect data, and analyze it with the help of an advisor. You have a very high chance of accomplishing something, and meanwhile you can try to learn how your work fits into a larger picture. Most mathematical projects are more risky. It takes a lot of work as an advisor to create a project approachable by an above average mathematics major which has a good chance to produce new results the student can write up. Much of mathematics does not involve programming, but many projects designed for short-term results are programming exercises which may give a distorted picture of mathematics.

This is not to say that undergraduate research is not worth the attempt. It is one way to see that mathematics is alive and exciting, which may be hard to see from courses. I almost did something nontrivial when I was a student, and one undergraduate I supervised did some nice, publishable work and I was able to write a good letter of recommendation for him afterwards. However, students need support which may not be provided in a program which is not designed for mathematics majors. You need an advisor who will put a lot of effort in (even in choosing the topic) beyond what is needed in other areas. You should be aware that failing to solve the problem is often not a surprise, and that you may be severely handicapped with only one year of solid mathematics courses.

I think you should look at alternatives such as mathematics Research Experiences for Undergraduates (REUs) or setting up a reading course in which the goal is not to produce new research, but to understand some recent result or paper, perhaps to create a more accessible exposition of that topic.

• 7 $\begingroup$ After looking over the McNair website, I agree completely. I didn't see anything oriented towards math specifically, and this makes me skeptical -- the way math is done is quite different from that of other academic endeavors. I was especially concerned by something on the website that said that creation of a paper of "publishable quality" was a requirement of the program. In my opinion this is not a realistic goal for undergrad summer research. If I were the OP, I would consider either doing the summer research in something else, or, if math is truly of interest, applying for an REU. $\endgroup$ –  Pete L. Clark Nov 13, 2010 at 7:08
• 4 $\begingroup$ I ran out of space in my last comment, but let me give one reason an REU is a better bet for summer math study than this McNair Program: in REUs there is a guaranteed cohort of other students to interact with and derive support from. $\endgroup$ –  Pete L. Clark Nov 13, 2010 at 7:11
• $\begingroup$ Sorry I didn't mention that. It is pretty much an REU. It is a TRIO program and I do have a cohort, although they are not in my field. I will be living with them in the summer as well as students from other universities. $\endgroup$ –  user7504 Nov 14, 2010 at 3:08
• 3 $\begingroup$ @Metahominid: if the other students are not doing math, then they will be able to support you in some ways but not others -- for instance, you cannot ask them casual questions that you might not want to bother your mentor with. I think an REU would be better for this. $\endgroup$ –  Pete L. Clark Nov 14, 2010 at 7:41
• $\begingroup$ @Pete L. Clark I recognize this however I have already been accepted and I cannot do both this summer and I am not sure if I will be able to my senior year's summer. $\endgroup$ –  user7504 Nov 14, 2010 at 9:34

I think the earlier one starts doing research the better, even if it is a research in plane geometry.

I directed a few REU's and it was great fun; the only problem is that it is hard to do anything significant in 8 weeks. I am confident that many US undergrads can produce a publishable work after focusing on a problem for 1-2 years.

In the place where I went in college (Novosibirk, Russia, mid 80s) students went through abstract algebra and real analysis in the first two years and the best of them them could do nontrivial work in the 3rd year. Quite a few people were going to research seminars in 3-4th year, which they had to be doing since a (master) thesis with original research was expected at the end of 5th year. As it happens for many students this thesis was largely expository, but those who later become professional mathematicians usually got something publishable in the 5th year. My own research in the 4th year went nowhere, but in the 5th year I proved something I am not ashamed of.

• $\begingroup$ Although the OP is in his 3rd year, it doesn't sound like he would fit into the third year of the college you describe by the courses he is currently taking. In many US colleges, mathematics majors spend a lot of their first two years taking classes outside mathematics. $\endgroup$ –  Douglas Zare Nov 13, 2010 at 4:14
• 1 $\begingroup$ Douglas, I am fully aware that US system is different (having taught here for many years). My points are (a) those who want to become mathmaticians should try research long before they pass comprehensive and oral exams in grad school (b) with proper mentoring math research is quite doable. $\endgroup$ –  Igor Belegradek Nov 13, 2010 at 12:13
• $\begingroup$ Thank you Igor. One of my good friends in my math classes is from Russia. From what he has told me and I have read both the secondary and post-secondary schooling emphasizes math more and introduces many things earlier. I wish that were the case here. I also wish the culture for chess were the same. $\endgroup$ –  user7504 Nov 14, 2010 at 3:06
• $\begingroup$ @Metahominid, the system in Russia is different and I cannot say it is better, but it does prove that original research by undergraduates is quite possible. Some in this thread seem to argue that a student is better off by delaying research till grad school, and I disagree. $\endgroup$ –  Igor Belegradek Nov 14, 2010 at 12:32
• 1 $\begingroup$ There are two different issues being confounded throughout, for unfortunate semantic reasons. If "research" means "thinking" (as opposed to obeying, conforming, guessing what's on the final), well, yes, this is a fundamental scientific/intellectual trait. But if "research" means to do better than all existing professionals on an issue meaningful and interesting to them... well, let's think... maybe this is not something to be counted-on on a regular basis. Sometimes kids are in stark, low-energy situations, which is bad, but let's not deceive them about the larger world. Yes, thinking is good. $\endgroup$ –  paul garrett Aug 27, 2013 at 1:22

