The Colloquium is held on Mondays at 4pm in Deady 208.

Fall Quarter, 2019

• October 21, Brittany Erickson (University of Oregon, CS)
  A Computational Framework for Earthquake Sequences in Nonlinear Media
  Abstract: Deficient understanding of the earthquake cycle is the single greatest barrier to minimizing the devastating effects of earthquakes on society and the human environment. An ongoing goal of earthquake science is to synthesize physical models with observational data in order to assess seismic hazards and promote societal resilience. In this talk I will share my recent work in the development of a new computational framework for physically robust models of the earthquake cycle. One of the greatest challenges in earthquake modeling stems from the fact that earthquakes are not stand-alone, independent events, but rather are influenced by a millenial-long history of plate motion and a cycle of past earthquakes. Computations are further complicated because faults contain geometric and frictional complexities, and the solid Earth is heterogeneous and behaves nonlinearly. I will discuss details of the applied mathematical techniques that form the computational methods, while providing a context within the larger scope of earthquake science.
• October 28, Phillip Wood (UC Berkeley)
  Singularity of Discrete Random Matrices
  Abstract: Consider an n by n square matrix where n is large. For each entry, flip a fair coin, making the entry +1 if the coin comes up heads, and -1 if the coin comes up tails. What is the probability that the matrix has determinant equal to zero? This talk will highlight work of Terence Tao and Van Vu, Rudelson and Vershynin, joint work with Jean Bourgain and Van Vu, and a recent breakthrough by Konstantin Tikhomirov. One idea that runs through all the work is quantifying structure in random objects in order to understand their behavior.
• November 11, Matthew Gursky (Notre Dame)
  Asymptotically hyperbolic Einstein metrics: existence, obstructions, and generalizations
  Abstract: In this talk I will describe an interesting singular boundary value problem in differential geometry that has important connections to theoretical physics. The basic example of a solution is the Poincare ball model of hyperbolic space, so I will begin by reviewing some well known properties of hyperbolic space as a way of introducing the general boundary value problem. After describing some important existence results, I will spend some time at the end talking about obstructions to solving the problem, and some generalizations of the existence question that have been the focus of considerable attention lately.
• November 25, Aaron Lauda (USC)
  A new look at quantum knot invariants
  Abstract:

  In this talk we will explain how Lie theory leads to interesting families of invariants for knots and links that can all be defined in an elementary diagrammatic fashion.

  The Reshetikhin-Turaev construction associated knot invariants to the data of a simple Lie algebra and a choice of irreducible representation. The Jones polynomial is the most famous example coming from the Lie algebra sl(2) and its two-dimensional representation. In this talk we will explain Cautis-Kamnitzer-Morrison’s novel new approach to studying RT invariants associated to the Lie algebra sl(n). Rather than delving into a morass of representation theory, we will show how two relatively simple Lie theoretic ingredients can be combined with a powerful duality (Howe duality) to give an elementary and diagrammatic construction of these invariants. We will explain how this new framework solved an important open problem in representation theory, proves the q-holonomic conjecture in knot theory (joint with Garoufalidis and Lê), and leads to a new elementary approach to `categorifying’ these knots invariants to link homology theories.

• December 2, Julie Rowlett (Chalmers University)
  The mathematics of “hearing the shape of a drum”
  Abstract: Have you heard the question, “Can one hear the shape of a drum?” Do you know the answer? In 1966, M. Kac’s article of the same title popularized the inverse isospectral problem for planar domains. Twenty six years later, Gordon, Webb, and Wolpert demonstrated the answer, but many naturally related problems remain open today. We will discuss old and new results inspired by “hearing the shape of a drum.”

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