Number theorists have long been interested in questions about distribution of arithmetic objects: what is the frequency that the class group of a random quadratic imaginary field contains (Z/3Z)^r? How many extensions K/Q with Galois group S_6 have discriminant less than N? How many elliptic curves over Q are there with conductor at most N? In general, little is known about such questions, which include well-known conjectures of Cohen-Lenstra and Bhargava. It turns out that the analysis of such questions over function fields over finite fields reveals surprising connections with problems of current interest in topology and algebraic geometry. In particular, we explain how a purely topological theorem about stabilization of cohomology for moduli spaces of covers of curves (or, the group theory language, families of congruence subgroups of braid groups) would imply function-field versions of several distributional conjectures in number theory (and several other statements which have not previously been conjectured), and we describe some partial progress towards the desired topological result. (joint work with Akshay Venkatesh and Craig Westerland)

We consider finite time blow up solutions for the -critical nonlinear Schrodinger equation. For solutions with initial data, there is a minimal amount of concentration of norm at the blow up time. This result was first obtained by Merle and Tsutsumi in the case of radial initial data. The radial assumption was later removed by Nawa using concentration compactness techniques. We present a proof of this result in the general case following the argument of Hmidi and Keraani. A similar mass concentration result was later proved by Bourgain in dimension 2 for general initial data. This is a practice for my candidacy talk.

The Porous Media Equation and Fast Diffusion Equation are non-linear versions of the heat equation that are degenerate parabolic, their diffusion becomes degenerate near the regions where the concentration being diffused becomes zero; this is because the diffusion coefficient is like a power of the concentration itself. These equations not only model very interesting physical phenomena (e.g. diffusion in a porous medium, plasma physics and microscopic droplets), they also have interesting mathematical properties, one of these being the finite speed of propagation which gives rise to free boundaries. In this talk I will talk a little about the scaling of these equations, self-similar solutions and general asymptotic behavior and in each of these instances I will draw a constrat with what happens with the heat equation

Using recently calculated virial coefficients of hard spheres up to 10th order, it is shown that the singularity in the bulk modulus of a hard sphere fluid is a simple pole. The singularity occurs as the density approaches the close-packed packing fraction, . After the pole is isolated and its residue evaluated (), a very accurate expression is obtained for the bulk modulus. Integrating the modulus yields an equation of state (EOS) and entropy that both contain logarithmic singularities. These singularities are consistent with the probability of a successful random insertion of a sphere vanishing as . An analysis of density fluctuations in the grand canonical ensemble also argues for the existence of a simple pole singularity in the modulus. The new EOS accurately describes extant molecular dynamics data up to the fluid-solid transition (). Using an insertion probability argument, it is shown why the EOS can be successfully extended to the metastable fluid branch up to some unique density associated with the glass transition. The entropy of fusion (melting) is estimated to be 1.2 k. This is in excellent agreement with the melting entropy of several close-packed metals (aka Richard's rule). Significantly, the modulus shows no trace of a zero (infinity compressibility) associated with the fluid-solid transition. By appealing to the Yang and Lee theory of phase transitions, it is argued that this singularity should be undetectable in a virial EOS.

Khovanov Homology is a link invariant which generalizes the Jones polynomial. It associates a bigraded chain complex to an oriented link diagram in such a way that the Jones polynomial of the link is equal to the graded Euler characteristic of the chain complex. After describing this construction, we'll talk about a recent paper by Liam Watson in which distinct knots with the same Khovanov homology are produced. This is an option talk.

In solving the time-dependent wave equation numerically, it appears that working on operators, instead of wavefields, is the only known way to break classical complexity barriers such as CFL timestepping. Re-thinking wave computation is important in the scope of large-scale inverse scattering problems such as reflection seismology, where large datasets need to be processed. In this talk I will discuss discrete symbol calculus, an algorithmic framework to represent operators as pseudodifferential symbols via adequate quadrature and interpolation techniques. This new tool provides an efficient preconditioner for the Helmholtz equation, a method for taking near-exact square roots of elliptic operators, and will possibly allow to perform upscaled timestepping for the wave equation. Joint work with Lexing Ying from UT Austin.

I'll present some computations for a volume preserving map that has been studied recently by H. Lomeli and J. Meiss. This map arises as a normal form for quadratic volume preserving maps with quadratic inverse. The theory of Meiss and Lomeli relates several dynamical properties of the map to some line integrals over the heteroclinic connections between it's fixed points. I'll go over a scheme for computing these connections that uses the parameterization method developed by X. Cabre, E. Fontich and R. de la Llave.

Using tropical methods, we study the compactifaction of the moduli space of six lines in . In particular, the log canonical model is Kapranov's Chow quotient compactification. This is a thesis defense.

We consider finite time blow up solutions of the -critical Nonlinear Schrodinger equation. For solutions with initial data, there is a minimal amount of concentration of norm at the blow up time. This result was first obtained by Merle and Tsutsumi in the case of radial initial data. The radial assumption was removed by Nawa using concentration compactness techniques. We present a proof of this result in the general case following the argument of Hmidi and Keraani. A similar mass concentration result was later proved by Bourgain in dimension 2 for general initial data. This is a candidacy talk.

The space of the algebraic maps from P^1 to p^n is a quasi-projective variety. A good compactification of this space is a nonsingular projective variety containing this quasi-projective variety such that the boundary is a divisor with normal crossings. In this talk, we will construct such a good compactification by successive blow-ups along subvarieties following the original work of Yi Hu, Jiayuan Lin and Yijun Shao. This is a candidacy talk.

Abstract: Nekrasov conjecture predicts a relation between the N=2 SUSY gauge theory partition function and the SW prepotential. This conjecture was proven for instantons on R4 by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. I will report on joint work with Melissa Liu, where we prove the conjecture for instantons on toric surfaces.

We consider the full regularity of a suitable weak solution of the incompressible Navier Stokes equation in 3D when we have enough control on the second derivative of the velocity along the direction of the streamlines. If time permits, we present a log improvement of the Prodi-Serrin criteria in homogeneous case of p=q=5. This is a thesis defense.

We will introduce Ramsey Theory by proving Ramsey's Theorem, and then go on to discuss generalizations and consequences. We will finish by discussing Rado's Theorem, which characterizes which linear systems of equations can be solved in a single color class under arbitrary finite colorings.

This is a Ph.D candidacy talk.