## Jessica Fintzen : Frontiers in Mathematics Lecture 1: Representations of p-adic groups

- Presentations ( 294 Views )The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of certain matrix groups, called p-adic groups. In my talk I will introduce p-adic groups and provide an overview of our understanding of their representations, with an emphasis on recent progress. I will also briefly discuss applications to other areas, e.g. to automorphic forms and the global Langlands program.

## Kirsten Wickelgren : An arithmetic count of the lines on a cubic surface

- Presentations ( 292 Views )A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. By contrast, over the real numbers, the number of real lines depends on the surface. A classification was obtained by Segre, but it is a recent observation of Benedetti-Silhol, Finashin-Kharlamov, Horev-Solomon and Okonek-Teleman that a certain signed count of lines is always 3. We extend this count to an arbitrary field k using an Euler number in A1-homotopy theory. The resulting count is valued in the Grothendieck-Witt group of non-degenerate symmetric bilinear forms. (No knowledge of A1-homotopy theory will be assumed in the talk.) This is joint work with Jesse Kass.

## Troy Schaudt : Mathematica 11 in Education and Research

- Presentations ( 292 Views )This technical talk will show live calculations in Mathematica 11 and other Wolfram technologies relevant to courses and research. Specific topics include: * Enter calculations in everyday English, or using the flexible Wolfram Language * Visualize data, functions, surfaces, and more in 2D or 3D * Store and share documents locally or in the Wolfram Cloud * Use the Predictive Interface to get suggestions for the next useful calculation or function options * Access trillions of bits of on-demand data * Use semantic import to enrich your data using Wolfram curated data * Easily turn static examples into mouse-driven, dynamic applications * Access 10,000 free course-ready applications * Utilize the Wolfram Language's wide scope of built-in functions, or create your own * Get deep support for specialized areas including machine learning, time series, image processing, parallelization, and control systems, with no add-ons required Current users will benefit from seeing the many improvements and new features of Mathematica 11 (https://www.wolfram.com/mathematica/new-in-11/), but prior knowledge of Mathematica is not required.

## Miklos Racz : From trees to seeds: on the inference of the seed from large random trees

- Presentations ( 266 Views )I will discuss the influence of the seed in models of randomly growing trees; in particular, I will focus on the preferential attachment and uniform attachment models. In both of these models, different seeds lead to different distributions of limiting trees from a total variation point of view. I will discuss the differences and similarities in proving this for the two models. This is based on joint work with Sebastien Bubeck, Ronen Eldan, and Elchanan Mossel.

## David Shea Vela-Vick : The equivalence of transverse link invariants in knot Floer homology

- Presentations ( 259 Views )The Heegaard Floer package provides a robust tool for studying contact 3-manifolds and their subspaces. Within the sphere of Heegaard Floer homology, several invariants of Legendrian and transverse knots have been defined. The first such invariant, constructed by Ozsvath, Szabo and Thurston, was defined combinatorially using grid diagrams. The second invariant was obtained by geometric means using open book decompositions by Lisca, Ozsvath, Stipsicz and Szabo. We show that these two previously defined invariant agree. Along the way, we define a third, equivalent Legendrian/transverse invariant which arises naturally when studying transverse knots which are braided with respect to an open book decomposition.

## Steven Sam : Noetherianity in representation theory

- Presentations ( 242 Views )Abstract: Representation stability is an exciting new area that combines ideas from commutative algebra and representation theory. The meta-idea is to combine a sequence of objects together using some newly defined algebraic structure, and then to translate abstract properties about this structure to concrete properties about the original object of study. Finite generation is a particularly important property, which translates to the existence of bounds on algebraic invariants, or some predictable behavior. I'll discuss some examples coming from topology (configuration spaces) and algebraic geometry (secant varieties).

