Classical Brill-Noether theory addresses the existence of special divisors on a general curve of genus g. In particular, Griffiths, Harris, Kleiman and Laksov combined to show that a general curve of genus g has a divisor of degree d and rank r if and only if g is at least (r+1)(g-d+r). Baker and Norine introduced a theory of ranks of divisors on graphs and metric graphs that is closely linked to ranks of divisors on algebraic curves, through specialization. Furthermore, Baker conjectured that suitable versions of Brill-Noether theorems should exist in tropical geometry and showed that some Brill-Noether type theorems on algebraic curves could be reproved tropically. In this talk, I will present recent joint work with Filip Cools, Jan Draisma, and Elina Robeva exploring these topics. The speaker is an assistant professor candidate.

We formulate an explicit conjecture that relates the degree of the Jones polynomial of a knot (and its parallels) with the slopes of incompressible surfaces in the knot complement. We will give ample motivation, and a proof in some botanically chosen classes of knots: alternating, 9 crossing or less, (-2,3,n) pretzel, and torus knots.

Analogies between the integers and the ring F_q[t] have been long known. However, from an arithmetic combinatorics perspective, these analogies have been little and only recently explored. As it turns out, in many cases existing methods can be transfered directly to F_q[t], while at times extra difficulties will arise. In this talk, I will discuss about analogs of some well-known results in this setting, including: -Green-Tao theorem for F_q[t]: The irreducible polynomials in F_q[t] contain affine spaces of abitrarily high dimension. -Sarkozy's theorem for F_q[t]: In any subset of positive density in F_q[t], we can find polynomials f,g such that f-g = h^2 for some nonzero polynomial h in F_q[t].

In this talk we will formulate the Monge-Kantorovich problem, mention some applications to economics and probability, and prove a few basic results.

We will discuss mirror symmetry from the perspective of the Strominger-Yau-Zaslow conjecture, focusing on the construction of pairs of mirror manifolds from Lagrangian torus fibrations, both in the Calabi-Yau case and in more general situations. The main goal in the first part will be provide general motivation for the constructions, and to explain how the mirror to a Kahler manifold with effective anticanonical class can be built from a (special) Lagrangian torus fibration and enumerative geometry data (weighted counts of holomorphic discs).

The second part will focus more specifically on examples where "instanton corrections" from Maslov index 0 holomorphic discs have to be taken into account. If time permits, we will discuss joint work with Mohammed Abouzaid and Ludmil Katzarkov about mirror symmetry for blowups of toric varieties along codimension 2 subvarieties. This construction allows one to recover geometrically the predicted mirrors in various interesting settings: pairs of pants, genus 2 curves, etc. More generally, it suggests how mirror symmetry might work for varieties of general type. (This is a two hour talk).

These talks will be about the C*-algebra approach to index theory and K-theory that was proposed by Atiyah and worked out in detail by Kasparov. Atiyah pointed out that an elliptic operator on a manifold can be viewed as cycles for the homology theory that is dual to K-theory. This led him to suggest a functional-analytic definition for K-homology, and Kasparov later worked out the complete theory. In the first talk I shall tell more of this story.

Abstract: For roughly the first half of the semester we will study recent developments in the geometry of nonabelian N=2 supersymmetric gauge theories in four dimensions as well as their dimensional reductions to three dimensions. Some recent progress results from regarding these as compactifications of a six-dimensional theory which does not have a classical description but is somehow a theory of nonabelian gerbes. The first lecture or two will cover some background material focused on the abelian version of these theories. Note: After this first lecture the seminar will move to RLM 8.136.

What is K-homology good for? I'll try to answer with examples in this talk and the next. 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.

The Langlands program relates complex representations of GL_n(Q_p) to Galois representations. For n = 1 this is explained by class field theory and for n = 2 this is closely related to the theory of modular forms. For general n, this is now understood by the work of Harris-Taylor and Henniart. In the last decade, a mod-p (as well as a p-adic) version of the Langlands program have been emerging, and they have already played an important role in some recent progress in number theory. But so far understanding has been limited to n = 1 and 2. We survey some of the known story in the classical and in the mod p case, and then discuss some recent progress on the classification of mod p representations of GL_n(Q_p).

We consider Klein-Gordon equations coupled to U(1)-invariant nonlinear oscillators. The solitary waves of the coupled nonlinear system form two-dimensional submanifold in the Hilbert phase space of finite energy solutions. Our main results read as follows: Theorem: Let all the oscillators be strictly nonlinear. Then any finite energy solution converges, in the long time limit, to the solitary manifold in the local energy seminorms. The investigation is inspired by Bohr’s postulates on transitions to quantum stationary states. The results are obtained for: • 1D KGE coupled to one oscillator; • 1D KGE coupled to finite number of oscillators; • nD KGE coupled to one oscillator via mean field interaction.

Matrix polynomial functions and which satisfy are called a Lax pair. I will present a construction of Hitchin which gives a concrete way of interpreting linear flows on the space of line bundles over a Riemann surface as solutions to Lax pair equations. This is a candidacy talk.

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle and make the scheme uniformly high order in space and time. One remarkable property of this approach is that it is straightforward to extend the method to two and higher dimensions. The same limiter can be shown to preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported. This is a joint work with Xiangxiong Zhang.

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.