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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. 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
Bryna Kra

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.

 
Immersing Manifolds in Euclidean Space
Paul Goerss

Manifolds are spaces which look locally like Euclidean space, and include many of the most basic objects we study in modern geometry and topology. Examples include the circle, the sphere, the torus, and many of geometric objects we draw on the board. Of course, when we draw a sphere or a torus on the board we are drawing a $2$-dimensional picture of something which most naturally embeds in Euclidean $3$-space. Embedding and immersion theory make a systematic study out of this observation. For example, we could ask, for a given abstract manifold $M$, what is the smallest Euclidean space in which $M$ embeds?

This question turns out to be hard, because we are  asking for a global property to be true. Immersions, by contrast, allow  self-intersections and the analogous problem – find the smallest immersion dimension – is much simpler because we can work locally. This allow us to use calculus which, in this area, means studying the properties on the tangent space of the manifold, which allows us, in turn, to use techniques and tools from algebraic topology. This typically gives lower bounds for immersion dimensions, which at times turn out to be upper bounds as well. 

This project introduces and studies many of the most central and powerful concepts ideas in current research mathematics, including differentiable manifolds, vector bundles, and characteristic classes. It is intended for the motivated and advanced student considering graduate school in differential geometry, algebraic geometry, or algebraic topology. 

Prerequisites: MATH 330-1 or MATH 331-1, MATH 344-1.
Co-requisite: MATH 440-1.

 
Fourier Series and Representation Theory
Eric Zaslow

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.

 
The Lefschetz Fixed Point Theorem
Paul Goerss

Given a topological space $X$ and continuous self-map $f:X \to X$ a very basic question we can ask is whether or not $f$ has fixed point: is there a point $x \in X$ with the property that $f(x) = x$? If so, then we can then ask if there is some way to continuously deform $f$ to another self-map $g:X \to X$ which does not have a  fixed point. In the very simple case when $X$ is a compact, convex subset of Euclidean space, this is covered by the Brouwer Fixed Point Theorem: every self map of such an $X$ must have a fixed point. The Lefschetz Fixed Point Theorem is a remarkable generalization which applies to many of the spaces which arise from basic problems in geometry: spheres, surfaces, projective spaces, and so on.

Here is the idea. We assign to the space $X$ a sequence of vector spaces $H_nX$. For each nonnegative integer $n$, the vector space $H_nX$ is known as that the $n$th homology group of $X$, and the map $f$ induces a corresponding linear transformation $f_n:H_nX \to H_nX$.The Lefschetz number of $f$ is the alternating sum of the traces of these maps: \[\Lambda(f) = \sum_{n \geq 0} (-1)^n \mathrm{tr}(f_n). \] (Recall that if we write out a linear map in matrix form, the trace is the sum of the diagonal elements.) This sum looks infinite, but our assumptions say $H_nX = 0$ for all $n$ large enough, so it is in fact a finite sum.  In practice it is highly computable. For example, if $X = S^2$ is the $2$-sphere of unit vectors in Euclidean $3$-space and $f$ is the identity, then $\Lambda(f) = 2$.  

The Lefschetz fixed point theorem then says that if this number is not zero, then the map $f$, and even any continuous deformation of $f$, must have a fixed point. Specifically, there is no way to deform the identity on $S^2$ to a map without a fixed point. This has many geometric implications. For example, it's not  hard to deduce the "Hairy Ball Theorem": you can't comb the hair on a billiard ball. Put in more precise language, this means there is no continuous non-vanishing vector field on $S^2$. 

At first glance, the Lefschetz Fixed Point Theorem seems impossible: why should an invariant built from such simple data as the traces of linear maps have such powerful geometric implications? The answer to  this meta-mathematical question lies in a deeper exploration of the ideas of basic algebraic topology. 

Prerequisites: MATH 330-1 or MATH 331-1, MATH 344-1.

Recommended: MATH 344-2.

 
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
Santiago Cañez 

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
Santiago Cañez 

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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