## James Bremer : Improved methods for discretizing integral operators

- Presentations ( 218 Views )Integral equation methods are frequently used in the numerical solution of elliptic boundary value problems. After giving a brief overview of the advantages and disadvantages of such methods vis-a-vis more direct techniques like finite element methods, I will discuss two problems which arise in integral equation methods. In both cases, I take a contrarian position. The first is the discretization of integral operators on singular domains (e.g., surfaces with edges and curves with corners). The consensus opinion holds that integral equations given on such domains are exceedingly difficult to discretize and that sophisticated analysis, often specific to a particular boundary value problem, is required. I will explain that, in fact, the efficient solution of a broad class of such problems can be effected using an elementary approach. Exterior scattering problems given on planar domains with tens of thousands of corner points can be solved to 12 digit accuracy on my two year old desktop computer in a matter of hours. The second problem I will discuss is the evaluation of the singular integrals which arise form the discretization of weakly singular integral operators given on surfaces. Exponentially convergent algorithms for evaluating these integrals have been described in the literature and it is widely regarded as a "solved" problem. I will explain why this is not so and describe an approach which yields only algebraic convergence, but nonetheless performs better in practice than standard exponentially convergent methods.

## Justin Curry : Studying Stratified Maps a la MacPherson

- Presentations ( 217 Views )This talk is motivated by the question "Given a stratified map, how are path components of the fiber organized?" Studying path components necessitates cosheaves, but the stratified assumption provides an elegant combinatorial description using MacPherson's entrance path category, which also controls the associated Leray sheaves. One of the goals of this talk will be to provide a self-contained exposition of these ideas, using a minimal amount of mathematical background. The talk will follow loosely a recent paper with Amit Patel, which is available on the arXiv as http://arxiv.org/abs/1603.01587 Connections with applied topology will also be described.

## Jayce Getz : An approach to nonsolvable base change

- Presentations ( 211 Views )In the 1970's, inspired by the work of Saito and Shintani, Langlands gave a definitive treatment of base change for automorphic representations of the general linear group in two variables along prime degree extensions of number fields. To give some idea of the depth and utility of his work, one need only remark that some consequences of it were crucial in Wiles' proof of Fermat's last theorem. In this talk we will report on work in progress on base change for automorphic representations of GL(2) along nonsolvable Galois extensions of number fields. We will attempt to explain this assuming only a little algebraic number theory.

## Dr. Andrew Barnes : Risk Measurement and Capital Allocation for large loan portfolios

- Presentations ( 207 Views )Calculation of portfolio loss distributions is an important part of credit risk management in all large banking institutions. Mathematically, this calculation is tantamount to efficiently computing the probability distribution of the sum of a very large number of correlated random variables. Typical Monte Carlo aggregation models apply brute force computation to this problem and suffer from two main drawbacks: lack of speed and lack of transparency for further credit risk analysis. I will describe an attempt to ameliorate these drawbacks via an asymptotic probabilistic method based on the Central Limit Theorem. I will next describe capital allocation, a process of attributing risk to individual transactions or subportfolios of a given portfolio. In so doing, I will state axioms for coherent risk measures. These axioms place the notion of risk measurement and diversification on a firm mathematical foundation. I will then describe axioms for capital allocation via coherent risk measures, and illustrate the ideas with efficient computational formulae for allocating capital based on a couple of commonly used risk measures. In the course of this talk, which will be geared towards graduate students, I will attempt to give a flavor of industrial research and role of applied mathematics in industry.

## Robert Lipshitz : The Jones polynomial as Euler characteristic

- Presentations ( 206 Views )We will start by defining the Jones polynomial of a knot, and discussing some of its applications. We will then explain a refinement of the Jones polynomial, called Khovanov homology, and give some applications of this refinement. We will conclude by discussing a further refinement, called a Khovanov homotopy type; this part is joint work with Sucharit Sarkar.

