- EventsExpandEvents Submenu
- AboutExpandAbout Submenu
- UndergraduateCollapseUndergraduate Submenu
- CoursesExpandCourses Submenu
- First Year Focus
- Advising
- Program RequirementsExpandProgram Requirements Submenu
- MENUExpandMENU Submenu
- School of Professional StudiesExpandSchool of Professional Studies Submenu
- Academic SupportExpandAcademic Support Submenu
- Prizes, Competitions, and OrganizationsExpandPrizes, Competitions, and Organizations Submenu
- Undergraduate Research
- Study Abroad
- Post-Undergraduate PlanningExpandPost-Undergraduate Planning Submenu
- Resources
- Remote LearningExpandRemote Learning Submenu

- GraduateExpandGraduate Submenu
- PeopleExpandPeople Submenu

Recommended: MATH 308-0.

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.

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.

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.

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.

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.

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.

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.

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.