Most major topics have been covered in discussion, so just two remarks/experiences:

While director of graduate studies at Northwestern (2007-2010), I led a committee which valued undergraduate preparedness over research experience. So at least as far as Northwestern was concerned during that time frame, research (especially research for which an undergrad may not be fully prepared) did not help as much as you might have thought.

In trying to use RTG funds toward undergraduates, rather than try to simulate a research environment, I and my co-PI's created an "undergraduate conference" to try to offer a supplement to standard undergraduate curricula, without yet getting on toward research. Here is the link: http://www.math.northwestern.edu/summerconference/ Maybe we'll do it again next year?

• 1 $\begingroup$ I can see why research experience is not a good way to select grad students, and that good bases are fundamental. But its usefulness is upstream: a serious research experience is a very good way for a student to figure out if they want to go to graduate school in the first place. In that respect, I think it plays a very educational role (I wish all of our secondary-math majors did REU-like intensive research, if only to have teachers out there with an understanding of what professional math is like.) $\endgroup$ –  Thierry Zell Nov 16, 2010 at 18:33
• 1 $\begingroup$ We need a different name for "undergraduate mathematics research"... I am very enthusiastic about getting people out of textbook/school-math/artificial/adversarial settings, but "exploration" is not "research", or else the latter term has become meaningless. I rant endlessly to my students about the evils of "school-math", and also about accidentally believing that one's voyage of discovery is "research" that should be published... as necessary as this voyage is. Confounding substantially different things is not helpful. $\endgroup$ –  paul garrett Aug 27, 2013 at 1:03

Yesterday I proved a small fact and asked a follow-up question in this answer:

Algebras over the little disks operad

It's fairly elementary, I'd be interested to know more about it, and I have never seen anything like it in the literature (although I have not searched). So you could look at that if you wanted to. More generally, there are plenty of problems like this, but they are not always easy to find. If you just start reading books and looking for things to research there is a danger that you will just be led along the best-travelled paths where enormous amounts of work have already been done. So I would second the advice to ask several professors for suggestions before deciding on a topic.

I have to disagree with the sentiment that undergraduate research (that is, research done by students who are actually at the undergraduate level in their studies, like the OP) is premature or somehow not worthwhile. A month into my first abstract algebra class, I approached my professor to talk about research and what I should do to get to that level. Luckily for me, this professor went a step further and actually gave me a choice of things to work on.

Now granted, I doubt this is the norm. My college did not have a graduate program, which made undergraduates more of the center of attention and moreover this particular professor has a keen interest in fostering undergraduate research. However, I think it is worthwhile to a student to pursue it (even if your professors aren't interested or don't have a problem at your level to give you, as has been mentioned there are always REUs) for the following reasons:

You see a different facet of mathematics then you typically see in a course or textbook. Those are realms of proven things (generally speaking; I am sure there are exceptions, but the usual undergraduate topics tend to be fully developed in my experience). On the other hand, research is messy, with missteps and "mathematicians block" and the thrill of showing something new. I think the potential mathematician should see that as soon as possible!

You gain a wealth of valuable insight and skills. At least for me, I grew very comfortable with TeX, got a good deal of experience with presenting mathematics, and learned a lot about effectively explaining mathematics on paper as well.

If you are lucky like I was, this initial collaboration can lead to more research, hence more time honing your intuition, research habits, and paper-writing skills.

In other words, sure, you are probably not going to see an undergraduate solve a particularly interesting problem, but surely it is worthwhile to promote growth of these skills as well (not to mention that, frankly, in the competitive world we live in it wouldn't hurt to have your name on some papers and some professors seeing you talk at workshops and things). On a personal note, I have to say I found that there was feedback between positive research and my courses: the more I learned, the more tools I had to attack problems obviously but the research aspect really improved my ability to see the solutions to exercises and grasp the larger picture of the courses I took.

Bottom line, for an aspiring mathematician there is nothing to lose and everything to gain, so you should definitely see what is out there and try to get involved with some research.

