## Curtis Porter : CRash CouRse in CR Geometry

- Graduate/Faculty Seminar,Uploaded Videos ( 2252 Views )CR geometry studies real hypersurfaces in complex vector spaces and their generalizations, CR manifolds. In many cases of interest to complex analysis and PDE, CR manifolds can be considered ``curved versions" of homogeneous spaces according to Elie Cartan’s generalization of Klein’s Erlangen program. Which homogeneous space is the ``flat model" of a CR manifold depends on the Levi form, a tensor named after a mathematician who used it to characterize boundaries of pseudoconvex domains. As in the analytic setting, the Levi form plays a central role in the geometry of CR manifolds, which we explore in relation to their homogeneous models.

## Yiming Zhong : Fast algorithm for Radiative transport

- Graduate/Faculty Seminar,Uploaded Videos ( 991 Views )This talk consists of two aspects about solving the radiative transport through the integral formulation. The radiative transport equation has been numerically studied for many years, the equation is difficult to solve due to its high dimensionality and its hyperbolic nature, in recent decades, the computers are equipped with larger memories so it is possible to deal with the full-discretization in phase space, however, the numerical efficiency is quite limited because of many issues, such as iterative scheme, preconditioning, discretization, etc. In this talk, we first discuss about the special case of isotropic scattering and its integral formulation, then walk through the corresponding fast algorithm for it. In the second part, we try to trivially extend the method to anisotropic case, and talk about the method’s limitation and some perspectives in both theory and numerics.

## Sarah Schott : Computational Complexity

- Graduate/Faculty Seminar ( 274 Views )What does it mean for a problem to be in P, or NP? What is NP completeness? These are questions, among others, that I hope to answer in my talk on computational complexity. Computational complexity is a branch of theoretical computer science dealing with analysis of algorithms. I hope to make it as accessible as possible, with no prior knowledge of algorithms and running times.

## Aaron Pollack : Modular forms on exceptional groups

- Graduate/Faculty Seminar ( 238 Views )Classically, a modular form for a reductive group G is an automorphic form that gives rise to a holomorphic function on the symmetric space G/K, when this symmetric space has complex structure. However, there are very interesting groups G, such as those of type G_2 and E_8, for which G/K does not have complex structure. Nevertheless, there is a theory of modular forms on these exceptional groups, whose study was initiated by Gross-Wallach and Gan-Gross-Savin. I will define these objects and describe what is known about them.

## Hubert Bray : What do Black Holes and Soap Bubbles have in common?

- Graduate/Faculty Seminar ( 212 Views )We will begin with the idea of General Relativity, which Einstein called his "happiest thought," and then proceed with a qualitative and quantitative discussion of the curvature of space-time. We will describe the central role of differential geometry in the subject and the important role that mathematicians have played proving the conjectures of the physicists, as well as making a few conjectures of our own. Finally, we will describe the geometry of black holes and their relationship to soap bubbles.

## Pam Miao Gu : Factorization tests and algorithms arising from counting modular forms and automorphic representations

- Graduate/Faculty Seminar ( 203 Views )A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on~$\Gamma_0(N)$ to a simpler function of $k$ and~$N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on~$\Gamma_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found to not be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input. (Joint work with Greg Martin.)

## Spencer Leslie : Intro to crystal graphs and their connections with number theory

- Graduate/Faculty Seminar ( 199 Views )I will review some basics of crystal bases for highest-weight representations for a semisimple Lie algebra. I will also point to some connections with number theory through Fourier coefficients of Eisenstein series, mostly in type A.

## Tom Witelski : Perturbation analysis for impulsive differential equations: How asymptotics can resolve the ambiguities of distribution theory

- Graduate/Faculty Seminar ( 196 Views )Models for dynamical systems that include short-time or abrupt forcing can be written as impulsive differential equations. Applications include mechanical systems with impacts and models for electro-chemical spiking signals in neurons. We consider a model for spiking in neurons given by a nonlinear ordinary differential equation that includes a Dirac delta function. Ambiguities in how to interpret such equations can be resolved via perturbation methods and asymptotic analysis of delta sequences.

## Aubrey HB : Persistent Homology

- Graduate/Faculty Seminar ( 185 Views )Persistent Homology is an emerging field of Computational Topology that is developing tools to discover the underlying structure in high-dimensional data sets. I will discuss the origins and main concepts involved in Persistent Homology in an accessible way, with illustrations and comprehensive examples. If time allows, I will also describe some current, as well as, future applications of Persistent Homology.

## Mike Gratton : Coarsening of thin liquid films

- Graduate/Faculty Seminar ( 144 Views )Coarsening is the phenomenon where many objects (water drops, molecular islands, particles in a freezing liquid) becoming a smaller number of objects in an orderly way. This talk will examine modeling one such system, tiny liquid drops, through three models: a PDE for the fluid, a coarsening dynamical system for the drops, and an LSW-type ensemble model for the group of drops. We will study self-similarity in the dynamics and extensions of the model to examine very long times when drops grow large enough that gravity distorts their shape.

## Loredana Lanzani : Practical uses of Complex Analysis

- Graduate/Faculty Seminar ( 113 Views )The notion of conformal mapping is of fundamental importance in complex analysis. Conformal maps are used by mathematicians, physicists and engineers to change regions with complicated shapes into much simpler ones, and to do so in a way that preserves shape on a small scale (that is, when viewed up close). This makes it possible to ``transpose a problem that was formulated for the complicated-looking region into another, related problem for the simpler region(where it can be easily solved) -- then one uses conformal mapping to ``translate'' the solution of the problem over the simpler region, back to a solution of the original problem (over the complicated region). The beauty of conformal mapping is that its governing principle is based on a very simple idea that is easy to explain and to understand (much like the statement of Fermat's celebrated last theorem) . In the first part of this talk I will introduce the notion of conformal mapping and will briefly go over its basic properties and some of its history (including a historical mystery going back to Galileo Galilei). I will then describe some of the many real-life applications of conformal maps, including: cartography; airplane wing design (transonic flow); art (in particular, the so-called ``Droste effect in the work of M. C. Escher). Time permitting, I will conclude by highlighting a 2013 paper by McArthur fellow L. Mahadevan that uses the related notion of quasi-conformal mapping to link D'Arcy Thompson's iconic work On Shape and Growth (published in 1917) with modern morphometric analysis (a discipline in biology that studies, among other things, how living organisms evolve over time). No previous knowledge of complex analysis is needed to enjoy this talk.