Clouds and precipitation are among the most challenging aspects of weather and climate prediction. Moreover, our mathematical and physical understanding of clouds is far behind our understanding of a "dry" atmospheric where water vapor is neglected. In this talk, in working toward overcoming these challenges, we present new results on clouds and precipitation from two perspectives: first, in terms of the partial differential equations (PDEs) for atmospheric fluid dynamics, and second, in terms of stochastic models. A new asymptotic limit will be described, and it leads to new PDEs for a precipitating version of the quasi-geostrophic equations, now including phase changes of water. Also, a new energy will be presented for an atmosphere with phase changes, and it provides a generalization of the quadratic energy of a "dry" atmosphere. Finally, it will be shown that the statistics of clouds and precipitation can be described by stochastic differential equations and stochastic PDEs. As one application, it will be shown that, under global warming, the most significant change in precipitation statistics is seen in the largest events -- which become even larger and more probable -- and the distribution of event sizes conforms to the stochastic models.

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.

A Diophantine equation is a polynomial equation in several variables, generally with integer coefficients, like x³ + y³ = z³. Provably finding all integer solutions of a Diophantine equation is a storied mathematical problem that is easy to state and notoriously difficult to solve. The method of Chabauty--Coleman is one particularly successful technique for ruling out extraneous solutions of a certain class of Diophantine equations. The method is p-adic in nature, and involves producing p-adic analytic functions that vanish on all integer-valued solutions. I will discuss work with Katz and Zureick-Brown on finding uniform bounds on the number of rational points on a curve of fixed genus, defined over a number field, subject to a (conjecturally weak) restriction on its Jacobian. The same technique also makes progress on the uniform Manin-Mumford conjecture on the size of torsion packets on curves of fixed genus.

This work is motivated by a fundamental problem in sensor networks -- the need to aggregate redundant sensor data across a network. We focus on a simple problem of enumerating targets with a network of sensors that can detect nearby targets, but cannot identify or localize them. We show a clear, clean relationship between this problem and the topology of constructable sheaves. In particular, an integration theory from sheaf theory that uses Euler characteristic as a measure provides a computable, robust, and powerful tool for data aggregation.

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.

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.

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.

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.

"Finding and Measuring Stratified Spaces Using Persistent Intersection and Local Homology"

"Numerical Approximation of Layer Potentials Along Curve Segments"

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.

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.

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.

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.

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.

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.

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.

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.

Generalized complex structures, introduced by Hitchin in 2003, interpolate between symplectic and complex structures. In this talk, I will discuss a reduction procedure unifying symplectic reduction and holomorphic quotients. In general, this reduction procedure presents new features that will be illustrated in examples (e.g., the generalized reduction of a symplectic structure can be complex, and 3-form twists can appear in the quotient). If time permits, I will also discuss a super-geometric viewpoint to generalized reduction.

A real matrix is totally nonnegative if every minor in it is nonnegative. The classical Edrei-Thoma theorem classifies totally nonnegative infinite Toeplitz matrices, and is related to problems in representation theory, combinatorics and probability. I will discuss progress towards two variations on this theorem to block-Toeplitz matrices, and to finite Toeplitz matrices. Both of these variations connect the classical theory to loop groups.

This paper reports the results of using two sets of ranking data, one from actual elections and the other from surveys of voters, to examine whether the outcomes of three-candidate vote-casting processes follow a discernible pattern. Six statistical models that make different assumptions about such a pattern are evaluated. Both data sets suggest that a spatial model describes an observable pattern much better than any of the other five models. The results imply that any conclusions about the probability of voting events reached on the basis of models other than the spatial model—for example, on the basis of the “impartial anonymous culture”—are suspect. (Joint work with Florenz Plassmann)

The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class. In this metric, typical pairs of points are connected by a unique geodesic. However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs. I will also discuss some connections with other models of random geometry.