You might want to start looking for a mentor before you get too deeply involved in developing your project. It's great to have some broad ideas, but it isn't a good idea to box yourself in so far that your project isn't a good fit for those on the faculty who might be interested in mentoring you over the summer. Also, potential mentors might have some ideas for projects that would be a good fit for both you and the mentor.

Before you start approaching potential mentors, be sure to check with your McNair program to find out what the program expects of the mentor (for example, the mentor might be expected to write a brief biweekly status report commenting on your progress, as well as validating that you have met milestones for your project). You might want to develop your project proposal with your mentor and to start working together on a realistic set of milestones. The McNair program here tied a large chunk of the summer stipend to meeting milestones.

Have there been other McNair scholars in math at your school in previous years? You might check with your school's McNair program, your department head, and/or your coursework advisor about this. If there have been others, you might also get some tips about potential mentors. You might also check with your coursework advisor or your department head about which faculty members have mentored undergraduate research students. Did you submit letters of recommendation as part of your application? If so, you might want to share your good news with your letter writers and ask their advice concerning potential mentors.

Please don't worry about finding a big result. This is an opportunity to get a taste of research and to learn about some new topics. You will probably be expected to write up what you learned at the end of the summer and present at a conference for McNair scholars. Good luck!

I cannot speak from the point of view of a Math major in US since I never was one. I completed my undergraduate studies in engineering and currently pursuing a Ph.D. in pure mathematics. In my opinion, applied mathematics (though admittedly this quite a generic term) would be more accessible to an undergraduate considering research than pure Mathematics. I ended up publishing two single author papers in respected journals while in my senior year. I had started working on both these problems during my junior and both of them were picked by me. When I though I had a good insight into the problems, I approached the faculty within my university for suggestions. I think it is safe to say that a lot of problems in applied mathematics require less sophisticated machinery than is used by most pure mathematicians. Many of my engineering friends started working on their Ph.D. thesis problems fresh out of a Bachelors in areas which could be termed as applied mathematics. This contrasts with most pure math grad students I know who usually spend between 1 to 3 years of coursework before starting to work on a concrete research problem. So it seems that "undergraduate level coursework" would be sufficient in handling a good number of applied math problems. So if you are advanced undergraduate student with a good background in one such allied area, I think it might be worthwhile to explore this possibility. After all you can gain valuable experience doing research even if you do decide to pursue some other area of math in your graduate life.

• 8 $\begingroup$ I was not implying that anyone who wishes to become a pure math researcher should consider applied mathematics research as an undergrad. This was specifically directed to the OP since the OP mentioned taking a class in numerical methods, an interest in game theory and having considered applied mathematics as a research option. $\endgroup$ –  Timothy Wagner Nov 12, 2010 at 22:52
• 1 $\begingroup$ It is also worth considering that part of the purpose of summer REU's is to discover what sort of research one might or might not want to do later. Besides, knowing some applied math is almost certainly bound to help even the most "pure" mathematician, since the origins of much of the best mathematics lie in very real problems. (I spent two summers trying to learn quantum field theory under the ultimately mistaken impression I wanted to do physics, and I would not say the time was wasted.) $\endgroup$ –  Dave Anderson Nov 13, 2010 at 4:47
• 1 $\begingroup$ I cannot agree more. To add to that, the number of first rate pure mathematicians who have made significant progress in applied math areas keeps increasing. For e.g. Tao (Compressed sensing), Mumford (computer vision, pattern theory) and those with a non math background, Raoull Bott (Electrical Network theory), Harish Chandra (theoretical physics) come to mind. $\endgroup$ –  Timothy Wagner Nov 13, 2010 at 5:11
• 10 $\begingroup$ Harry- That's a fairly ridiculous position; intellectual development and careers do not proceed in an entirely linear manner. I'm not sure I would strongly recommend applied math research to someone who ultimately was planning on going into pure math, but it can surely still be a valuable experience for them. But more to the point, no undergrad should proceed with their life as though it was absolutely certain that they would end up in a particular career. Given the job situation at the moment, I bet a lot of people trained in pure mathematics would be happy to go back in time and (cont'd) $\endgroup$ –  Ben Webster ♦ Nov 14, 2010 at 8:45
• 4 $\begingroup$ spend more time getting experience with applied mathematics as an undergrad. $\endgroup$ –  Ben Webster ♦ Nov 14, 2010 at 8:46

Since you are interested in game theory, one area you could consider is "Algorithmic Game Theory" (basically Algorithm Design + Game Theory) It is a now a fairly hot area in theoretical computer science but still seems relatively approachable to an undergraduate with knowledge of game theory. If you can find someone willing/able to mentor you in this area I think there is good potential for a productive experience.

