On a smooth Riemannian manifold , the Ricci flow is the evolution equation . One expects solutions of Ricci flow to encounter finite-time singularities, as seen in the example of Ricci flow on a unit sphere. It is then natural and interesting to ask: given Ricci flow solutions , where and is finite, what is the weakest geometric condition that allows us to extend the solution past time ? In this talk, we will survey some results that answer this question. This is a practice for my candidacy talk.

Applications of filtering and data assimilation arise in engineering, geosciences, weather forecasting, and many other areas where one has to make predictions based on uncertain models supplemented by a stream of data with noise. For nonlinear problems filtering can be very expensive. In this talk, a particle-based nonlinear filtering scheme will be presented. This algorithm is based on implicit sampling, a new sampling technique related to chainless Monte Carlo method. Its main features are that the posterior densities are represented by pseudo-Gaussians and a resampling based on normalization constants. This filter is designed to focus particle paths sharply so as to reduce the number of particles needed for nonlinear problems. Examples will be given.

Quantum field theory is notoriously hard to understand, or even define. One reason for this is the lack of precise mathematical definitions of concepts such as Feynman measures, renormalizations, and so on. This talk will try to show that, with the right definitions, QFT becomes simple. Or at least simpler than it used to be.

What is K-homology good for? I'll continue trying to answer with examples in this talk. There is an obvious connection to the Atiyah-Singer index theorem, and roughly speaking K-homology provides a context in which to consider the index theorem's many elaborations. A general theme is that while two operators may look rather different, for example when studied using a symbol calculus, functional analysis and the framework of K-homology can sometimes give a means to identify their Fredholm index theories.

We will introduce the cohomology of profinite (Galois) groups and then survey some of its arithmetic applications.

Note the new room for the seminar this semester.

In this talk, we will show how optimal transport theory can be used to prove a sharp stability estimate for the Wulff shape of a given surface tension energy. This is a candidacy talk.

Poincaré's theorem on fundamental polyhedra is a classical result regarding the action of a discrete group by isometries on a metric space, and the geometry and topology of the corresponding quotient space. In this talk, I'll discuss the basics of the theorem, and try to explain why I find it useful.

Curves and surfaces are examples of manifolds, one of the most important types of spaces arising in mathematics. These are frequently encountered in applications - the earth's surface, the universe and the space of configurations of a mechanical system are all examples. We will explore the theory of manifolds at the most fundamental level, namely their topology, from visualizing examples to investigating what is known regarding their classification.

See http://www.ma.utexas.edu/seminar/sallie.pdf

In this lecture I'll discuss the most famous application of Kasparov's work - to the Novikov higher signature conjecture and the Baum-Connes conjecture. I'll sketch the proof of both conjectures for Gromov's a-T-menable groups (these are groups that act properly on an infinite-dimensional Euclidean space, and include amenable groups, free groups, Coxeter groups and others). The argument uses an interesting noncommutative C*-algebra that serves as a proxy for the commutative algebra of continuous functions on a Euclidean space (which isn't itself very useful when the space is infinite-dimensional). This algebra, together with a closely related Bott-Dirac operator on the Euclidean space, may have other applications.

In joint work with Riad Masri, we study the asymptotic growth of p(n), the number of partitions of a positive integer n, using the equidistribution of Galois orbits of Heegner points on the modular curve X_0(6). We obtain a new asymptotic formula for p(n) with an effective error term. We then use this asymptotic formula to sharpen the classical bounds of Hardy and Ramanujan, Rademacher, and Lehmer on the error term in Rademacher's exact formula for p(n). In a separate work using similar methods, we establish an asymptotic formula for the traces of CM values of a general class of Maass-Poincare series that generalize the modular j-function.

On a smooth Riemannian manifold , the Ricci flow is the evolution equation . One expects solutions of Ricci flow to encounter finite-time singularities, as seen in the example of Ricci flow on a unit sphere. It is then natural and interesting to ask: given Ricci flow solutions , where and is finite, what is the weakest geometric condition that allows us to extend the solution past time ? In this talk, we will survey some results that answer this question. This is a Ph.D. Candidacy talk.

To model carrier transport in semiconductor devices different techniques, from macroscopic to microscopic scale, have been developed and applied. After an elementary introduction on the main features of molecular dynamics, stochastic and deterministic models for electron transport in semiconductors are described and compared. Numerical schemes for deterministic models, based on the Boltzmann kinetic equation, are depicted and results are shown for both models in the case of Silicon and Gallium Nitride semiconductors.

I will solve the quantum harmonic oscillator using the raising and lowering operators. These operators are sums of unbounded, self-adjoint operators in L^2 and are useful for computing position and momentum expectation values.