mathematics research projects for undergraduates

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Undergraduate Research Projects

Northwestern undergraduates have opportunities to explore mathematics beyond our undergraduate curriculum by enrolling in math 399-0 independent study, working on a summer project, or writing a senior thesis under the supervision of a faculty member. below are descriptions of projects that our faculty have proposed.  students interested in one of these projects should contact the project adviser. this should not be taken to be an exhaustive list of all projects that are availalbe, nor as a list of the only faculty open to supervising such projects. contact the director of undergraduate studies  for additional guidance. these projects are only available to northwestern undergraduates., combinatorial structures in symplectic topology, eric zaslow, symplectic and contact geometry describe the mathematics of phase space for particles and light, respectively.  they therefore are the mathematical home for dynamical systems arising from physics.  a noteworthy structure within contact geometry is that of a legendrian surface, closely related to the wavefront of propagating light.  these subspaces sometimes have combinatorial descriptions via graphs.  the project explores how well the combinatorial descriptions can distinguish legendrian surfaces, just as in knot theory one might explore whether the jones polynomial can distinguish different knots. , prerequisites:  math 330-1 or math 331-1, math 342-0. recommended: math 308-0., complexity and periodicity, the simplest bi-infinite sequences in $\{0, 1\}^{\mathbb z}$ are the periodic sequences, where a single pattern is concatenated with itself infinitely often. at the opposite extreme are bi-infinite sequences containing every possible configuration of $0$'s and $1$'s. for periodic sequences, the number of substrings of length $n$ is bounded, while in the second case, all substrings appear and so there are $2^n$ substrings of length $n$. the growth rate of the possible patterns is a measurement of the complexity of the sequence, giving information about the sequence itself and describing objects encoded by the sequence. symbolic dynamics is the study of such sequences, associated dynamical systems, and their properties. an old theorem of morse and hedlund gives a simple relation between this measurement of complexity and periodicity: a bi-infinite sequence with entries in a finite alphabet $\mathcal a$ is periodic if and only if there exists some $n\in\mathbb n$ such that the sequence contains at most $n$ words of length $n$. however, as soon as we turn to higher dimensions, meaning a sequence in $\mathcal a^{\mathbb z^d}$ for some $d\geq 2$ rather than $d=1$, the relation between complexity and periodicity is no longer clear.  even defining what is meant by low complexity or periodicity is not clear.  this project will cover what is known in one dimension and then turn to understanding how to generalize these phenomena to higher dimensions.   prerequisite: math 320-3 or math 321-3., finite simple groups, ezra getzler, finite simple groups are the building blocks of finite groups. for any finite group $g$, there is a normal subgroup $h$ such that $g/h$ is a simple group: the simple groups are those groups with no nontrivial normal subgroups.  the abelian finite simple groups are the cyclic groups of prime order; in this sense, finite simple groups generalize the prime numbers.  one of the beautiful theorems of algebra is that the alternating groups $a_n$ (subgroups of the symmetric groups $s_n$) are simple for $n\geq 5$. in fact, $a_5$ is the smallest non-abelian finite simple group (its order is $60$). another series of finite simple groups was discovered by galois. let $\mathbb f$ be a field.  the group $sl_2(\mathbb f)$ is the group of all $2\times2$ matrices of determinant $1$. if we take $\mathbb f$ to be a finite field, we get a finite group; for example, we can take $\mathbb f=\mathbb f_p$, the field with $p$ elements. it is a nice exercise to check that $sl_2(\mathbb f_p)$ has $p^3-p$ elements. the center $z(sl_2(\mathbb f_p))$ of $sl_2(\mathbb f_p)$ is the set of matrices $\pm i$; this has two elements unless $p=2$. the group $psl_2(\mathbb f)$ is the quotient of $sl_2(\mathbb f)$ by its center $z(sl_2(\mathbb f))$: we see that $psl_2(\mathbb f_p)$ has order $(p^3-p)/2$ unless $p=2$. it turns out that $psl_2(\mathbb f_2)$ and $psl_2(\mathbb f_3)$ are isomorphic to $s_3$ and $a_4$, which are not simple, but $psl_2(\mathbb f_5)$ is isomorphic to $a_5$, the smallest nonabelian finite simple group, and $psl_2(\mathbb f_7)$, of order $168$, is the second smallest nonabelian finite simple group. (when $\mathbb f$ is the field of complex numbers, the group $psl_2(\mathbb c)$ is also very interesting, though of course it is not finite: it is isomorphic to the lorentz group of special relativity.)  the goal of this project is to learn about generalizations of this construction, which together with the alternating groups yield all but a finite number of the finite simple groups. (there are 26 missing ones called the sporadic simple groups that cannot be obtained in this way.) this mysterious link between geometry and algebra is hard to explain, but very important: much of what we know about the finite simple groups comes from the study of matrix groups over the complex numbers. prerequisite: math 330-3 or math 331-3., fourier series and representation theory, fourier series allow you to write a periodic function in terms of a basis of sines and cosines.  one way to think of this is to understand sines and cosines as the eigenfunctions of the second derivative operator – so fourier series generalize the spectral theorem of linear algebra in this sense.  there is another viewpoint that is useful:  periodic functions can be thought of as functions defined on a circle, which is itself a group.  the connection between group theory and fourier series runs deeper, and this is the subject of this project. moving up a dimension, functions on a sphere can be described in terms of spherical harmonics.  while the sphere is not a group, it is the orbit space of the unit vector in the vertical direction.  thus it can be constructed as a homogeneous space:  it is the group of rotations modulo the group of rotations around the vertical axis.  we can therefore access functions on the sphere via functions on the group of rotations.  the peter-weyl theorem describes the vector space of functions on the group in terms of its representation theory.  (a representation of a group is a vector space on which group elements act as linear transformations [e.g., matrices], consistent with their relations.)  the entries of matrix elements of the irreducible representations of the group play the role that sines and cosines did above.  indeed, we can combine sines and cosines into complex exponentials and these are the sole entries of the one-by-one matrices (characters) representing the abelian circle group.  finally, we will connect spherical harmonics to polynomial functions relevant to geometric structures described in the borel-weyl-bott theorem.  students will explore many examples along with learning the foundations of the theory. prerequisites:  math 351-0 or math 381-0., linear poisson geometry, santiago cañez , a poisson bracket is a type of operation which takes as input two functions and outputs some expression obtained by multiplying their derivatives, subject to some constraints. for instance, the standard poisson bracket of two functions $f,g$ on $\mathbb r^2$ is defined by $\{f,g\} =\frac{\partial f}{\partial x} \frac{\partial g}{\partial y} - \frac{\partial f}{\partial y} \frac{\partial g}{\partial x}$. such objects first arose in physics in order to describe the time evolution of mechanical systems, but have now found other uses as well. in particular, a linear poisson bracket on a vector space turns out to encode the same data as that of a lie algebra, another type of algebraic object which is ubiquitous in mathematics. this relation between linear poisson brackets and lie algebra structures allows one to study the same object from different perspectives; in particular, this allows one to better understand the notion of coadjoint orbits and the hidden structure within them., the goal of this project is to understand the relation between linear poisson brackets and lie algebras, and to use this relation to elucidate properties of coadjoint orbits. all of these structures are heavily used in physics, and gaining a deep understanding as to why depends on the relation described above. moreover, this project will bring in topics from many different areas of mathematics – analysis, group theory, and linear algebra – to touch on areas of modern research., prerequisites: math 320-1 or math 321-1, math 330-1 or math 331-1, math 334-0 or math 291-2., noncommutative topology, given a space $x$, one can consider various types of functions defined on $x$, say for instance continuous functions from $x$ to $\mathbb c$. the set $c(x)$ of all such functions often comes equipped with some additional structure itself, which allows for the study of various geometric or topological properties of $x$ in terms of the set of functions $c(x)$ instead. in particular, when $x$ is a compact hausdorff space, the set $c(x)$ of complex-valued continuous functions on $x$ has the structure of what is known as a commutative $c^*$-algebra, and the gelfand-naimark theorem asserts that all knowledge about $x$ can be recovered from that of $c(x)$. this then suggests that arbitrary non-commutative $c^*$-algebras can be viewed as describing functions on "noncommutative spaces," of the type which arise in various formulations of quantum mechanics. the goal of this project is to understand the relation between compact hausdorff spaces and commutative $c^*$-algebras, and see how the topological information encoded within $x$ is reflected in the algebraic  information encoded within $c(x)$. this duality between topological and algebraic data is at the core of many aspects of modern mathematics, and beautifully blends together concepts from analysis, algebra, and topology. the ultimate aim in this area is to see how much geometry and topology one can carry out using only algebraic means. prerequisites: math 330-2 or math 331-2, math 344-1., simple lie algebras, a lie algebra is a vector space equipped with a certain type of algebraic operation known as a lie bracket, which gives a way to measure how close two elements are to commuting with one another. for instance, the most basic example is that of the space of all $n \times n$ matrices, where the "bracket" operation takes two $n \times n$ matrices $a$ and $b$ and outputs the difference $ab-ba$; in this case the lie bracket of $a$ and $b$ is zero if and only if $a$ and $b$ commute in the usual sense. lie algebras arise in various contexts, and in particular are used to describe "infinitesimal symmetries" of physical systems. among all lie algebras are those referred to as being simple, which in a sense are the lie algebras from which all other lie algebras can be built. it turns out that one can encode the structure of a simple lie algebra in terms of purely combinatorial data, and that in particular one can classify simple lie algebras in terms of certain pictures known as dynkin diagrams. the goal of this project is to understand the classification of simple lie algebras in terms of dynkin diagrams. there are four main families of such lie algebras which describe matrices with special properties, as well as a few so-called exceptional lie algebras whose existence seems to come out of nowhere. such structures are now commonplace in modern physics, and their study continues to shed new light on various phenomena. prerequisites: math 330-2 or math 331-2, math 334-0 or math 291-2., the spectral theory of polygons, jared wunsch, we can study, for any domain the plane, the eigenfunctions of the laplace-operator (with boundary conditions) on this domain: these are the natural frequencies of vibration of this drum head. students might want to read mark kac's famous paper "can you hear the shape of a drum" as part of this project, and there is lots of fun mathematics associated to this classical question and its negative answer by gordon-webb-wolpert.   an ambitious direction that this could possibly head in would be the theory of diffraction of waves on surfaces. in the plane, this is a classical theory, going back to work of sommerfeld in the 1890's, but there's still a remarkable amount that we don't know.  the mathematical story is more or less as follows: a wave (i.e. a solution to the wave equation, which could be a sound or electromagnetic wave, or, with a slight change of point of view, the wavefunction of a quantum particle) is known to reflect nicely off a straight interface.  at a corner, however, something quite interesting happens, which is that the tip of the corner acts as a new point source of waves.  this is the phenomenon of diffraction, and is responsible for many fascinating effects in mathematical physics.  the student could learn the classical theory in the 2d context, starting with flat surfaces and possibly (if there is sufficient geometric background) curved ones, and then work on a novel project in one of a number of directions, which would touch current research in the field., prerequisites: math 320-1 or math 321-1, math 325-0 or math 382-0. more ambitious parts of this project might require math 410-1,2,3..

