Teodoro Collin

Title: The Mathematical Theory of the Finite Element Method from a
Computational Prospective

Abstract:The finite element method is a numerical technique for solving partial differential equations. We will describe the mathematical theory of the finite element method. Using the computational origin and basis of the finite element method, we will provide motivation for the definitions and insight into the proofs. If there is enough time, we will discuss how the mathematical theory of the finite element method impacted its implementation.c


William Golding

Title: Well-Posedness Theory for ODEs.

Abstract: While linear ODEs are explicitly solvable and well-behaved, non-linear ODEs are not always so nice. For example, non-linear ODEs may only exist for a short time or have multiple solutions. We will show that under quite mild constraints, non-linear ODEs are well-posed. 


Eric Chen

Title:  Combinatorial structure of MV Cycles

Abstract: The affine Grassmannian of a reductive group G is a central object of study in geometric representation theory, whose geometry controls the representation theory of the Langlands dual group \hat{G}. Mirkovic and Vilonen's proof of this result calls to attention certain subvarieties (now known as MV cycles) that serve as geometric avatars for the highest weight  theory of \hat{G}. In this talk, we will describe certain interplays between representation theory and geometry made possible by various combinations of results due to Mirkovic, Vilonen, Anderson, Kamnitzer, and many others.


Bat Dejean

Title: Zeta functions

Abstract: Zeta-functions are of great interest to number theorists. These functions, while usually defined as sums convergent on a half-plane, nevertheless extend meromorphically to the entire complex plane, which is classically done by establishing a functional equation. For a particular class of zeta-functions, Hecke used rather horrific methods to give such a functional equation for the class of "Hecke zeta-functions." About 30 years later, Tate bypassed most of Hecke's thorny bramblepatch and used simple Fourier analysis to derive the same functional equation. T.he Langlands program, an active area of number theory, generalizes Tate's ideas. I will sketch Tate's derivation in the special case of Dirichlet L-series. Due to time constraints, I will be unable to give Tate's thesis justice, but I can hopefully give listeners a reasonable 15-minute introduction.


Charles Hudgins

title: Understanding Noether's Theorem with Symplectic Geometry

Abstract: Noether's Theorem is one of the most celebrated results in mathematical physics. Informally, it states that every continuous symmetry of a physical system corresponds to a quantity which is conserved in time. In spite of its fame, few undergraduate physicists learn to apply or even formulate this result. The goal of this talk will be to motivate and explain the symplectic geometry perspective in analytical dynamics and, having done this, provide a mathematically precise statement of Noether's theorem on symplectic manifolds.


Alberto San Miguel Malaney

Title: Characteristics of vector bundles

Abstract: We will begin with a basic exposition of vectors bundles and explain how they are classified by the Grassmanian. Then we will compute the Stiefel-Whitney classes for a few simple bundles and show how the cohomology of the Grassmanian is freely generated by its Stiefel-Whitney classes. Lastly we will relate this to cobordism classes of manifolds using Stiefel-Whitney numbers.


Catherine Wolfram

Title:  Mathematical linguistics

Abstract: Languages have lots of structure. Can some of that structure be studied mathematically? This project is a prototype that explores some of the ways this could be done, in particular by looking at phonetic structure (sounds). One of the primary ways that I have attempted to analyze this data is using persistent homology (which requires a metric), so part of this talk will be about persistent homology. Given a language, we can look at a list of words in that language written phonetically, and consider each phonetic sound. We can look at the set of ?contexts" of each sound?i.e. the sounds that come immediately before and after it in all of its occurrences in the data. From this, we can construct a distance between sounds, where two sounds are close if they are used in similar contexts, and far apart if they are not. This at least distinguishes between vowels and consonants, which is interesting. This talk will be an overview of how this metric works, including why it might or might not be a good way to quantify this information, and some of the ways I have used this metric to try to analyze the data, including persistent homology.