There is a free textbook online and the blog of Noam Nisan (a leader in the field) is a good place to follow the latest developments.

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• Department of Mathematics

Undergraduate Studies

• Undergraduate Research

Undergraduate Mathematics Research

Approaching a uh math professor about supervising an undergraduate research project.

• Choose a professor that you know or have had a class from, and who you think you would enjoy working with. You may want to take a look at their webpage to see what their research entails, but it's not particularly important that it matches your interests (or that you even understand what it is). Unlike the sciences, in an undergraduate research project in mathematics, you will probably not be working on the professor's own research projects.
• Undergraduate Research Projects are usually done by juniors or seniors after they have completed the Calculus Sequence, the 2000-level Linear Algebra, and the Transition to Advanced Math Class. It also helps if you've had some exposure to upper-level mathematics in other 3000-level or 4000-level courses.
• If possible, go to the professor's office and talk to them in person rather than sending an email. Bring a copy of your unofficial transcripts (either printed from PeopleSoft or just a list of courses you've taken with grades received). Tell the professor you are interested in doing an undergraduate research project, and ask if they would be willing to supervise you. You don't need to have an idea of what you want to work on --- if you want to express some general interests (e.g., "I enjoy Linear Algebra", or "I liked my Real Analysis class") that is fine, but not necessary. It is also a good idea to tell the professor how much time you can contribute to the project. (This may vary, but something like one hour per week meeting with the professor plus 3--6 hours per week of work on your own seems reasonable.)
• The professor may ask you some questions about your background and what you may want to work on. They may ask for some time to see if they can come up with a project that is suitable. They may give you one possible project or they might give you a few possible projects you can choose from. You can then discuss the potential project(s) together to decide if you are interested.

Support for Undergraduate Research at UH

• The Provost's Undergraduate Research Scholarship Program (PURS) provides part-time support during the semester.
• The Summer Undergraduate Research Fellowship provides full-time support during the summer.
• UH Students who are members of the Math Alliance can apply for funding form the Math Alliance to assist with undergraduate research projects. Contact Dr. Will Ott if you are interested in becoming a student member of the Math Alliance.
• The Scholar Enrichment Program (SEP) sometimes has funds to support undergraduate research projects. Contact them to inquire if funding is available.
• Enroll in an Independent Study course [ MATH 4198 ] by contacting Dr. David Blecher (Director of Undergraduate Studies) or Dr. Nicholas Leger (Assoc. Director of Undergraduate Studies)
• Enroll in the Honors College's Senior Honors Thesis Program

Typically, receiving course credit for an undergraduate research project is meant for advanced students or students who have taken a wide range of background courses.  It is not intended as a means to fulfill major requirements.  Enrolling in an independent study course (e.g., Math 4198) requires a petition and approval of the department chair, and should not be used to fulfill a 4000-level course requirement for the major.  The Senior Honors Thesis Program is run by the Honors College, and has GPA and other requirements.

In general, you can receive either financial reimbursement or academic credit for a research project, but not both. Of course, you can also do an undergraduate research project with a UH professor without any funding or credit, so if you'd prefer to do it purely for the benefit of the experience, and on a voluntary basis without payment or course credit, that is fine too.

General Undergraduate Research at UH

• Office of Undergraduate Research at UH
• Facebook page for Undergraduate Research at UH
• Houston Undergraduate Research Network (HURN)
• Scholarships for Undergraduate Research at UH

Research Opportunities and Math Programs at Other Schools

• REU Programs: [ NSF ] [ AMS ] [ MAA ]
• Summer Programs for Math Majors: [ MAA ] [ AMS ] 
• Programs for Math Majors to obtain Teaching Experience:  https://uh.edu/nsm/teachhouston/
• Programs for Women, Minorities, or Members of Under-Represented Groups: [ NSF ]

A Project-Based Guide to Undergraduate Research in Mathematics pp 287–302 Cite as

Researching in Undergraduate Mathematics Education: Possible Directions for Both Undergraduate Students and Faculty

• Milos Savic 5  
• First Online: 18 April 2020

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Part of the book series: Foundations for Undergraduate Research in Mathematics ((FURM))

Research in Undergraduate Mathematics Education (RUME) is a new field to both mathematics and mathematics education. It borrows theory and methodology from other disciplines including psychology, sociology, and neurology. At its core, RUME is attempting to find out about the teaching and learning of undergraduate mathematics education in order to improve it. In this book chapter, I attempt to give a quick overview on how to conduct RUME with undergraduate students. I pull from my experiences as a mentor of ten undergraduate projects. There is also a suggested timeline of RUME in a semester, some ways to generate RUME open questions, and a large amount of open questions conjectured by others. My hope is that this book chapter has information for both mentors and undergraduates alike.