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Research projects suitable for undergraduates

What follows is a sample, provided by members of the faculty, of mathematical research projects where undergraduate students in the honors program in mathematics could participate. Interested students should contact either the faculty members directly, one of the honors advisors: professors Al Novikoff or Steve Childress .

A joint research project of Helmut Hofer and Esteban Tabak studies the behavior of Hamiltonian flows on a prescribed energy surface. Computer experiments using symplectic integrators could give some new insight. Such a project would be ideal for a team of an undergraduate and a graduate student. Codes would be developed and experiments would be conducted, shedding new light on the intriguing dynamics of these flows.

Charles Newman has recently studied zero-temperature stochastic dynamics of Ising models with a quenched (i.e., random) initial configuration. When the Ising models are disordered (e.g., a spin glass), there are a host of open problems in statistical physics which could be profitably investigated via Monte Carlo simulations by students (graduate and undergraduate) without an extensive background in the field. For example, on a two-dimensional square lattice, in the +/- J spin glass model, it is known that some sites flip forever and some don't; what happens in dimension three?

Current experiments in the Applied Mathematics Laboratory (WetLab/VisLab) include one project on dynamics of friction, and another involving the interaction of fluid flow with deformable bodies. Gathering data, mathematical modeling, and data analysis all provide excellent opportunities for undergraduate research experiences. In the friction experiment of Steve Childress, for example, the formulation and numerical solution of simplified models of stick/slip dynamics gives exposure to modern concepts of dynamical systems, computer graphics and analysis, and the mathematics of numerical analysis.

Marco Avellaneda's current research in mathematical finance demands econometric data to establish a basis for mathematical modeling and computation. The collection and analysis of such data could be done by undergraduates. The idea is to get comprehensive historical price data from several sources and perform empirical analysis of the correlation matrices between different price shocks in the same economy. The goal of the project is to map the ``principal components'' of the major markets.

