mathematics research projects for undergraduates

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DEPARTMENT OF MATHEMATICS

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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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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

Undergraduate Research Projects

Undergraduate Research and Reading Programs

Undergraduate students at MIT Mathematics Department have several opportunities to participate in mathematical research and directed reading. Four programs dedicated to cultivating research with the guidance of graduate students and faculty are:

• SPUR - Summer Program in Undergraduate Research
• DRP - Directed Reading Program during IAP in January
• UROP - Undergraduate Research Opportunities Program
• UROP+ - Supervised UROP Summer Program
• MSRP - MIT Summer Research Program

Undergraduates also have an opportunity to lead recitations, grade, and tutor in mathematics. For more information visit Math Academic Services .

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• Part B Students

Undergraduate Projects

Why choose to do a project.

There are several reasons why you may choose to do a project.

• Gaining research experience : projects give you a taste of what mathematical research is like.
• Developing transferable skills : the ability to collect material, organise it, expound it clearly and persuasively will be useful to many students in their future careers.
• Pursuing your interests : a project allows you to pursue topics which interest you, whether that be a particular area of mathematics or a subject related to mathematics, such as the history of mathematics.
• Demonstrating your understanding : the department recognises that some students might prefer to show their mathematical understanding and progress via a sustained piece of exposition rather than in a timed examination.

Further information about the projects

• Extracurricular Projects

Opportunities to do Project Work

On Course You can choose to undertake project work as one or more of the optional courses studied in Part B and the dissertation is compulsory at Part C. These options cover the whole spectrum of mathematics and include topics related to mathematics, such as the history of mathematics and mathematics education. Information on the different options is given below.

Extracurricular Projects There are also opportunities to undertake extracurricular projects, such as summer projects, during your studies. These projects are an excellent opportunity to gain experience of mathematical research.

Students' Experience of Projects

Summer Projects : "My summer project has been a fantastic opportunity to experience mathematical research, and has given me a really valuable taste of what it would be like to do postgraduate study. It's been great to be able to get engrossed in an area I knew relatively little about beforehand- probabilistic models used in genetics- and to be able to explore it in a less prescribed way than I am used to with lecture courses. The project was more varied than I expected: reading papers, posing new questions, working on these questions, discussing ideas with my supervisor and DPhil students and writing up what I'd done. I would very much recommend doing a summer project!"

"I really enjoyed doing a summer project as it's a completely different way of working compared to term time lectures and problem sheets. Not only was I introduced to some interesting theory about elliptic curves, I also got some general experience writing code and working with the command line. Our project involved about 10 people so it was incredibly useful to help each other and share ideas."

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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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