2019-2020 Mathematics Colloquium Schedule

September 26

3:30 PM - SDRICH 207

Changfeng Gui (University of Texas, San Antonio)
Title: The Sphere Covering Inequality and its Applications
Abstract: In this talk, I will introduce a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two distinct surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. Other applications of this inequality include the classification of certain Onsager vortices on the sphere, the radial symmetry of solutions to the Gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori and the standard sphere, etc. The resolution of several open problems in these areas will be presented. Some generalizations of the inequality to include singular terms or more general surfaces will also be presented.

October 15

4:00 PM - BSB D110

John Ewing (Math for America)
Title: Is there an education crisis?
Abstract: American education seems to be in permanent crisis. News reports tell us that schools and teachers are failing; international comparisons show American students are near the bottom; major corporations complain they cannot find qualified workers. Politicians and policymakers urge us to take immediate and radical action to address the crisis. Are things really so dire? The evidence for an education crisis is surprisingly ambiguous. What drives this apocalyptic view of education? What are the consequences of manufacturing a crisis where there isn’t one? And how can we solve education’s real problems with less melodrama and more common sense?

October 17

4:30 PM - MMSCI 301 (Baylor Lecture Series in Math; Colloquium)

Luis Caffarelli (University of Texas, Austin)
Title: The Interaction of Local and Non-Local Phenomena
Abstract: Often, infinitesimal and integral diffusion processes interact with each other: solids and gas, bids and calls, elastic membranes involving solids. The interactions may be due to phase transitions, transmission conditions or surface - solid conditions. After a short overview, we will discuss some examples and the mathematics involved.

October 18

4:00 PM - BSB D110 (Baylor Lecture Series in Math; Public Lecture)

Luis Caffarelli (University of Texas, Austin)
Title: Diffusion-type Equations: From the Heat Equation to Long Distance Interactions
Abstract: The heat equation (an equation that describes how heat propagates along a solid, for instance alog a metal rod) was proposed by Fourier in 1807. It describes how temperature evolves in time given the influx of heat from its infinitesimal surrounding. It was soon realized that the pointwise evolution of other modeled systems (speed of a fluid, density of deformation of an elastic body, price of goods, populations) adjusts and reverst to its surroundings revealing a universality property that made it fundamental to science. On the other hand, in many cases, diffusion processes involve long range interactions including population dynamics, pricing, and atmospheric events. We will give an overview of the mathematics involved.

November 7

3:30 PM - SDRICH 207

Li Gao (Texas A&M University)
Title: Quantum entropy and noncommutative Lp norms
Abstract: Entropy and its variants are important measures of information in both classical- and quantum information theory. For quantum systems, entropy quantities such as von Neumann entropy naturally relates to noncommutative Lp-norms. In this talk, we will discuss connection between various quantum entropies and Lp-norms. Such connections have found many applications in quantum information theory. I will talk about applications of Lp-norms in estimating quantum channel capacity and entropic uncertainty relations.

November 14

3:30 PM - SDRICH 207

David Damanik (Rice University)
Title: The topological structure of the spectrum of almost periodic Schrödinger operators
Abstract: In this talk we discuss the topological structure of the spectrum of almost periodic Schrödinger operators, both in one dimension and in higher dimensions. The problem is quite well understood in the one-dimensional case and the talk will briefly describe some of the known results. The question is significantly less well understood in higher dimensions. The Bethe-Sommerfeld conjecture for periodic potentials serves as a guiding principle for the different mechanisms and phenomena that should be expected to play a role. Passing from periodic to non-periodic almost periodic potentials, we discuss both positive and negative results in the spirit of the Bethe-Sommerfeld conjecture.

December 5

4:00 PM - BSB D110 (Undergraduate Lecture Series; Public Lecture)

Douglas Arnold (University of Minnesota)
Title: Computational Mathematics Simulating the World
Abstract: In the late 20th century, science underwent a revolution as computational science emerged as the third mode of scientific exploration alongside experiment and theory. Computer simultation of physical reality has played an equally transformative role in virtually all areas of technology, affecting many aspects of modern life. We now depend on simulation to design, predict, and optimize natural and engineered systems of all sorts, ranging from mechanical to chemical to electronic, and at scales ranging from atomic to terrestrial to cosmological. Mathemtical algorithms have been crucial to these advances, even more so than advances in computer technology. In this talk we will encounter some of the mathematical ideas that have emerged and the ongoing challenges facing computational mathematics in simulating the world.

December 6

4:00 PM - MMSci 301 (Undergraduate Lecture Series; Colloquium)

Douglas Arnold (University of Minnesota)
Title: Finite Element Exterior Calculus
Abstract: Finite element exterior calculus, or FEEC, is a prime example of a strucutre-preserving discretization method in which key mathematical structures of the continuous problem are exactly captured at the discrete level. In the case of FEEC these structures arise from differential complexes and their cohomology, and FEEC applies geometry, topology, and analysis in order to design and analyze stable and accurate numerical methods for the differential equations related to the complexes. We will present an accessible overview of FEEC and some of its applications.

February 5

Time: 2:30 PM - Room: SDRICH 325 (C*-day talk)

Mehrdad Kalantar (Houston)
Title: Noncommutative ergodic theory
Abstract: We give a soft introduction to several aspects of applications of operator algebras in dynamics and ergodic theory of groups. For a group G we consider several spaces including the space Sub(G) of subgroups of G, the space of positive definite functions, and state spaces of operator algebras generated by unitary representations of G, and review their connections and their applications in the study of actions of G. I intend to make it accessible to a very general audience, only assuming advanced undergraduate algebra, linear algebra, and analysis.

February 5

Time: 3:30 PM - Room: SDRICH 325 (C*-day talk)

Michael Brannan (Texas A&M)
Title: Quantum information theory and quantum symmetry groups of graphs
Abstract: In this talk I will give a light introduction to the theory of quantum groups by describing a concrete class of examples: the quantum symmetry groups of finite graphs. As the terminology suggests, the algebraic structure that we dub the ``quantum symmetry group'' of a graph ought to describe some sort of ``quantized symmetries'' of the given graph (in the physical sense of quantum mechanics). I will explain how recent ideas from the theory of non-local games in quantum information theory (QIT) provide this appropriate interpretation of quantum symmetry groups of graphs as ``physically realizable'' symmetries. Time permitting, I will also highlight some striking applications of ideas from QIT and quantum group theory to problems in graph theory and operator algebras.

February 6

Time: 3:30 PM - Room: SDRICH 207 (C*-day talk)

David Blecher (University of Houston)
Title: Noncommutative linear analysis in quantized function theory
Abstract: Since this talk is aimed at a general audience we begin by reviewing some general techniques for dealing with spaces and algebras of Hilbert space operators (that is, with the `quantum analogue' of functions and function spaces and algebras). There will be an emphasis on the theories of operator spaces, the *-algebraic approach to quantum physics, positivity, and on noncommutative measure and integration theory. We end with some new results.

February 13

3:30 PM - SDRICH 207

Stefan Friedenberg (HOST, Hochschule Stralsund)
Title: TBD
Abstract: TBD

March 26

3:30 PM - SDRICH 207

Jussi Behrndt (TU Graz)
Title: TBD
Abstract: TBD

April 16

Time: TBD, Room: TBD (AWM Lecture)

Melinda Gann (Mississippi College)
Title: TBD
Abstract: TBD

April 23

3:30 PM - SDRICH 207

Andrzej Swiech (Georgia Tech)
Title: TBD
Abstract: TBD

April 30

3:30 PM - SDRICH 207

Svetlana Jitomirskaya (UC Irvine)
Title: TBD
Abstract: TBD


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