Joel Spencer is studying the enumeration of connected graphs with given numbers of vertices and edges. The approach turns asymptotically into certain questions about Brownian motion. Much of the asymptotic calculation is suitable for undergraduates, while the subtleties of going to the Brownian limit would need a more advanced student.

A joint project of David McLaughlin, Michael Shelley, and Robert Shapley (Professor, Center for Neural Science, NYU) is developing a computer model of the area V1 of the monkey's primary visual cortex. Simplifications of this complex network model can provide projects for advanced undergraduate students, giving excellent exposure to mathematical and computational modeling, as well as to biological experiment and observation.

Peter Lax has carried out many numerical experiments with dispersive systems, and with systems modeling shock waves. The basic theory of these equations is well within the grasp of interested undergraduates, and calculations can reveal new phenomena.

A joint research project of David Holland and Esteban Tabak investigates ocean circulation at regional, basinal and global scales. Their approach is based on a combination of numerical and analytical techniques. There is an opportunity within this framework for undergraduate and graduate students to work together to further develop the simplified analytical and numerical models so as to gain insight into various mechanisms underlying and controlling ocean circulation.

Aspects of Lai-Sang Young's work in dynamical systems, chaos, and fractal geometry are suitable for undergraduate research projects. Simple analytic tools for iterations are accessible to students. Research in this area brings together material the undergraduate student has just learned from his or her classes. With proper guidance, this can be a meaningful scientific experience with the possibility of new discoveries.

David McLaughlin and Jalal Shatah's work on dynamical systems provides opportunities for undergraduate research experiences. For instance, the study of normal forms and resonances can be simplified to require only calculus and linear algebra. Thus undergraduate students can study analytically what is resonant in a given physical system, as well as its concrete consequences on qualitative behavior.

Leslie Greengard and Marsha Berger's work on adaptive computational methods plays an increasingly critical role in scientific computing and simulation. There are a number of opportunities for undergraduate involvement in this research. These range from designing algorithms for parallel computing to using large-scale simulation for the investigation of basic questions in fluid mechanics and materials science.

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School of Mathematics and Natural Sciences

Sample Undergraduate Research Projects

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Here is a list of recent undergraduate research projects. When available, we have added images that give you a flavor of some of the topics studied. If the student wrote a thesis, you can look it up at USM's library .

• Samuel Dent , "Applications of the Sierpiński Triangle to Musical Composition", Honors Thesis 
• Brandon Hollingsworth, "A time integration method for nonlinear ordinary differential equations", undergraduate research thesis.
• Haley Dozier, "Ideal Nim", undergraduate research project.
• Sean Patterson, "Generalizing the Relation between the 2-Domination and Annihilation Number of a Graph", Honors Thesis.
• Elyse Garon, "Modeling the Diffusion of Heat Energy within Composites of Homogeneous Materials Using the Uncertainty Principle", Honors Thesis.
• Brandi Moore, "Magic Surfaces", Mathematics Undergraduate Thesis.
• Amber Robertson, "Chebyshev Polynomial Approximation to Solutions of Ordinary Differential Equations", Mathematics Undergraduate Thesis
• Kinsey Ann Zarske, " Surfaces of Revolution with Constant Mean Curvature H=c in Hyperbolic 3-Space H 3 (- c 2 )", Undergraduate Student Paper Competition, 2013 meeting of the LA/MS Section of the MAA.

• Benjamin Benson, "Special Matrices, the Centrosymmetric Matrices", Undergraduate Thesis, 2010.

• Christopher R. Mills, "Method of approximate fundamental solutions for ill-posed elliptic boundary value problems", Honors Thesis, 2009.
• Ashley Sanders, "Problems in the College Math Journal", Undergraduate Project, 2009.

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Undergraduate Research Opportunities, Resources, and Support

Click on the links below to learn more about undergraduate student research opportunities in mathematics:.

Berkeley Mathematics Directed Reading Program (DRP)  

The Directed Reading Program provides undergraduates with the opportunity to work closely with UC Berkeley Mathematics graduate students in an independent reading project in the fall and spring semesters. The aim of the program is to equip students with the tools necessary to delve into sophisticated mathematics, to foster relationships between undergraduates and graduate students, and to provide students with a valuable opportunity to practice presenting mathematical ideas, both in conversation and public presentations. 

Berkeley Summer Undergraduate Research Fellowships (SURF)  

The SURF program consists of five paid summer research fellowship programs: 1) SURF L&S 2) SURF Rose Hills 3) SURF-SMART 4) UROC-Gates Foundation Fellows 5) SURF Math Team Fellowship. In addition to receiving funding, summer fellows from all five programs are assigned to a small peer group, which meets periodically throughout the summer. Each small group is led by a SURF Advisor. Although primary guidance in research will come from individual faculty/graduate student mentors, the small group meetings build community within the SURF cohort, connecting students with peers who are grappling with similar issues. SURF fellows also benefit from attending professional development and skills-building workshops. International and undocumented students are eligible to participate. 

Haas Scholars Program  

The Haas Scholars Program was founded in 1997 through the generous vision of Robert and Colleen Haas. Each year, twenty highly qualified, academically talented undergraduates with financial need come together to build a supportive intellectual community during their final year at UC-Berkeley. Applicants are evaluated primarily on the merit and originality of their proposal for an independent research or creative project that will serve as the basis for a senior or honors thesis. Once selected, Haas Scholars receive close mentoring from members of the UC-Berkeley faculty, seminars and workshops to assist them in the research and writing process, the opportunity to present their work at a professional conference, and up to $13,800 each in financial support. International students and undocumented students are welcome and encouraged to apply.  

Undergraduate Research Apprentice Program (URAP)  

 The Undergraduate Research Apprentice Program (URAP) is designed to involve Berkeley undergraduates more deeply in the research life of the University. The Program provides opportunities for you to work with faculty and staff researchers on the cutting edge research projects for which Berkeley is world-renowned. Working closely with mentors, you will deepen your knowledge and skills in areas of special interest, while experiencing what it means to be part of an intellectual community engaged in research. 

New research opportunities are open at the start of each semester; student applications are due the second week of instruction.  

Mathematical Sciences Research Institution Undergraduate Program (MSRI-UP)

MSRI-UP is a comprehensive summer program designed for talented undergraduate students, especially those from groups underrepresented in the mathematical sciences, who are interested in mathematics and make available to them meaningful research opportunities, the necessary skills and knowledge to participate in successful collaborations, and a community of academic peers and mentors who can advise, encourage and support them through a successful graduate program. At this time only U.S. citizens and permanent residents are eligible to apply. 

