The fifth talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Andrew Lee will be on the analysis of (generalised) Dirac operators. It will roughly follow the fifth chapter of the book.

The sixth talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Rustam Antia-Riedel and Javier Morales will be on Hodge theory, a bridge between cohomology and harmonic analysis. It will roughly follow the sixth chapter of the book.

The seventh talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Rahul Shah will continue on from the previous talks on spin structures on Riemannian manifolds. It will roughly follow the fourth chapter of the book.

The seventh talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Rahul Shah will continue on from the previous talks on spin structures on Riemannian manifolds. It will roughly follow the fourth chapter of the book.

The eighth talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Valentin Zakharevich will be on the heat and wave equations. It will roughly follow the seventh chapter of the book.

The ninth talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Cagri Karakurt will be on trace-class operators and spectral geometry, arriving at a formula of Weyl connecting the spectrum of the Laplacian associated to a manifold to the geometry of the manifold. It will roughly follow the eighth chapter of the book.

The tenth talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Richard Hughes will be on the functional calculus and heat kernel for the harmonic oscillator operator. Following this (time permitting), a geometric construction of Witten this gives rise to similar operators will be described, and some of the general theory for the functional calculus of Dirac operators on non compact manifolds will be discussed. It will roughly follow the ninth chapter of the book.

The eleventh talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Laura Starkson will be on proving a Lefschetz formula for geometric endomorphisms of a Dirac complex. It will roughly follow the tenth chapter of the book.

The twelfth talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Valentin Zakharevich will introduce the index problem and do some preliminary working with the super trace of the heat equation solution operator. It will roughly follow the eleventh chapter of the book.

The thirteenth talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Richard Hughes will be on the Getzler calculus, a sophisticated way of describing the order of a differential operator. It will roughly follow the twelfth chapter of the book.

The fourteenth talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Valentin Zakharevich will be on proving the index theorem, using the Getzler calculus to 'pick' an exact formula for the correct term out of the asymptotic expansion. It will roughly follow the twelfth chapter of the book.

This is a thesis defense.

The fifteenth (and final) talk for the Atiyah-Singer Index Theorem seminar, following John Roe's book "Elliptic operators, topology and asymptotic methods". This talk by Lee Cohn is on the Hirzebruch-Riemann-Roch Theorem (as an application of the Index Theorem); the thirteenth chapter of the book is a good reference (though it will not be followed closely)/

The usefulness of the various Riemann-Roch formulas as computational tools is well documented in literature. Grothendieck-Riemann-Roch is a commutative diagram relating push-forward in K-theory to the push-forward of associated Chow invariants for locally complete intersection (l.c.i.) morphisms. We extend this notion to quasi-smooth morphisms between derived schemes, this is the ``derived" analog of l.c.i. morphisms and it encompasses relative perfect obstruction theories. We will concentrate on the naturality of the construction from the standpoint of pure intersection theory and how it interacts with the virtual Gysin homomorphism defined by Behrend-Fantechi. Time permitting we will discuss the relationship with existing formulas, i.e., Ciocan-Fonanine, Kapranov, Fantechi, and Goettsche.

We begin by proving an index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with cyclic quotient singularities and then present two applications of this theorem. The first is to study deformations of a particular family of anti-self-dual ALE spaces and the second is to study anti-self-dual deformations of the canonical anti-self-dual Kahler metric on any weighted complex projective plane.

The first half of this talk will be an accessible introduction/review of the absolute logarithmic height function on algebraic numbers, its properties, and why we care about it.  In the second half I will introduce the notion of a (relative) Bogomolov extension of fields of algebraic numbers.  This is a new generalization of the Bogomolov property for a field K of algebraic numbers, which is satisfied the height of non-torsion points of K* is bounded away from 0.  An extension L/K is Bogomolov if the height of points of L*\K* is bounded away from zero.  We will describe a criterion for this property to be satisfied, which yields many examples.

This is the first of three lectures (by me, Distler, and Neitzke) on the following paper: http://arxiv.org/abs/1106.5698

It seems likely that every open 4-manifold admits uncountably many smooth structures, but we are still far from resolving this in full generality. The goal of this talk is to explore what is understood, answering this question positively in many cases. I will focus on an approach that generalizes constructions of exotic smooth structures on R^4 to a large set of open 4-manifolds. After presenting these and other related techniques, I will introduce examples illustrating where known methods differ and conclude with a simple example where all seem to fail. This is a practice candidacy talk.