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How one reads mathematics education literature is, in itself, an open RUME question!

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Ellis, J., Fosdick, B. K., Rasmussen, C.: Women 1.5 times more likely to leave STEM pipeline after calculus compared to men: Lack of mathematical confidence a potential culprit. PLoS One, 11 (7), e0157447 (2016)

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Acknowledgements

Thank you to the editors for even considering me; it was an honor. Thank you to Emily Cilli-Turner and Estrella Johnson for reading and making comments prior to submission while always being supportive. I am always indebted to my advisors for their support and care for my professional well-being, while allowing me to be myself throughout this academic journey. Finally, to my family; they are my energy, life, and love.

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Savic, M. (2020). Researching in Undergraduate Mathematics Education: Possible Directions for Both Undergraduate Students and Faculty. In: Harris, P., Insko, E., Wootton, A. (eds) A Project-Based Guide to Undergraduate Research in Mathematics. Foundations for Undergraduate Research in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-37853-0_10

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research for undergraduates

Many deadlines for research programs at other universities occur during Winter Quarter.

• The Blackwell Summer Research Institute is designed to provide research experience to talented undergraduate students who aspire to obtain PhD’s in the mathematical sciences, and guide them to a path towards the most competitive graduate programs in the country. Our long-term goal is to increase the number of talented researchers and teachers in mathematical and computer sciences.
• A six-week competitive summer activity designed to provide research experience to talented undergraduate students. Students will conduct research in Applied Probability, Analysis, or Computer Science, under the supervision of faculty members who are experts in these areas
• Will take place on the UCLA and UC Berkeley campuses. Learn more at:   https://ww3.math.ucla.edu/david-harold-blackwell-summer-research-institute/
• DIMACS REU:  Research Experience for Undergraduates
• UCLA graduate students in mathematics recently started a directed reading program, which pairs undergraduate students with graduate students to study a topic that is not typically covered in the undergraduate curriculum. While the DRP is not research based, it helps develop skills necessary to do research, such as reading and learning more advanced mathematics independently. Find more information here:  https://www.math.ucla.edu/~drp/ .
• The Institute for Pure and Applied Math (IPAM), which is located on UCLA’s campus, typically hosts several industry-oriented REU programs.
• NSF (National Science Foundation) REU (Research Experience for Undergraduates)
• These are research programs that occur during the summer and are specifically intended to expose undergraduates to research. These programs can be very competitive. If you are hoping to participate in an REU, we recommend applying to a number of different programs. The National Science Foundation has a list of many of these programs:  https://www.nsf.gov/crssprgm/reu/
• An Applied Math REU is typically hosted at UCLA. The following link describes some of the 2018 REU projects:  https://www.marcusroper.org/2018/.
• The Institute for Pure and Applied Math (IPAM), which is located on UCLA’s campus, typically hosts several industry-oriented REU programs ( https://www.ipam.ucla.edu/programs/student-research-programs/research-in-industrial-projects-for-students-rips-2019/ )
• UCLA has an undergraduate Research Center. They have additional resources and programs aimed at promoting undergraduate research. 

General information about undergraduate research at UCLA can also be found at  http://hass.ugresearch.ucla.edu/getting-started/portal/ .

• Visit the website by young scientists for young scientists at  sciencecareers.sciencemag.org .

College of Science to launch Center for Advancing Undergraduate Science Education

27 Mar 2024

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Estrella Johnson

The College of Science has created the Center for Advancing Undergraduate Science Education (CAUSE), an effort to enhance educational excellence within Virginia Tech’s science community.

Rooted in the belief that education is a scientific endeavor, the center embodies a vision where educational research informs instructional practices. By integrating research and practice, the center aims to foster a culture of continuous improvement and innovation in undergraduate science and mathematics education.

“The bottom line for this new center is to elevate student learning outcomes,” said Kevin Pitts, dean of the College of Science. “The launch of CAUSE marks a significant milestone in the College of Science's ongoing commitment to educational excellence.”

Estrella Johnson, associate professor in the Department of Mathematics and the college’s assistant dean for inclusion and diversity, saw an opportunity for Virginia Tech to become a leader in innovative teaching methods in undergraduate science and math.

"At the heart of CAUSE is the acknowledgment that education is a science that can be improved through research and practice," said Johnson. "Through collaborative efforts and interdisciplinary engagement, we want to empower faculty as professional educators and enhance student learning experiences."

The center's dual approach emphasizes both research to practice and practice to research. The research to practice initiative will support faculty in implementing evidence-based instructional strategies, while the practice to research program will facilitate professional development in educational research methodologies.