National Science Foundation Research Experiences for Undergraduate Students (NSF 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 a US or foreign location. To find Mathematical Sciences REU sites  click here .  

Interested in publishing your research? 

Berkeley Scientific Journal  

Berkeley Scientific is the undergraduate science journal of the University of California, Berkeley. Every semester, the undergraduate staff publishes independent research done by undergraduates at UC Berkeley, interviews with faculty members, reviews of recent scientific publications (books), and articles on current issues in science. All research papers are faculty-reviewed, and all interviews are conducted by the staff. The focus of the journal is broad, spanning scientific disciplines from ecology to engineering, from astronomy to biochemistry. 

Need additional support?

Have questions about other research opportunities on and off campus? Visit the  Office of Undergraduate Research and Scholarships  (OURS) for additional resources and support.  

OURS is UC Berkeley’s hub for undergraduate research and prestigious scholarships.  Established in 1997, OURS seeks to integrate undergraduates more fully into the dynamic and diverse research life of UC Berkeley. The center does so through a wide range of programs, workshops, partnerships, and communication platforms.

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.

A Project-Based Guide to Undergraduate Research in Mathematics

Starting and Sustaining Accessible Undergraduate Research

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• Pamela E. Harris 0 ,
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• Provides students with all the tools and information they will need to pursue undergraduate research within concise and accessible chapters
• Guides faculty through creating sustainable research programs by offering helpful hints and tips
• Contains a unique chapter on pursuing research in mathematics education

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

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Mentoring Undergraduate Research in Mathematical Modeling

Proving the “Proof”: Interdisciplinary Undergraduate Research Positively Impacts Students

A brief summary of my scientific work and highlights of my career

• Undergraduate research in mathematics
• Phylogenetic networks

Tropical Geometry

• Numerical Semigroups
• Squigonometry
• Mathematical Epidemiology
• combinatorics

Table of contents (11 chapters)

Front matter, folding words around trees: models inspired by rna.

• Elizabeth Drellich, Heather C. Smith

Phylogenetic Networks

• Elizabeth Gross, Colby Long, Joseph Rusinko
• Ralph Morrison

Chip-Firing Games and Critical Groups

• Darren Glass, Nathan Kaplan

Counting Tilings by Taking Walks in a Graph

• Steve Butler, Jason Ekstrand, Steven Osborne

Beyond Coins, Stamps, and Chicken McNuggets: An Invitation to Numerical Semigroups

• Scott Chapman, Rebecca Garcia, Christopher O’Neill

Lateral Movement in Undergraduate Research: Case Studies in Number Theory

• Stephan Ramon Garcia

Projects in (t, r) Broadcast Domination

• Pamela E. Harris, Erik Insko, Katie Johnson

Squigonometry: Trigonometry in the p -Norm

• William E. Wood, Robert D. Poodiack

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

• Milos Savic

Undergraduate Research in Mathematical Epidemiology

• Selenne Bañuelos, Mathew Bush, Marco V. Martinez, Alicia Prieto-Langarica

Editors and Affiliations

Pamela E. Harris

Aaron Wootton

About the editors

Bibliographic information.

Book Title : A Project-Based Guide to Undergraduate Research in Mathematics

Book Subtitle : Starting and Sustaining Accessible Undergraduate Research

Editors : Pamela E. Harris, Erik Insko, Aaron Wootton

Series Title : Foundations for Undergraduate Research in Mathematics

DOI : https://doi.org/10.1007/978-3-030-37853-0

Publisher : Birkhäuser Cham

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : Springer Nature Switzerland AG 2020

Hardcover ISBN : 978-3-030-37852-3 Published: 18 April 2020

Softcover ISBN : 978-3-030-37855-4 Published: 18 April 2021

eBook ISBN : 978-3-030-37853-0 Published: 17 April 2020

Series ISSN : 2520-1212

Series E-ISSN : 2520-1220

Edition Number : 1

Number of Pages : XI, 324

Number of Illustrations : 61 b/w illustrations, 81 illustrations in colour

Topics : Combinatorics

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Purdue REU Opportunities

Vertically Integrated Projects (VIP) are research projects in directed by faculty and aimed at providing research opportunities in mathematics for undergraduate students at all levels. Below are a list of projects that are currently accepting applications from students. Students who are interested in one of the projects below should send an e-mail to the professor supervising the project  with a resume, a list of courses taken (or transcript), and a personal statement explaining their reason for wanting to participate in the project.

• Project:  Exceptional Affine Standard Lyndon Words Through Coding Project available Fall 2023
• Project: 2-Parameter BCD-Type Quantum Affine Algebras Project available Fall 2023
• Project: Polyhedral Development and Polygon Foldng Project available Spring 2024
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Undergraduate Research

There are several ways students can become involved in research while still an undergraduate at Duke. After these programs, students can continue their research as a senior research project to earn  Graduation with Distinction .

Math+ (formerly DOmath) is an eight-week collaborative summer research program in mathematics, open to all Duke undergraduates. The program consists of groups of 2-4 undergraduate students working together on a single project. Each project is led by a faculty mentor assisted by a graduate student.

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PRUV provides financial support for six weeks of summer research with a faculty mentor in the Duke Mathematics Department. PRUV fellows, usually in the summer after their junior year, receive a stipend and live on campus with other students in the program.

Data+ is a ten-week summer research experience open to Duke undergraduates of all majors who are interested in exploring new data-driven approaches to interdisciplinary challenges. Students join small project teams, working alongside other teams in a communal environment. They learn how to marshal, analyze, and visualize data while gaining broad exposure to the modern world of data science.

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We offer seminar courses that offer the opportunity to begin a research project within a small class setting and earn course credit. These include MATH 305S, Number Theory; MATH 323S, Geometry; MATH 361S, Mathematical Numerical Analysis; MATH 451S, Nonlinear Ordinary Differential Equations; and MATH 476S, Mathematical Modeling.

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Students can pursue a research project with a faculty mentor and earn course credit in a Research Independent Study .  Often such an independent study continues research begun in one of the above programs.  Examples of recent projects include "Neural Coding in the Barn Owl", "Higher Genus Modular Forms", "Computing and Kidneys", "Yang-Mills flow on 4-manifolds", "Image Processing for Art Investigation", and "Random Unitary Transformations"..

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The Mathematics Department has offered a summer research program for undergraduates for many years. Formerly an REU and now called SPUR, the 8-week program provides an opportunity for undergraduate students of mathematics to participate in leading-edge research. We offer three projects every summer, each led by an expert in the field.