Let E be an elliptic curve defined over with supersingular reduction at p. By work of Kobayashi and Iovita-Pollack we know that if F is a totally ramified -extension of then E(F) splits as the sum of two submodules consisting of certain points defined over even layers and odd layers of the -extension F. We prove an analogous result for the anticyclotomic -extension of a quadratic extension of and use it to analyze CM points of E.

Photoacoustic tomography is an imaging technique which combines optical tomography with ultrasound to reconstruct an object's interior optical properties from acoustic boundary data. Since the technique involves both an optical and acoustic process, the inverse problem is typically solved in two steps as well. In this talk we focus on the optical process. There are two acceptable mathematical models for the physics: the diffusion equation and radiative transfer equation. Uniqueness, stability, and methods for reconstruction for the radiative transfer model will be discussed.

In the theory of electrostatics, a natural situation to consider is that of a discontinuous change in permittivity of a medium across an interface (this happens, for example, in the neighborhood of the boundary between two different homogeneous materials). It is a classical fact that in such situations, the electric potential will be continuous across the interface, while the electric field will form a specific and fully describable discontinuity. This is known in PDE literature as a transmission problem. I will consider a limiting version of this problem from the perspective of regularity theory of elliptic equations. Our main goal will be a uniform Holder estimate proven via the method of De Giorgi. Time permitting, I will mention some extensions, including a "nonlocal transmission problem" for which questions of regularity can be approached from this perspective.

Abstract: Determinantal Point Process is a special class of Point Process that present repulsion. Heuristically, given a point at a certain location in space, the conditional probability of seeing another point nearby is less than the unconditional one, i.e., points repel each other. These processes are defined not in terms of distributions, but in terms of their "moments" (higher order moment measures), which in turn are defined by determinants (hence the name determinantal). I will present the rigorous definitions and give conditions for existence and uniqueness.

It seems likely that every open 4-manifold admits uncountably many smooth structures, but we are still far from resolving this in full generality. The goal of this talk is to explore what is understood, answering this question positively in many cases. I will focus on an approach that generalizes constructions of exotic smooth structures on R^4 to a large set of open 4-manifolds. After presenting these and other related techniques, I will introduce examples illustrating where known methods differ and conclude with a simple example where all seem to fail. This talk will be expository, including some background material for a future talk by Bob Gompf presenting recent work addressing this same topic. This is a candidacy talk.

We will examine the existence of self-dual orbifolds with 2-dimensional singular sets. After reviewing the definition of self-duality and defining orbifold-cone metrics, I will begin by discussing an example of these metrics on S^4 with singular set an embedded RP^2 and use this to motivate a series of questions regarding the existence of a type of self-dual orbifold-cone metric on S^4 with all possible singular sets. In the remainder of the talk I will describe answers to these questions.

We compute the volume of the idele class group of a number field, and sketch how it is related to the residue of the corresponding Dedekind -function at s=1.

This is the second in a series of three talks aimed at understanding the paper 1106.5698 by Moore-Tachikawa ( http://arxiv.org/abs/1106.5698 )

We first present the asymptotic analysis for a kinetic Cucker-Smale model. The asymptotic system is a form of the pressureless Euler equations with nonlocal force. In particular, the nonlocal term is derived from the flocking force of Cucker-Smale model. We also study the global-in-time existence of classical solution to the asymptotic system with initial compact support.

The pants graph of a surface S is a graph that encodes combinatorial data about curves on S and has important connections to hyperbolic geometry. This talk will present a result, joint with A. Zupan, about geometrically significant subgraphs of the pants complex. No background beyond the classification of surfaces will be assume, and this talk will serve as an introduction to active field of surface topology.

Two groups are quasi-isometric if they have the same ``asymptotic geometry''. Analogously, two groups are measure-equivalent if they behave the same from the point of view of groups actions on measure spaces. I will survey this area and discuss recent results on measure-equivalence rigidity for hyperbolic lattices due to Bader-Furman-Sauer.

Let K be an algebraic extension of the field Q of rational numbers. Write F(K) for the multiplicative group of K modulo its torsion subgroup, E(K) for the rational vector space generated by F(K), and X(K) for the closure of E(K) with respect to the metric induced in E(K) by the Weil height. We will describe some recent results about the algebraic and topological structure of these objects. The space X(K) is a real Banach space, and if K is a finite extension of Q, we describe a special Schauder basis in X(K). Using this Schauder basis we prove a new, seemingly elementary, result about points in E(K) with small height. This is joint work with R. Grizzard.

Lusztig stated a conjecture in 1980 for the characters of the irreducible representations of reductive algebraic groups over a field of positive characteristics. It was shown in 1990 that for a fixed group scheme the conjecture holds given that the characteristic is 'big enough', i.e. up to finitely many exceptions. We do now have a bound on the exceptions, which is an enormously big number. But recent results of Geordie Williamson indicate that the exceptions might grow exponentially with the rank of the root system. This came as a big surprise to representation theorists and currently we do not know what might be going on for 'small primes'.