With approximately 425 teaching faculty within the College of Science, the center will be a vital resource for faculty across all ranks and positions. Through an array of activities, including informational sessions, workshops, and speaker series, the center will foster a supportive community where educators can collaborate and exchange ideas.

"We recognize the importance of building a welcoming community in our initiatives," said Johnson. "Our activities are designed to accommodate faculty interests and schedules, ensuring that all members of the academic community feel welcomed and supported in reaching their instructional goals."

The Center for Advancing Undergraduate Science Education also will offer faculty the opportunity to deepen their engagement through titles such as scholar and fellow, acknowledging their contributions and dedication to advancing undergraduate science education. These titles come with benefits that reflect the college’s commitment to elevating and valuing the work as educators.

In addition to workshops and grants, the center will serve as a central entity for faculty and leaders in undergraduate science education to collaborate and exchange innovative teaching practices and research insights.

The center will work with campus partners such as the School of Education , the Department of Engineering Education , the Center for Excellence in Teaching and Learning , Technology-enhanced Learning and Online Strategies , and others to further advance the educational mission across campus, Johnson said.

"As we build this community together, CAUSE invites faculty to join us in shaping the future of undergraduate science education," Johnson said. "Together, we can harness the power of research and practice to create meaningful learning experiences for our students."

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Undergraduate Research

Engaging in research is the most effective way of learning how real science is performed, and undergraduate research has become an increasingly important component of graduate school applications. Working in a lab is a great way to develop the experience and skills necessary for both graduate school and industry. The UW Physics Department aims to provide research opportunities for all Physics majors regardless of financial need.

University of Washington faculty perform internationally recognized research across a very wide range of areas. From the highest energy particle collisions to single ions for quantum computing, from gravity to dark energy to the universe’s first stars, from quantum materials to batteries for green energy, from the evolution of SARS-Cov-2 and HIV to measuring faint magnetic signals from the brain, from neutron stars to dark matter, from quantum gravity to quantum chaos, there are diverse opportunities for undergraduate students to become involved in ground-breaking research.

Getting involved in research

The first step is to find a faculty research mentor. A list of Physics faculty who serve as undergraduate research mentors, organized by research area, may be found on MyPhys under Student Information . Before you approach a faculty member to ask about research opportunities, please read over the student research guide, available on MyPhys, and be prepared with good answers to the questions. Because lab openings change and some research requires specific skills, you will likely need to approach a number of faculty to find a research opportunity that matches your interests and current skills. Be patient, open-minded, and persistent. If you would like advice on which research areas and groups might be a good fit, you are encouraged to schedule an office hours visit with the Undergraduate Research Coordinator . Once you have found a research mentor, you will work with them quarter by quarter to agree on how many hours per week you will work, plan your schedule, and discuss whether your effort will earn Phys 499 credit, be performed as a volunteer, or be compensated as part of Work Study or as an hourly employee.

Undergraduate Research Coordinator

The Physics Department Undergraduate Research Coordinator is Prof. Miguel Morales . Feel free to send email to [email protected] or arrange an office hour visit to discuss questions about the department’s undergraduate research programs.

Work Study Program

The Physics Department has allocated significant resources to enable students to use Work Study hours to perform undergraduate research. If you have Work Study as part of your financial aid package, you may arrange to be paid for your research. Once you have found a Physics faculty member to serve as your research mentor, simply go to the physics front office with your Work Study confirmation email and, contingent on available funds, staff will arrange for you to be hired as an undergraduate researcher . As an employee you will submit your hours bi-weekly for approval by your research mentor. The number of hours you work will be agreed upon with your research mentor up to the maximum provided by the Work Study award.

Can I sign up for both research credit (499) and Work Study? No. School and employment are legally separate, so it is not possible to obtain credit for the same hours you are paid.

I would like to be part of this program, but no Work Study hours were included in my financial aid award. Every financial aid award is unique, but in cases when there is a particularly promising opportunity (like research) it is sometimes possible to adjust a financial aid package to include Work Study hours. Please talk with your financial aid counselor to see if Work Study hours can be added to your financial aid package.

Other research access programs

In addition to the Work Study program the physics department has a number of additional programs designed to broaden access to undergraduate research. Please explore the following to see if they are a good match for you.

Louis Stokes Alliance for Minority Participation (LSAMP)

A wide range of internship, mentorship, and leadership programs for under-represented STEM students.

Physics Program for Advanced Training in Hands-on Science (PATHS)

A Community College transfer program using the power of research. Community College students can be paid to start research before they transfer to UW, seeing what real research is like and building strong interpersonal connections at UW.

INT Undergraduate Research Network (INTURN)

Both school year and summer research positions working with members of the University of Washington’s internationally famous Institute for Nuclear Theory.