This program provides the opportunity for undergraduate students of mathematics to participate in leading-edge research. Each year, we offer three different projects, each led by an expert in the field. This program is sponsored by the US National Science Foundation (award DMS-1156350), and is open to US citizens and permanent residents who are currently enrolled in an undergraduate program.

See the SPUR 2017 program announcement for information about this year's program, including project descriptions and application instructions.

Previous Years' Projects and Some Results

• Analysis on Fractals, directed by Robert Strichartz
• Topological Methods in Discrete Geometry, directed by Florian Frick
• On Reay's relaxed Tverberg conjecture and generalizations of Conway's thrackle conjecture
• Nonlinear Heat Equations, directed by Xiaodong Cao
• Harmonic Analysis on Stiefel Manifolds, directed by Raul Gomez
• Graphs, Chip-Firing Games, and Algebraic Geometry, directed by Farbod Shokrieh
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• Generating Sets for Finite Groups, directed by Keith Dennis
• High Dimensional Data Analysis, directed by Matthew Hirn
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• Packings and Tilings, directed by Robert Connelly
• The Heat Equation, directed by Xiaodong Cao
• Combinatorics of Triangulations, directed by Ed Swartz
• Geometric Differential Equations, directed by Xiaodong Cao
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Collections of undergraduate research projects [closed]

I would like to compile a "big list" of undergraduate research projects in the following areas of mathematics:

• abstract algebra;
• linear algebra;
• number theory;
• mathematical physics;
• game theory;
• graph theory, combinatorics, probability, statistics;
• algorithms, computability, computational complexity, and other computer-science-related topics;
• et cetera...

To be precise, I'm searching for some references ( books , web-pages , online notes , etc.) that collect

• the statement of the problems (which are the most important thing);
• and preferably hints, guidance, or complete solutions ( if they have been found ) of the problems, and the actual complete projects.

I will start the list myself by mentioning the well-known book Student Research Projects in Calculus .

• reference-request
• soft-question
• $\begingroup$ If they have solutions, how can they be research problems? $\endgroup$ –  Matt Samuel Commented Dec 23, 2014 at 14:52
• 4 $\begingroup$ @MattSamuel: well, mathematicians do happen to solve problems sometimes :) $\endgroup$ –  Dal Commented Dec 23, 2014 at 15:00
• $\begingroup$ @MattSamuel Anyway, I've edited the question. $\endgroup$ –  Dal Commented Dec 23, 2014 at 15:00
• 2 $\begingroup$ Some departments have REU pages which list past/present projects and sometimes include writeups by past students. $\endgroup$ –  Antonio Vargas Commented Dec 25, 2014 at 19:48
• $\begingroup$ Please avoid the tags undergraduate-research and research . We are trying to remove them. $\endgroup$ –  Caleb Stanford Commented Jul 24, 2016 at 19:45

2 Answers 2

I looked into the question and found some projects that may be of interest:

Calculus and Differential Equations

Simple Differential Equations and the Growth and Decay of Ice Sheets :

In this project we will re-visit and expand upon a project of John Imbrie of the University of Virginia and his daughter Katherine matching periodic earth temperatures reflected in ice cores to when the earth axis tilt wobbled and the planets relative annual position to sun. We will investigate how key aspects of the ice-age record (such as shifts in dominant periodicities) follow from simple ordinary differential equations capturing the essential physics of the growth and decay of ice sheets.

The Gamma Function :

The theory of the gamma function was developed in connection with the problem of generalizing the factorial function of the natural numbers. The gamma function is defined as a definite, improper integral, and the notion of factorials is extended to complex and real arguments. This function crops up in many unexpected places in mathematical analysis, such as finding the volume of an n-dimensional “ball”. In this project we develop and explore the basic properties of this function.

Conic Sections via Cones :

In this project we will work our way through Conics of Apollonius of Perga ca. 262 BC – ca. 190 BC. In this work Apollonius develops simple and not so simple properties of conic sections, many of which we now only know through calculus. We will also attempt to illustrate the propositions in Conics using the powerful mathematics software Mathematica.

Linear Algebra

Affine Transformations and Homogeneous Coordinates :

In this project we will look at geometric transformations using homogeneous coordinates and matrices. Affine transformations include translation, rotation, reflection, shear, expansion/contraction, and similarity transformations. This will show the student the relationship between high school geometry, linear algebra, and group theory. We will also illustrate properties in geometry and linear algebra using the powerful mathematics software Mathematica or MATLAB.

Approximation of Functions with Simpler Functions :

In this project we will use functions with "nice" properties to approximate other functions. An example that might be familiar to students is using polynomials to approximate certain functions via Taylor/Maclaurin series. Eventually we will look at families of so-called “orthogonal functions” and how they are used to approximate other functions. We will use the powerful mathematics software Mathematica to illustrate the approximations.

Least is the Best :

A common concern in industry is optimization: minimizing the cost, maximizing the profit, optimizing resource utilization, and so on. Students learn basic optimization techniques in calculus courses. But to what is it applied? What if the objective function is non-differentiable? What if variables are discrete? In this project, students can choose their preferred "no-so-nice" application and explore heuristic approaches to estimate the optimum and the optimizer.

The kinematics of rolling. (Riemannian geometry/Non-holonomic mechanical systems) :

On a smooth stone, draw a curve beginning at a point p, and hold the stone over a flat table with p as the point of contact. Now roll the stone over the plane of the table so that at all times the point of contact lies on the curve, being careful not to allow the stone to slip or twist. We may equally well think that we are rolling the plane of the table over the surface of the stone along the given curve. Mechanical systems with this type of motion are said to have "non-holonomic" constraints, and are common fare in mechanics textbooks.

Now imagine a tangent vector to the plane at p. This rolling of the plane over the surface provides a way to transport v along the curve, keeping it tangent at all times. The resulting vector field over the curve is said to be a "parallel" vector field. Show that there is a unique way to carry out this parallel translation. (Find a differential equation that describes the parallel vector field and use some appropriate existence and uniqueness theorem.) Let c be a short path joining p and q, whose velocity vector field is parallel. Show that c is the shortest path contained in the surface that joins p and q.

Whether or not you fully succeed, this mechanical idea will give you a concrete way of thinking about ideas in differential geometry that might seem a bit abstract at first, such as Levi-Civita connection, parallel translation, geodesics, etc. Also look for an engineering text on Robotic manipulators and explain why such non-holonomic mechanical systems are important in that area of engineering.

