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Series: Research Horizons Seminar

Hot fluid expands. Expansion makes a fluid ``parcel'' lighter, causing it to rise. Cold, dense patches of fluid sink. And there we have the thermally induced motion of a fluid sitting on a hot plate... A longstanding open problem in applied analysis is the scaling of the Nusselt number (with respect to the Rayleigh number or, equivalently, system height) in thermal convection. The goal is a fundamental understanding of the effect of buoyancy-induced convection on heat transport in chaotic systems. The commonly held belief that the Nusselt number scales like (Ra)^(1/3) has eluded analytical proof for decades. We will describe the nature of the questions involved, the way that they can be framed (and reframed) mathematically, and the partial successes so far, including a recent preprint by Otto and Seis and a work in progress by the same authors

Series: Research Horizons Seminar

Hosts: Amey Kaloti and Ricardo Restrepo

By means of examples, I will illustrate the connection between orthogonal hypergeometric polynomials which satisfy interesting spectral and self-dual properties and representations of Lie algebras.

Series: Research Horizons Seminar

Hosts: Amey Kaloti and Ricardo Restrepo.

In this talk we will connect several different areas of mathematical analysis: complex analysis, harmonic analysis and functiontheory all in the hopes of gaining a better understanding of Carleson measures for certain classes of function spaces.

Series: Research Horizons Seminar

This talk is an introduction to using variational approaches for image reconstruction and segmentation. This talk will start with Total Variation minimization (TV) model and discuss Mumford-Shah and Chan-Vese model for image segmentation. I will mainly focus on multiphase segmentation and its extensions.

Series: Research Horizons Seminar

Hosts: Amey Kaloti and Ricardo Restrepo

This talk gives an overview of the mathematics of service processes, with a focus on several problems I have been involved in. In many service environments, resources are shared and delays arise as a result; examples include bank tellers, data centers, hospitals, the visa/mortgage application process.I will discuss some frequently employed mathematical tools in this area. Since randomness is inherent to many service environments, I will focus on stochastic processes and stochastic networks.

Series: Research Horizons Seminar

An elliptic curve is the set of solutions to a cubic equation in two variables and it has a natural group structure---you can add two points to get another. I'll explain why this is so, give some examples of the different types of groups that can arise (depending on the ground field), and mention some recent results on curves with many points. The are some nice thesis problems in this area which will be discussed in a follow-up talk later this semester in the algebra seminar.

Series: Research Horizons Seminar

In this talk the general setting for stochastic perturbation for dynamical systems is given. Recent research direction are given for the case in which the perturbation is non-linear. This is a generalization of the well known theory of Freidling Wentzell and Large deviations, which will be summarized during the talk.As always pizza and drinks will be served. Hosts: Amey Kaloti and Ricardo Restrepo.

Series: Research Horizons Seminar

Four dimensions is unique in many ways. For examplen-dimensional Euclidean space has a unique smooth structure if andonly if n is not equal to four. In other words, there is only one wayto understand smooth functions on R^n if and only if n is not 4. Thereare many other way that smooth structures on 4-dimensional manifoldsbehave in surprising ways. In this talk I will discuss this and I willsketch the beautiful interplay of ideas (you got algebra, analysis andtopology, a little something for everyone!) that go into proving R^4has more that one smooth structure (actually it has uncountably manydifferent smooth structures but that that would take longer toexplain).

Series: Research Horizons Seminar

Hosts: Amey Kaloti and Ricardo Restrepo

I will state two different inequalities which are poorly understood, even utterly mysterious. They are stated purely in terms of conditional expectations over dyadic intervals. Motivations and connections carry one into the area of Furstenberg's multilinear ergodic averages.

Series: Research Horizons Seminar

Hosts: Amey Kaloti and Ricardo Restrepo

Gauge theory is a beautiful subject that studies the space of connections on a vector bundle. It is also the natural language in which theories of particle physics are formulated. In fact, the word "gauge" has its origins in electromagnetism, and in this talk, we explore the basic geometric objects of gauge theory and show how one explicitly recovers the classical Maxwell's equations as a special case of the equations of gauge theory . Next, generalizing Maxwell's equations to a ``nonabelian" setting, we obtain the Yang-Mills equations, which describe the electroweak force in nature. Surprisingly, these equations were used by Simon Donaldson in the 1980s to prove spectacular results for the topology of smooth four-manifolds. We conclude this talk by describing some of the beautiful geometry and analysis behind gauge theory that goes into the work of Donaldson (for which we awarded a Fields Medal), and time permitting, we hope to say a bit about other gauge-theoretic applications to low-dimensional topology, for instance, instanton Floer homology.