## Ravi Vakil : Murphys Law in algebraic geometry: Badly-behaved moduli spaces

- Presentations ( 183 Views )We consider the question: ``How bad can the deformation space of an object be?'' (Alternatively: ``What singularities can appear on a moduli space?'') The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space can be arbitrarily bad.'' We show this for a number of important moduli spaces. More precisely, up to smooth parameters, every singularity that can be described by equations with integer coefficients appears on moduli spaces parameterizing: smooth projective surfaces (or higher-dimensional manifolds); smooth curves in projective space (the space of stable maps, or the Hilbert scheme); plane curves with nodes and cusps; stable sheaves; isolated threefold singularities; and more. The objects themselves are not pathological, and are in fact as nice as can be. This justifies Mumford's philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. I will begin by telling you what ``moduli spaces'' and ``deformation spaces'' are. The complex-minded listener can work in the holomorphic category; the arithmetic listener can think in mixed or positive characteristic. This talk is intended to be (mostly) comprehensible to a broad audience.

## Dragos Oprea : Theta divisors on moduli spaces of bundles over curves

- Presentations ( 179 Views )The Jacobian of any compact Riemann surface carries a natural theta divisor, which can be defined as the zero locus of an explicit function, the Riemann theta function. I will describe a generalization of this idea, which starts by replacing the Jacobian with the moduli space of higher rank bundles. These moduli spaces also carry theta divisors, described via "generalized" theta functions. In this talk, I will describe recent progress in the study of generalized theta functions.

## Matthias Heymann : Computing maximum likelihood paths of rare transition events, and applications to synthetic biology

- Presentations ( 177 Views )Dynamical systems with small noise (e.g. SDEs) allow for rare transitions from one stable state into another that would not be possible without the presence of noise. Large deviation theory provides the means to analyze both the frequency of these transitions and the maximum likelihood transition path. The key object for the determination of both is the quasipotential, V(x,y) = inf S_T(phi), where S_T(phi) is the action functional associated to the system, and where the infimum is taken over all T>0 and all paths phi:[0,T]->R^n leading from x to y. The numerical evaluation of V(x,y) however is made difficult by the fact that in most cases of interest no minimizer exists.

In my work I prove an alternative geometric formulation of V(x,y) that resolves this issue by introducing an action on the space of curves ( i.e. this action is independent of the parametrization of phi). In this formulation, a minimizer exists, and we use it to build a flexible algorithm (the geometric minimum action method, gMAM) for finding the maximum likelihood transition curve.

In one application I show how the gMAM can be useful in the newly emerging field of synthetic biology: We propose a method to identify the sources of instabilities in (genetic) networks.

This work was done in collaboration with my adviser Eric Vanden-Eijnden and is the core of my PhD thesis at NYU.

## Andrei Caldararu : The Pfaffian-Grassmannian derived equivalence

- Presentations ( 175 Views )We argue that there exists a derived equivalence between Calabi-Yau threefolds obtained by taking hyperplane sections (of the appropriate codimension) of the Grassmannian G(2,7) and the Pfaffian Pf(7). The existence of such an equivalence has been conjectured in physics for almost ten years, as the two families of Calabi-Yau threefolds are believed to have the same mirror. It is the first example of a derived equivalence between Calabi-Yau threefolds which are provably non-birational.

## Roman Vershynin : Randomness in functional analysis: towards universality

- Presentations ( 174 Views )The probabilistic method has redefined functional analysis in high dimensions. Random spaces and operators are to analysis what random graphs are to combinatorics. They provide a wealth of examples that are otherwise hard to construct, suggest what situations we should view as typical, and they have far-reaching applications, most notably in convex geometry and computer science. With the increase of our knowledge about random structures we begin to wonder about their universality. Is there a limiting picture as the dimension increases to infinity? Is this picture unique and independent of the distribution? What are deterministic implications of probabilistic methods? This talk will survey progress on some of these problems, in particular a proof of the conjecture of Von Neumann and Goldstine on random operators and connections to the Littlewood-Offord problem in additive combinatorics.