I don't know of many places where these things are explained in a simple way. Perhaps Geometric Control Theory by Velimir Jurdjevic is a place to start. In the engineering literature, A Mathematical Introduction to Robotic Manipulation is a particularly good reference.

Geometry in very high dimensions. (Convex geometry)

Geometry in very high dimensions is full of surprises. Consider the following easy exercise as a warm-up. Let $B(n,r)$ represent the ball of radius $r$, centered at the origin, in Euclidian n-space. Show that for arbitrarily small positive numbers $a$ and $b$, there is a big enough $N$ such that $(100 - a)\%$ of the volume of $B(n,r)$ is contained in the shell $B(n,r) - B(n,r - b)$ for all $n > N$.

Here is a much more surprising fact that you might like to think about. Let $S(n-1)$ denote the sphere of radius 1 in dimension $n$. (It is the boundary of $B(n,1 )$.) Let f be a continuous function from $S(n-1)$ into the real line that does not increase distances, that is, $| f(p) - f(q) |$ is not bigger than $| p - q |$ for any two points $p$ and $q$ on the sphere. ($f$ is said to be a "1-Lipschitz" function.) Then there exists a number $M$ such that, for all positive $a$, no matter how small, the set of points $p$ in $S(n-1)$ such that $| f(p) - M |>a$ has volume smaller than $\exp(-na^2 / 2 )$. In words, this means that, taking away a set with very small volume (if the dimension is very large), $f$ is very nearly a constant function, equal to $M$.

This is much more than a geometric curiosity. In fact, such concentration of volume phenomenon is at the heart of statistics, for example. To make the point, consider the following. Let $S(n-1, n^{0.5})$ be the sphere in n-space whose radius is the $\sqrt{n}$. Let f denote the orthogonal projection from the sphere to one of the $n$ coordinate directions, which we agree to call the x-direction. Show that the part of the sphere that projects to an interval $a < x < b$ has volume very nearly (when $n$ is big) equal to the integral from $a$ to $b$ of the standard normal distribution. (This is easy to show if you use the central limit theorem).

For a nice introduction to this whole subject, see the article by Keith M. Ball in the volume Flavors of Geometry, Cambridge University Press, Ed.: S. Levy, 1997.

Hodge theory and Electromagnetism. (Algebraic topology/Physics)

Electromagnetic theory since the time of Maxwell has been an important source of new mathematics. This is particularly true for topology, specially for what is called "algebraic topology". One fundamental topic in algebraic topology with strong ties to electromagnetism is the so called "Hodge-de Rham theory". Although in its general form this is a difficult and technical topic, it is possible to go a long way into the subject with only Math 233. The article "Vector Calculus and the Topology of Domains in 3-Space", by Cantarella, DeTurck and Gluck (The American Mathematical Monthly, V. 109, N. 5, 409-442) is the ideal reference for a project in this area. (It has as well some inspiring pictures.)

Another direction to explore is the theory of direct current electric circuits (remember Kirkhoff's laws?). In fact, an electric circuit may be regarded as electric and magnetic field over a region in 3-space that is very nearly one dimensional, typically with very complicated topology (a graph). Solving circuit problems implicitly involve the kind of algebraic topology related to Hodge theory. (Hermann Weyl may have been the first to look into electric circuits from this point of view.) The simplification here is that the mathematics involved reduces to finite dimensional linear algebra. A nice reference for this is appendix B of The Geometry of Physics (T. Frankel), as well as A Course in Mathematics for Studentsof Physics vol. 2, by Bamberg and Sternberg.

Symmetries of differential equations. (Lie groups, Lie algebras/Differential equations)

Most of the time spent in courses on ODEs is devoted to linear differential equations, although a few examples of non-linear equations are also mentioned, only to be quickly dismissed as odd cases that cannot be approached by any general method for finding solutions. (One good and important example is the Riccati equation.) It turns out that there is a powerful general method to analyze nonlinear equations that sometimes allows you to obtain explicit solutions. The method is based on looking first for all the (infinitesimal) symmetries of the differential equation. (A symmetry of a differential equation is a transformation that sends solutions to solutions. An infinitesimal symmetry is a vector field that generates a flow of symmetries.) The key point is that finding infinitesimal symmetries amounts to solving linear differential equations and may be a much easier problem than to solve the equation we started with.

Use this idea to solve the Riccati equation. Choose your favorite non-linear differential equation and study its algebra of infinitesimal symmetries (a Lie algebra). What kind of information do they provide about the solutions of the equation? Since my description here is hopelessly vague, you might like to browse Symmetry Methods for Differential Equations - A Beginner's Guide by Peter Hydon, Cambridge University Press. It will give you a good idea of what this is all about.

Riemann surfaces and optical metric. (Riemannian geometry/Optics)

Light propagates in a transparent medium with velocity $c/n$, where c is a constant and n is the so called "refractive index" -- a quantity that can vary from point to point depending on the electric and magnetic properties of the medium. For a given curve in space, the time an imaginary particle would take to traverse its length, having at each point the same speed light would have there, is called the "optical length" of the curve. Therefore, the optical length is the line integral of n/c along the curve with respect to the arc-length parameter. According to Fermat's principle, the actual path taken by a light ray in space locally minimizes the "optical length". It is possible to use the optical length (for some given function n) to defined a new geometry whose geodesic curves are the paths taken by light rays. This is a particular type of Riemannian geometry, called "conformally" Euclidian. All this also makes sense in dimension 2.

One of the most famous paintings of Escher show a disc filled with little angels and demons crowding towards the boundary circle. What refractive index would produce the metric distortions shown in that picture?

A fundamental result about the geometry of surfaces states that, no matter what shape they have, you can always find a coordinate system in a neighborhood of any point that makes the surface conformally Euclidian. Why is this so? (This will require that you learn something about so called "isothermal coordinates".)

Failure of von Neumann's inequality.

Von Neumann proved that if A is a contractive matrix (has operator norm $\leq 1$) and $p(z)$ is a complex polynomial, then $p(A)$ has operator norm bounded by the supremum of $p$ on the unit circle. A two variable version of this result is true (Andô's inequality) but the three variable version is false. Counterexamples can be shown to exist either through probabilistic arguments (i.e. a random polynomial will fail the inequality) and there are also a few examples constructed through ad hoc methods. This project would involve trying to construct more interesting families of counterexamples to the three variable von Neumann inequality in order to understand "how badly" the inequality fails.

