This talk is aimed (mostly) at undergraduate students. 

Abstract: In the 1930’s and 1940’s, Murray and von Neumann developed a theory of operators on Hilbert spaces, which heuristically may be thought of as infinite matrices acting on infinite dimensional vector spaces. Their works include a procedure which starts with an infinite group, a discrete object, and generates a von Neumann algebra, an analytic object which is a continuous analog to the n × n matrices. Much of the active research in this field has been generated by the following question: are structural properties of groups able to classify the resulting algebras? Obtaining a satisfactory resolution to this problem has been surprisingly difficult since standard group invariants are often not invariants of the algebras. We give a brief survey of the evolution of this problem, the surprising broader impacts including the emergence Jones polynomial, and the recent rapid progress in this classification program due to the advent of S. Popas deformation/rigidity theory. We close by describing recent developments in this program which have been made by my collaborators and myself. 

About the speaker: Rolando de Santiago is currently an Assistant Adjunct Professor and a UC Presidential Postdoctoral Fellow at UCLA working under S. Popa. His work is in the classification of type II1 von Neumann algebras, a subfield of functional analysis, and his research interests extend into group theory, topology, fractal geometry, and mathematical physics. He was born and raised up in the South-Eastern part of Los Angeles with 6 of his siblings. He spent 27 years studying at numerous public institutions including Pasadena City College and Cal Poly Pomona. After approximately 8 years of undergraduate work, he finally earned his B.S. in Mathematics. He completed his Masters in Mathematics at Cal Poly Pomona shortly thereafter. His mentors at Cal Poly, J. Rock and R. Wilson, strongly suggested that he pursue his Ph.D. After a significant amount of convincing, he threw all his belongings into a U-Haul, moved to Iowa City, and started grad school at the University of Iowa. He worked under of I. Chifan, the advisor who would help his launch his research career.

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Earthquakes produce seismic waves. They provide a way to obtain information about the deep structures of our planet. The typical measurement is to record the travel time difference of the seismic waves produced by an earthquake. If the network of seismometers is dense enough and they measure a large number of earthquakes, we can hope to recover the wave speed of the seismic wave from the travel time differences. In this talk we will consider geometric inverse problems related to different data sets produced by seismic waves. We will state uniqueness results for these problems and consider the mathematical tools needed for the proofs. The talk is based on joint works with: Maarten de Hoop, Joonas Ilmavirta, Matti Lassas and Hanming Zhou.

In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the boundary rigidity problem on manifolds with boundary (for instance, a domain in Euclidean space with a perturbed metric), i.e. determining a Riemannian metric from the restriction of its distance  function to the boundary. This corresponds to travel time tomography, i.e. finding the Riemannian metric from the time it takes for solutions of the corresponding wave equation to travel between boundary points. A version of this relates to finding the speed of seismic waves inside the Earth from travel time data, which in turn permits a study of the structure of the inside of the Earth.

This non-linear problem in turn builds on the geodesic X-ray transform on such a Riemannian manifold with boundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, so for instance in domains in Euclidean space along straight lines. The X-ray transform on symmetric tensors is similar, but one integrates the tensor contracted with the tangent vector of the geodesics. I will explain how, under suitable convexity assumptions, one can invert the geodesic X-ray transform on functions, i.e. determine the function from its X-ray transform, in a stable manner, as well as the analogous tensor result, and the connection to the full boundary rigidity problem.

Heterogeneous problems with high contrast, multiscale and possibly also random coefficients arise frequently in practice, e.g., reservoir simulation and material sciences. However, due to the disparity of scales, their efficient and accurate simulation is notorious challenging. First, I will describe some impor- tant applications, and review several state-of-the-art multiscale model reduction algorithms, especially the Generalized Multiscale Finite Element Method (GMsFEM). Then I will describe recent efforts on developing a mathematical theory for GMsFEM, and ongoing works on algorithmic developments and novel applications.

References

[1] Guanglian Li, On the Convergence Rates of GMsFEMs for Heterogeneous Elliptic Problems without Oversampling Techniques, submitted to Multiscale Modeling & Simulation, 2018.

[2] Shubin Fu, Eric Chung and Guanglian Li, Edge Multiscale Methods for elliptic problems with hetero- geneous coefficients, submitted to J. Comput. Phys, 2018.

Let G(D) be the set of all graphs with degree sequence d. The Erdos-Gallai conditions give the necessary and sufficient conditions for the existence of a graph with degree sequence d. If G(D) is nonempty, how do we approximate the size of all such graphs? I will discuss a maximum entropy approach to this problem. It involves considering G(D) as the set of integer points of a certain polytope in R^(n choose 2) and constructing a probability distribution that is constant on this set of points. Using a concentration result, we can use this distribution to approximate the size of G(D).