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# Archive: Inverse Problems and Analysis

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 Prof. Volker Runde, University of Alberta Prof. Volker Runde, University of Alberta EWG 336 Title: What is amenability - from the Banach-Tarski paradox to Banach algebras

Abstract: We give a survey of how the notion of amenability emerged from John von Neumann's work on measure theory, and how it moved from there to abstract harmonic analysis, Banach and operator algebras. 11/14/2018 6:30:00 PM 11/14/2018 7:30:00 PM False Dr. Shihua Gong, Penn State University Dr. Shihua Gong, Penn State University EWG 336 Title: Hybridized Mixed Finite Element Methods for Linear Elasticity and Optimal Multilevel Solvers

Abstract: Mixed finite element methods are important in solid mechanics since they avoid locking and provide a straightforward approximation for the stress, which is a symmetric tensor. The relevant existing conforming mixed finite element spaces for the symmetric stress are subject to continuity constraints at the element vertices, which is not natural for H(div) conformity and prohibits techniques like hybridization. This talk will relax the continuity at the element vertices and characterize the full special space for the stress. By this enrichment, the corresponding mixed finite element method is stable in some lower order cases. Moreover, hybridization techniques and multilevel solvers can be applied to solving the arising linear system. 11/9/2018 4:00:00 PM 11/9/2018 5:00:00 PM False Dr. Murat Akman, University of Connecticut Dr. Murat Akman, University of Connecticut EWG 336
Title: Perturbations of elliptic operators on domains with non-smooth boundaries

Abstract: In this talk, we study perturbations of elliptic operators on domains with rough boundaries. In particular, we focus on the following problem: suppose that we have good estimates'' for the Dirichlet problem for a uniformly elliptic operator $L_0$, under what optimal conditions, are those good estimates transferred to the Dirichlet problem for uniformly elliptic operator L which is a perturbation'' of L_0?
We prove that if discrepancy between L_0 and L satisfies certain smallness assumption then the elliptic measure corresponding to L is in the reverse H{\"o}lder class with exponent 2 with respect to the elliptic measure corresponding to $L_0$ when the domain is 1-sided NTA satisfying the capacity density condition (CDC).
This is a joint work in progress with Steve Hofmann, Jose Maria Martell, and Tatiana Toro. 10/24/2018 5:30:00 PM 10/24/2018 6:30:00 PM False Prof. Mehrdad Kalantar, University of Houston Prof. Mehrdad Kalantar, University of Houston EWG336 Title: A notion of topological boundary for unitary representations

Abstract: We introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group $G$ to the setting of a general unitary representation $\pi: G \to B(H_\pi)$. This generalized space, which we call the Furstenberg-Hamana boundary" is a $G$-invariant subspace of $B(H_\pi)$ that carries a canonical C*-algebra structure. In many natural cases, including when $\pi$ is a quasi-regular representation or, more generally, a Koopman representation, the Furstenberg-Hamana boundary of $\pi$ is commutative, hence of the form C(X) for a compact $G$-space X. We study several examples, and also give a few applications. This is joint work with Alex Bearden. 10/10/2018 5:30:00 PM 10/10/2018 6:30:00 PM False