Multilinear Bohnenblust-Hille inequality

This is a different kind of inequality for polynomials. Multilinear polynomials satisfy an inequality bounding certain little $l^p$ norms of their coefficients by the supremum norm of the polynomial. This project would also involve looking for interesting examples to test the sharpness of known versions of this inequality.

algebra project 1

If $a_1,a_2,...a_n$ are integers with $gcd = 1$, then the Eulidean algorithm implies that there exists a $n \times n$-matrix $A$ with integer entries, with first row $= (a_1,a_2,...,a_n)$, and such that $\det(A) = 1$. A similar question was raised by J.P. Serre for polynomial rings over a field, with the a's being polynomials in several variables. This fundamental question generated an enormous amount of mathematics (giving birth to some new fields) and was finally settled almost simultaneously by D. Quillen and A. A. Suslin, independently. Now, there are fairly elementary proofs of this which require only some knowledge of polynomials and a good background in linear algebra. This could be an excellent project for someone who wants to learn some important and interesting mathematics. (These results seem to be of great interest to people working in control theory.)

algebra project 2

A basic question in number theory and theoretical computer science is to find a "nice" algorithm to decide whether a given number is prime or not. This has important applications in secure transmissions over the internet and techniques like RSA cryptosystems. Of course, the ancient method of Eratosthenes (sieve method) is one such algorithm, albeit a very inefficient one. All the methods availabe so far has been known to take exponential time. There are probabilistic methods to determine whether a number is prime, which take only polynomial time. The drawback is that there is a small chance of error in these methods. So, computer scientists have been trying for the last decade to find a deterministic algorithm which works in polynomial time. Recently, this has been achieved by three scientists from IIT, Kanpur, India. A copy of their article can be downloaded from www.cse.iitk.ac.in A nice project would be to understand their arguments (which are very elementary and uses only a little bit of algebra and number theory) and maybe to do a project on the history of the problem and its ramifications.

• VT past projects
• UMD past projects
• UC Berkeley past projects
• NYU project ideas
• NSF research experience for undergrads
• Washington State past projects
• Cornell REU
• U of Mary Washington opportunities, conferences, and past projects
• A few examples from Stanford and how to enroll if you are at Stanford
• BYU mentor section at the bottom has projects ideas

Not an answer, just one contribution.

The William's College SMALL summer program is an impressive model, at the high-end. "Around 500 students have participated in the project since its inception in 1988."

There are six areas this (2015) summer: Arithmetic Combinatorics (Leo Goldmakher), Combinatorial Geometry (Satyan Devadoss), Commutative Algebra (Susan Loepp), Geometry (Frank Morgan), Hyperbolic Knots (Colin Adams) and Number Theory & Harmonic Analysis (Steven Miller and Eyvi Palsson).

Here is a link for the project abstracts . Past projects have resulted in an impressive number of published papers .

• $\begingroup$ Thank you, Professor. As always, your contributions are precious. If you happen to find out other similar collections, please, do not hesitate to add them. $\endgroup$ –  Dal Commented Dec 31, 2014 at 10:48

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How should I start Undergraduate Research in Mathematics?

I am a college sophomore in US with double majors in mathematics and microbiology. My algorithmic biology research got me passionate about number theory and analysis. I started pursuing a mathematics major this Spring semester. I have been independently self-studying the number theory textbooks written by Niven/Zuckerman/Montgomery, Apostol, and Ireland/Rosen during this semester. As this semester progressed, I discovered that I am more interested in pure mathematics than applied aspects (computational biology, cryptography, etc.). I want to pursue a career as analytic number theorist and prove the Collatz conjecture and Erdos-Straus conjecture.

I have been thinking about doing number-theory research in my university (research university; huge mathematics department). I have been self-studying NT by myself and also regularly attending professional and graduate seminars on number theory. But, I did not do any pure mathematics research as an undergraduate. Should I visit NT professors in my university and ask them if I can do undergraduate research under them? If research is not possible (perhaps due to me lacking maturity), should I request to do independent reading under them and later proceed with research? How should I ask them? What should I address? If even independent reading is not desirable to them, what should I ask them or do by myself?

As for my mathematical background, I have been taking Calculus II (computational) and discrete mathematics. I will be taking Calculus III (vector calc.) over Summer, followed by Analysis I, Probability, Theoretical Linear Algebra in Fall 2015. As for my self-studying on this semester, I have been studying NT textbooks (mentioned above), proof methodologies, and basic linear algebra.

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• 7 You are in a research university with a huge math department. Do they offer "Introduction to Number Theory" course? If so, take it as soon as possible. –  Nobody Commented May 3, 2015 at 6:45
• 1 The responses to this MSE question may help: Undergraduate Mathematics Research . –  Joseph O'Rourke Commented May 3, 2015 at 14:36

3 Answers 3

While it is possible for a highly competent mathematician to dole out a doable problem for an undergraduate to solve over the summer, empirical data has proven otherwise - i.e., it's rather hard for an undergrad to prove anything original in number theory if only given a few weeks during the summer.

For an REU in NT, it might be more realistic to expect to read some interesting topics in number theory or perform some numerical analysis. Take it as a bonus if you obtain any original result.

Number theory is known to be a very difficult topic to get into. Before you decide to commit, take courses in complex analysis and abstract algebra. They are crucial if you want to read more advanced texts in number theory.

This depends a bit on where you are. You mention that you are at a large research university, presumably with a large graduate program - in these places I expect that it is pretty rare for an undergraduate to do research directly with a faculty member unless they are exceptionally talented. Faculty members have their own research programs and their own graduate students to direct, and coming up with interesting yet tractable problems is hard! However, it can't hurt to ask. Since you've been regularly attending seminars, you should know who the regular attendees are - send an email to someone to set up a meeting to chat (or drop by an open office hour), tell them what you've told us, and ask if they have any suggestions for what you should do next. Sending the email first gives them the chance to think about your meeting ahead of time, and if they're not interested they could just send an email back saying so. As for what exactly to ask, I would recommend just asking for suggestions and see what they come up with. If they don't seem inclined to do so, asking for suggestions on what to read/which courses to take is a good plan.

In a smaller liberal arts place, perhaps surprisingly, there are more opportunities for undergraduate research, and just generally more opportunities for interacting with faculty members.

However, there are other opportunities to get into research other than working directly with faculty. In a large research university, there are probably lots of graduate students and postdocs (who probably also attend the same seminars as you). They, particularly if they are interested in a more teaching-focused direction, might be willing to talk to you on some regular basis about your readings. (I recently heard of the University of Chicago Directed Reading program , which sounds pretty cool.)