Numerical solutions to conservation laws give the evolution of cell averaged data over time. The techniques used for solving this class of problems require the knowledge of the data at precise points. In this talk I will give an overview of the traditional WENO, weighted essentially non-oscillatory, interpolation technique and explain its limitations. Then present a different integration based I-WENO scheme which overcomes many of the difficulties of the traditional scheme.

Starting from an optimal cash management problem and its formulation as a stochastic impulse control problem we will derive an obstacle problem where the obstacle exhibits non-local behavior. The goal of this talk will be to derive a regularity result for the solution when the operator is fully nonlinear and uniformly elliptic.

This is a weekly working group at the Department of Mathematics at UT Austin. It is opened to anybody who is interested in topics around the theme of random structures, such as networks, spatial processes, random graphs, random combinatorial objects, with applications in operations research, machine learning and computer science. This week, Mir Omid Mirsadegi will talk about Point-Shift and Palm Probabilities

The Todd genus appears in the Grothendieck-Riemann-Roch theorem as a discrepancy between the Chern character of the pushforward bundle and the pushforward of the Chern character. Since the Chern character is closely related to the loop space, the Todd genus has an interpretation as a volume form on the loop space. Just as the Todd genus is related to the complex K-theory and the loop space, the Witten genus is related to elliptic cohomology and the space of maps from an elliptic curve. Costello gave an interpretation of the Witten genus as a volume form on this space of maps which comes from a quantization of a certain conformal field theory. My goal will be to explain the necessary background material to understand this result, such as the language of factorization algebras and deformation quantization.

The representation of closed 3-manifolds by 4-colored graphs has been independently introduced in the late seventies by S. Lins and by Pezzana's research group in Modena, by using dual constructions. The attempt of extending the representation to 3-manifolds with boundary was performed by C. Gagliardi by using a slightly different class of colored graphs, but the result was not suitable for a satisfactory computer tabulation of non closed 3-manifolds. We show that the whole class of 3-manifolds with non-empty non-spherical boundary can be represented by 4-colored graphs as the closed ones. This gives the opportunity of starting a more efficient computer aided tabulation. Partial results about enumeration and classification according to the minimal number of vertices of the graphs have been obtained.

Estimating the Tree of Life will likely involve a two-step procedure, where in the first step trees are estimated on many genes, and then the gene trees are combined into a tree on all the taxa. There are many intriguing and difficult computational problems involved in the species tree estimation. For example, even if the gene trees are correctly estimated, finding the true compatibility tree that contains all the species is NP-hard. However, estimated gene trees may not be correct, making the estimation problem additionally challenging. Finally, the true gene trees may not agree with each other or with the species tree, due to biological processes such as deep coalescence, gene duplication and loss, and horizontal gene transfer. In this talk, I will present new algorithms for these problems. We focus on the case where the gene trees can differ from the species tree due to incomplete lineage sorting. For this problem we present algorithms for the MDC (minimize deep coalescence) problem, taking gene tree estimation error into account (Yu et al., RECOMB and J Comp Biol 2011, Bayzid et al., J Comp Biol 2012). We also explore the use of "bin-and-conquer" to improve the accuracy and/or scalability of coalescent-based species tree methods (Bayzid and Warnow, Bioinformatics 2013).

Kashiwara's equivalence, saying that the category of D-modules on a variety X supported on a smooth, closed subvariety Y is equivalent to the category of D-modules on Y, is a key result in the theory of D-modules. In this talk I will explain how one can generalize Kashiwara's result to modules for deformation-quantization algebras on a conic symplectic manifold. As an illustrative application, one can use this result to calculate the additive invariants, such as the K-theory and Hochschild homology, of these module categories. This is based on joint work with C. Dodd, K. McGerty and T. Nevins.

In this talk we will consider some arithmetic properties of a particular family of Chatelet surfaces defined over the rationals. We will introduce the necessary tools to describe the Integral Brauer-Manin obstruction and the Brauer-Manin set for a variety over a number field and describe what is known in our setting. We will then discuss some techniques for understanding obstructions to the Hasse principle and weak approximation for certain cubic forms, and briefly outline some possible approaches for how to generalize to this family of surfaces.

2n-dimensional Dehn twists can be defined for each n as diffeomorphisms of 2n-manifolds. We will take a closer look at 4-dimensional ones, show that their square is isotopic to the identity and investigate their importance in the theory of Lefschetz fibrations.