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Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

Many conservative PDE models can be written in a Hamiltonian form. They include Euler equations in fluids, Vlasov models for plasmas and galaxies, ideal MHD for plasmas, Gross–Pitaevskii equation for superfluids and Bose-Einstein condensates, and various water wave models (KDV, BBM, KP, Boussinesq systems etc). I will describe some dynamical problems of these models, from a more unifying point of view by using their Hamiltonian forms. They include: stability/instability of coherent states (steady solution, traveling waves, standing waves etc.), invariant manifolds near unstable states, and inviscid and enhanced damping in fluids and plasmas. It is a topic course that will be taught in the fall.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

Abstract: Certain materials form geometric structures called "grains," which means that one has distinct volumes filled with the same semi-solid material but not mixing. This can happen with semi-molten copper and something like this can also happen with liquid crystals (which are used in some calculator display screens). People who try to analyze such systems tend to be interested in the motion of the boundaries between grains (which are often modeled by mean curvature flow) and the motions of the exterior surfaces of grains (which are often modeled by surface diffusion flow). Surfaces of constant mean curvature are stationary for both flows and provide stationary or equilibrium configurations. The surfaces of constant mean curvature which are axially symmetric have been classified. Grain boundaries are not usually axially symmetric, but I will describe a model situation in which they are and one can study the resulting equilibria. I will give a very informal introduction to the flow problems mentioned above (about which I know very little) and then go over the classification of axially symmetric constant mean curvature surfaces (about which I know rather more) and some reasonable questions one can ask (and hopefully answer) about such problems.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

Compressive sensing is a (relatively) new paradigm in data analysis that is having a large impact on areas from signal processing, statistics, to scientific computing. I am teaching a special topics on the subject in the Fall term, in support of the GT-IMPACT program. The talk will list some basic principles in the subject, stating some Theorems, and using images, and sounds to illustrate these principles.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

Abstract: If P(z) is a polynomial, then log|P(z)| is a potential. We discuss some facets of this observation, and some gems in classical potential theory. A special topics course on potential theory will be offered in the fall.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

We shall introduce and discuss several notions from classical Convex geometry. In particular, covering number, separation number and illumination number shall be defined and explored. Another parameter, which has been studied in the recent years, the dilated covering number of a convex set shall be introduced. We shall present best known estimate on this number, which is a part of a joint work with K. Tikhomirov.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

Abstract: It is not necessary to know what étale, motivic, or homotopy mean for this talk. The talk is intended to advertise motivic homotopy theory, and introduce it a little too. To do this, we'll give an example of an elementary problem the field can be used to solve, and then describe some aspects of the field itself which make this possible. The part of this talk which is original is joint with Jesse Kass.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

In this talk, we will discuss: (1) How geometry plays a role in machine learning/data science? (2) What it's like being a mathematician at a software company.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

Weak Galerkin (WG) is a new finite element method for partial differential equations where the differential operators (e.g., gradient, divergence, curl, Laplacian etc) in the variational forms are approximated by weak forms as generalized distributions. The WG discretization procedure often involves the solution of inexpensive problems defined locally on each element. The solution from the local problems can be regarded as a reconstruction of the corresponding differential operators. The fundamental difference between the weak Galerkin finite element method and other existing methods is the use of weak functions and weak derivatives (i.e., locally reconstructed differential operators) in the design of numerical schemes based on existing variational forms for the underlying PDE problems. Weak Galerkin is, therefore, a natural extension of the conforming Galerkin finite element method. Due to its great structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations by providing the needed stability and accuracy in approximation. In this talk, the speaker will introduce a general framework for WG methods by using the second order elliptic problem as an example. Furthermore, the speaker will present WG finite element methods for several model PDEs, including the linear elasticity problem, a fourth order problem arising from fluorescence tomography, and the second order problem in nondivergence form. The talk should be accessible to graduate students with adequate training in computational mathematics.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar

In this seminar,we will explain why and how unpredictable (chaotic) dynamics arises in deterministic systems. Some open problems in dynamical systems, probability, statistical mechanics, optics, (differential) geometry and number theory will be formulated.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

In elementary calculus, we learn that (1+z/n)^n has limit exp(z) as n approaches infinity. This type of scaling limit arises in many contexts - from approximation theory to universality limits in random matrices. We discuss some examples.