Lastly, I know that you mentioned you wanted to do research at your own institution, but I'd like to ask you to reconsider. There are several summer research opportunities in mathematics, and they are a fantastic opportunity not only to learn math and get the experience of tackling a problem on your own (or in a small group), it's also good to just meet other people at your stage in life with similar interests, as well as mathematicians from other universities. If you're curious, here is a list of summer REUs (REU stands for Research Experiences for Undergraduates).

Should I visit NT professors in my university and ask them about if I can do undergraduate research under them? If research is not possible (perhaps due to my lacking maturity), should I request of doing independent reading under them and later proceed with the research? How should I ask them? What should I address? If even independent reading is not desirable to them, what should I ask to them or do in my own?

Don't be shy. Go talk to some professors! You can start the conversation by telling them what books you've been working with, and ask for some academic advising. This means the professor will go over your plan (which courses to take, when, and in what order) with you. The professor will likely confirm the wisdom of the tentative plan you came up with -- and then the conversation will blossom from there. A professor may make some specific suggestions for coursework and/or independent study. But maybe you should wait until you've got more coursework under your belt before proposing a research project. (Still, someone may surprise you!)

You can start with a short email saying that you have fallen in love with number theory, and would like to make an appointment for some academic advising.

It is always a pleasure when two people who love the same thing get together for a chat. You have nothing to fear.

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Directed research or senior project in mathematics, 2 or 4 units.

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Bridging the Gap Between High School and College: Frontiers of Science Institute

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Celebrating its 65th anniversary, the Frontiers of Science Institute (FSI) is reflecting on all of the research its current students and alumni have conducted over the years. The six week summer program fast tracks high school student's science, technology, engineering and mathematics (STEM) exploration.

July 8, 2024 | By Tamsin Fleming

Across the University of Northern Colorado campus, there is no shortage of new and exciting programs. For 65 years one program that has been causing excitement across campus: the Frontiers of Science Institute (FSI): a six week program for high school students to study a broad range of science, technology, engineering and mathematics (STEM) topics and perform their very own research.  

“This year they took wilderness medicine, and they did a snake class where they learned about the biochemistry of snake venom. Next week, they're learning about statistics and different math interpretations, and health and anatomy. They take all different kinds of topics and it’s mainly to give them a snippet of what a college course in some of these topics would be like,” said Victoria Duncan, Ph.D. ’24, director of FSI and assistant director of the Math and Science Teaching (MAST) Institute.  

Experiencing what it’s like to live and study on a college campus can be a life-changing experience for students. For many past participants who once had reservations about attending college and their ability to do so, participating in FSI gave them confidence that they could succeed in that type of setting, despite not having anyone in their immediate family who had earned a college degree. FSI is a way for high school students to participate in a trial-sized version of college in both atmosphere and substance.  

“Through this experience, I have become better prepared to face the challenges of college life. Having acquired personal readiness and a diverse range of extracurricular activities to enrich my resume, I am now filled with confidence that this will enhance my chances of gaining admission to the colleges I aspire to attend,” said a previous FSI participant.  

When students are accepted into the program, they begin working on a research project under the guidance of a mentor. The students will produce a paper, podcast or another method of communicating their findings, complete a 10-minute scientific talk on their findings, and present at a poster celebration for alumni and staff.  

Hyojae Lee is a senior high school student from Aurora, Colorado. Unsure of whether she’d like to stay in state or go out of state for college, FSI has allowed her to better understand the reality of being away from home. Encouraged by her teachers to apply for the program, Lee is now doing high-level research that she wouldn’t have been able to perform otherwise.  

“My group is looking at the impact of puberty blockers on the activity levels in rats,” said Lee.  

Another student, Nalan Rajan, a sophomore high school student from Palo Alto, California, is studying with graduate student Lani Irvin to see if antibiotics can balance aphid populations for improved crop yields.  

“I might be a co-author on a research paper that an undergraduate student is working on right now, which would help me a lot on my resume and college applications,” said Rajan, “I’m learning a lot here.”  

After all research is conducted, students present it on stage to one another. Presenting research findings is a common occurrence for a scientist. Through these experiences, students can more fully understand what to expect when seeking out research positions in college.  

All high school students are welcome to apply for the program, and while the majority of students historically are from Colorado, in recent years FSI has received some interest from out-of-state students. Historically, STEM fields have been dominated by men, but FSI has been pushing to increase representation and diversity among its participants.  

“Thinking of women in STEM, that is a long history of underrepresentation. We try specifically to recruit young women who have an interest in STEM. Last year, over half the cohort were  women, which is awesome,” said Duncan .  

FSI has received generous donor support for quite some time, and this year, over $114,000 was raised from corporate, foundation, and individual donors to support the program. With this generous support, every students in the program is receiving significant scholarship support, with multiple students receiving a full scholarship. The breadth of support for FSI is a testament to the importance of investing in STEM education and to the quality, uniqueness, and impact this longstanding program has had on hundreds of students.  

Celebrating its 65 th anniversary, FSI is welcoming its alumni back to campus to meet current students and research mentors at its culminating event on July 20. Alumni are encouraged to share about their experience with the program, what it has done for their career paths, and any of their favorite memories from their time in FSI. The banquet should be a perfect culmination of the 65 years of research and relationships fostered by FSI as students current and past connect and celebrate their shared love of STEM.  

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For first-year architecture students, an assignment of consequence.

For decades, the Jim Vlock First Year Building Project has offered students at the Yale School of Architecture the opportunity to design and build a house in New Haven, creating badly needed homes for individuals and families who would otherwise struggle to afford one.

The project recently launched a multi-year partnership with the Friends Center for Children, an early-childhood care and education in New Haven, offering to design and build five adjacent houses for two of the center’s educators and their families by 2027. The partnership is part of the Friends Center’s Teacher Housing Initiative, which addresses both the crisis in childcare and affordable housing by providing 20% of the center’s educators with rent-free homes, substantially increasing their take-home pay.

Last year, Yale students designed and built the first duplex dwelling, in the Fair Haven Heights neighborhood of New Haven. In this video, we follow the Yale students throughout the year-long process, from the first site visits, through design and construction, and ultimately to the celebration of the newly completed home.

View Slideshow 9 Photos

The project, a key facet of the curriculum in the school’s professional architecture degree program, was established in 1967 when the late Charles Moore, who directed Yale’s Department of Architecture from 1965 to 1971, sought to address students’ desire to pursue architecture committed to social action. The first-of-its-kind program is now emulated by many other architecture schools.

In its early years, students traveled to sites in Appalachia to build community centers and medical facilities. Since 1989, when the project switched its focus to building affordable housing in New Haven, first-year students have designed and built more than 50 homes in the city’s economically challenged neighborhoods.

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