UW Physics Research Experiences for Undergraduates (REU)

A 10 week summer program of intense research hosted at the University of Washington.

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Main navigation, stanford data science undergraduate research pathways (dsurp).

The Stanford Data Science Undergraduate Research Pathways program is an 8-week full-time research experience designed to provide students at institutions without access to research opportunities the chance to conduct a research project under the supervision of both a mentor and faculty member. This is an in-person experience held at Stanford from June 24 to August 16, 2024.

• The program is held during the Stanford summer quarter from June 24–August 16 (8 weeks).
• Participants will receive a stipend of $6000, but the program is not otherwise able to provide housing support.
• Available slots are limited and selection is competitive. Priority is given first to students from non-R1 universities, and also to those from backgrounds underrepresented in data science research.
• The program is not open to Stanford students.

How to apply

Applicants will need to provide the following by 11:59pm PST on March 3, 2024:

• Personal and demographic information.
• Resume and unofficial transcript.
• Demonstration of data science reasoning ability.

 Apply Now

Any questions should be directed by email to Daniel LeJeune, dlejeune @ stanford.edu.

Can AI Solve the Math Mysteries Stumping the Field’s Brightest Minds?

Artificial Intelligence already won a gold ribbon in the most elite high school math competition. It could help humans conduct boundary-pushing math research next.

Each year since 1959, high school math students from more than 100 countries have competed to solve a wide variety of math problems involving algebra , geometry , and number theory quickly and elegantly. Many IMO winners have secured prestigious math awards as adults, including the coveted Fields Medal .

In essence, IMO is a benchmark for students to see if they have what it takes to succeed in the field of mathematics. Now, artificial intelligence has aced the test—well, the geometry part at least.

In a paper published this January in Nature , a team of scientists from Google’s DeepMind have introduced a new AI called AlphaGeometry that’s capable of passing the geometry section of the International Math Olympiad without relying on human examples.

“We’ve made a lot of progress with models like ChatGPT … but when it comes to mathematical problems, these [large language models] essentially score zero,” Thang Luong , Ph.D., a senior staff research scientist at Google DeepMind and a senior author of the AlphaGeometry paper, tells Popular Mechanics . “When you ask [math] questions, the model will give you what looks like an answer, but [it actually] doesn’t make sense.”

For example, things get messy when AI tries to solve an algebraic word problem or a combinatorics problem that asks it to find the number of permutations (or versions) of a number sequence.

To answer math questions of this caliber , AlphaGeometry relies on a combination of symbolic AI—which Luong describes as being precise but slow—and a neural network more similar to large language models (LLMs) that is responsible for the quick, creative side of problem-solving.

Yet, math experts aren’t convinced that an AI made to solve high school-level math problems is ready to take off the training wheels and tackle more difficult subjects, e.g. advanced number theory or combinatorics, let alone boundary-pushing math research.

Why AI Struggles With Math

While LLM-powered AI tools have exploded in the past two years, these models have routinely struggled to handle math problems. This is part of what makes AlphaGeometry stand out from the crowd. But even so, that doesn’t necessarily mean it’s ready to tackle higher-level math yet.

.css-2l0eat{font-family:UnitedSans,UnitedSans-roboto,UnitedSans-local,Helvetica,Arial,Sans-serif;font-size:1.625rem;line-height:1.2;margin:0rem;padding:0.9rem 1rem 1rem;}@media(max-width: 48rem){.css-2l0eat{font-size:1.75rem;line-height:1;}}@media(min-width: 48rem){.css-2l0eat{font-size:1.875rem;line-height:1;}}@media(min-width: 64rem){.css-2l0eat{font-size:2.25rem;line-height:1;}}.css-2l0eat b,.css-2l0eat strong{font-family:inherit;font-weight:bold;}.css-2l0eat em,.css-2l0eat i{font-style:italic;font-family:inherit;} “The challenge of AI is that [it] cannot come up with new concepts.”

Marijin Heule , Ph.D., is an associate professor of computer science at Carnegie Mellon University whose work focuses on another kind of automated theorem prover called SAT solvers. In this case, “SAT” refers to a measure of validity called “satisfiability” and not the math section of the high school SAT.

“When it comes down to solving math problems or problems in general, the challenge of AI is that [it] cannot come up with new concepts,” Heule tells Popular Mechanics .

This limitation impacts symbolic AI and neural networks in different ways, Heule explains, but both stem from the issue that these AI rely on an existing bank of human knowledge. However, this isn’t exactly true for AlphaGeometry because it relies on synthetic data , which isn’t based on human examples but is made to mimic them.

While AIs might not be effective mathematicians on their own, that doesn’t necessarily mean they can’t be great apprentices to human mathematicians.

