## Sam Stechmann : Clouds, climate, and extreme precipitation events: Asymptotics and stochastic

- Presentations ( 298 Views )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.

## 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.

## 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).

## Wolfgang Gaim : Semiclassical approximations to quantum mechanical equilibrium distributions

- Presentations ( 227 Views )In his 1932 paper, Eugene Wigner introduced the now famous Wigner function in order to compute quantum corrections to classical equilibrium distributions. We show how to extend this program and compute semiclassical approximations to quantum mechanical equilibrium distributions for slow, semiclassical degrees of freedom coupled to fast, quantum mechanical degrees of freedom. The main examples are molecules and electrons in crystalline solids. Where we will focus on the thermodynamics of the Hofstadter model as an application of the general results. The semiclassical formulas contain, in addition to quantum corrections similar to those of Wigner, also modifications of the classical Hamiltonian system used in the approximation: The classical energy and the Liouville measure on classical phase space turn out to have non-trivial-expansions in the semiclassical parameter. This talk is based on joint work with Stefan Teufel.

## Robert Ghrist : Sheaves and Sensors

- Presentations ( 224 Views )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.

## Lan-Hsuan Huang : Positive mass theorems and scalar curvature problems

- Presentations ( 219 Views )More than 30 years ago, Schoen-Yau and later Witten made major breakthroughs in proving the positive mass theorem. It has become one of the most important theorems in general relativity and differential geometry. In the first part of the talk, I will introduce the positive mass theorem and present our recent work that extends the classical three-dimensional results to higher dimensions. In the second part, I will discuss how the observation from general relativity enables us to solve classical geometric problems related to the scalar curvature.

## 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.

## 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.

## 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.

## 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.