## Ben Weinkove : Symplectic forms, Kahler metrics and the Calabi-Yau equation

- Presentations ( 171 Views )Yau's theorem on Kahler manifolds states that there exists a unique Kahler metric in every Kahler class with prescribed volume form. This has many applications in complex geometry. I will discuss symplectic manifolds. In a different direction, I will talk about the problem of existence of constant scalar curvature Kahler metrics, which can also be considered a generalization of Yau's theorem.

## Nicholas Eriksson : Combinatorial methods in evolutionary biology

- Presentations ( 166 Views )My research focuses in three areas of evolutionary biology: the structure of viral populations, the evolution of drug resistance, and phylogenetics. Knowledge of the diversity of viral populations is important for understanding disease progression, vaccine design, and drug resistance, yet it is poorly understood. New technologies (pyrosequencing) allow us to read short, error-prone DNA sequences from an entire population at once. I will show how to assemble the reads into genomes using graph theory, allowing us to determine the population structure. Next, I will describe a new class of graphical models inspired by poset theory that describe the accumulation of (genetic) events with constraints on the order of occurrence. Applications of these models include calculating the risk of drug resistance in HIV and understanding cancer progression. Finally, I'll describe a polyhedral method for determining the sensitivity of phylogenetic algorithms to changes in the parameters. We will analyze several datasets where small changes in parameters lead to completely different trees and see how discrete geometry can be used to average out the uncertainty in parameter choice.

## Laura DeMarco : Complex dynamics and potential theory

- Presentations ( 165 Views )I will begin with the basics of the two subjects, with the goal of explaining how each has been used as a method to obtain results in the other. The first half will be devoted to foundational results, dating to the 1930s for potential theory and the 1980s for complex dynamics. The second half will be devoted to more recent developments.

## James Nolen : Reaction-Diffusion Fronts in Heterogeneous Media

- Presentations ( 160 Views )Reaction-diffusion equations are used in mathematical models of many physical and biological phenomena involving front propagation and pulse propagation. How do variations in the environment effect these phenomena? In this seminar, I will describe recent progress in understanding how fronts propagate in heterogeneous media. In particular, I will describe properties of generalized traveling waves for one-dimensional reaction-diffusion equations with variable excitation. I also will discuss multi-dimensional fronts in stationary random media, a model relevant to premixed-turbulent combustion. Along the way, I plan to describe interesting topics for future research.

## Jason Mireles-James : Adaptive Set-Oriented Algorithms for Conservative Systems

- Presentations ( 143 Views )We describe an automatic chaos verification scheme based on set oriented numerical methods, which is especially well suited to the study of area and volume preserving diffeomorphisms. The novel feature of the scheme is an iterative algorithm for approximating connecting orbits between collections of hyperbolic fixed and periodic points with greater and greater accuracy. The algorithm is geometric rather than graph theoretic in nature and, unlike existing methods, does not require the computation of chain recurrent sets. We give several example computations in dimension two and three.

## David Speyer : Matroids and Grassmannians

- Presentations ( 143 Views )Matroids are combinatorial devices designed to encoded the combinatorial structure of hyperplane arrangements. Combinatorialists have developed many invariants of matroids. I will explain that there is reason to believe that most of these invariants are related to computations in the K-theory of the Grassmannian. In particular, I will explain work of mine limiting the complexity of Hacking, Keel and Tevelev's "very stable pairs", which compactify the moduli of hyperplane arrangements. This talk should be understandable both to those who don't know matroids, and to those who don't know K-theory.

## Patrick Brosnan : Essential dimension and algebraic stacks

- Presentations ( 142 Views )Essential dimension is an invariant introduced by Buhler and Reichstein to measure how many parameters are needed to define an algebraic object such as a field extension or an algebraic curve over a field. I will describe joint work with Vistoli and Reichstein which studies essential dimension in the case where the algebraic objects are represented by a stack. I will also give examples of applications in the theory of quadratic forms.