“At least for the foreseeable future, [AI will] be mostly assisting,” Heule says. “One of the other things that these machines can do really well is they can tell you if there is an incorrect argument and [offer] a counterexample.”

These AI-powered nudges can help researchers distinguish research dead-ends from promising paths.

Why Geometry?

Of all the math fields the AlphaGeometry team could have tackled, Luong says there were a few factors that helped them zero in on geometry .

“I think geometry is visually appealing [and] we do geometry as kids,” he says. “And geometry is everywhere in design and architecture, so it’s very important.”

Geometry also offered a unique challenge as being one of the International Math Olympiad fields with the fewest number of proof examples written in a computer-friendly format (e.g. without pictures).

While Heule agrees that AlphaGeometry is “really cool work,” he admits that designing a geometry solver is one of the easier tasks for a math AI.

While human computer scientists did work behind the scenes to formalize geometry problems in a way that computers can reason about, Heule says the reasoning is pretty straightforward once that preparation work is complete.

In part, this is because the considerations of geometry problems (e.g. the relationship between angles, points, and lines) are fairly contained compared to more complex areas, he says.

Take for example Fermat’s Last Theorem . This number theory problem took over three centuries to solve, and Heule says it would be extremely difficult to explain its solution to AI, let alone ask AI to solve it.

“Large-scale fields of modern mathematics … are so big that any one of them contains multitudes,” says Heather Macbeth , Ph.D., an assistant professor of mathematics at Fordham University with a focus on geometry. “I think, maybe a more precise question would be to talk about the styles of problems, which might occur within any mathematical field that some of these AI systems are useful for,” she tells Popular Mechanics.

For example, AI could be useful for pattern recognition or so-called needle-in-a-haystack problems where mathematicians are looking for something with a very particular property, Macbeth says.

Toward General AI

While AI likely won’t be solving centuries-old math problems in the near future, Luong is confident there are still existing advancements on the horizon for AlphaGeometry and its ilk. Perhaps these models could even graduate high school and take on the Putnam Mathematical Competition for undergraduate students.

But beyond math tests themselves, Luong is hopeful about what models like AlphaGeometry could mean for the field of AI at large—in particular, researchers’ goals of designing a generalized AI.

“If we want to talk about building an artificial general intelligence, where we want the AI to be as smart as a human, I think the AI needs to be able to perform deep reasoning,” Luong says. “This means that the AI needs to be able to plan ahead for many, many steps [and] see the big picture of how things connect together … the IMO is the perfect test for that.”

Sarah is a science and technology journalist based in Boston interested in how innovation and research intersect with our daily lives. She has written for a number of national publications and covers innovation news at Inverse .

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UNM Department of Mathematics & Statistics hosts Southwestern Undergraduate Mathematics Research Conference

March 18, 2024

SUnMaRC brings together students and faculty across the southwest for a weekend of mathematics and is designed to provide opportunities for attending students to present mathematics research. The event consists of student talks and invited speakers from various areas including academia, industry, and government.

Beginning in 2004 as the Arizona Mathematics Undergraduate Conference, the conference changed to SUnMaRC in 2008 to recognize the participation of institutions throughout the southwest.

UNM participants can register using the   Individual UNM Participant Registration   link. Further information can be found on the   conference website.   Contact local organizers   Dimiter Vassilev   and   Janet Vassilev   with any questions. 

Read more in the UNM Newsroom

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Independence Complex of Kneser Graph

625 Thackeray Hall/Zoom:  https://pitt.zoom.us/j/97845560498

Abstract or Additional Information

An independence complex of a graph is the simplicial complex consisting of the independent sets of the graph. We’ll discuss the topological properties of the independence complexes of Kneser Graph $KG(n, k)$ using  Vietoris-Rips complexes. Barmak gives the homotopy types of the independence complex of $KG(2,k)$ which is a wedge sum of spheres. For $n\geq 3$, not much is known about the independence complex of $KG(n, k)$.   We’ll investigate the maximal simplices in such complexes and then provide a lower bound on some specific dimensional homology.  

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Research Experience for Undergraduates (REUs) are summer programs sponsored by the National Science Foundation (NSF). REUs usually consist of two parts: intensive study of topics through lecture and interaction, and student research on a question/questions. Travel costs are paid for as well as room and board. A stipend is given to participants. These are all available on a competitive basis. Students that participate in REUs often present their research at national meetings.

Finding and Applying for REUs

Both AMS and the NSF keep current lists of math REUs.  Most deadlines for the summer programs are in February and March.

AMS list of REU sites

NSF list of REU sites

More Information

Not sure what it would be like to participate in an REU? Take a look at this article from our archives:  Is an REU for you?

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