# Colloquium, Academic Year 17-18

Thursdays 4:10 pm in 5211 Stevenson Center, unless otherwise noted

Tea at 3:33 pm in 1425 Stevenson Center

Colloquium Chair (2017-2018): Ioana Suvaina

## Applied Random Matrix Theory

Joel A. Tropp, California Institute of Technology

Location: Stevenson 5211

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Akram Aldroubi)

## Calibrations, Potential Theories and the Dirichlet Problem

Blaine H. Lawson, Stony Brook University

Location: Stevenson 5211

I will begin by explaining the theory of calibrations and present some important examples. These examples are cases where one has large families of minimal (in fact, homologically volume minimizing) varieties. The basic case is K\”ahler geometry, where the varieties are the complex analytic ones. Now for a general calibrated manifold there is no analogue of the holomorphic functions. However, it turns out that there is always a complete analogue of the plurisubharmonic functions (those which are subharmonic on complex curves), and many of the results from several complex variables carry over. These functions are, in a sense, dual to the subvarieties. Furthermore, the inequality defining general plurisubharmonic functions also gives an equality, that is, a nonlinear partial differential equation. These associated equations are generalizations of the complex Monge-Amp\`ere equation in the K\”ahler case. We now have a good understanding of how to solve the Dirichlet Problem for solutions to these equations. An example some people may like concerns symplectic manifolds equipped with a Gromov metric. The associated plurisubharmonic functions are those whose Hessian has non-negative trace on every Lagrangian plane. The corresponding differential operator is a polynomial in tr$(D^2 u)$ and the skew-Hermitian part of $D^2 u$. Tea at 3:30 pm in Stevenson 1425. (Contact Person: Ioana Suvaina)

## Random Graphs and Applications to Geometric Group Theory

Jason Behrstock, CUNY Graduate Center

Location: Stevenson 5211

Erdos and Renyi introduced a model for studying random graphs of a given “density” and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will survey connections between this field and graph theory and describe new threshold theorems for random graphs and applications to the geometry of Coxeter groups. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Denis Osin)

## Perturbations of Operators and Commutants Mod Normed Ideals

Dan-Virgil Voiculescu, University of California, Berkeley

Location: Stevenson 5211

Normed ideals of compact operators are the infinitesimals of Alain Connes’ noncommutative geometry. In the study of operators modulo perturbations from these ideals a numerical invariant plays often a key role. Recently more structure is appearing in these questions from the commutant modulo the normed ideal. Connections with dynamical entropy, K-theory of operator algebras, supramenable groups and Banach space duality aspects will also be discussed. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Dietmar Bisch)

## Rigidity for Group von Neumann Algebras

Adrian Ioana, University of California at San Diego

Location: Stevenson 5211

Any countable group $\Gamma$ gives rise to a von Neumann algebra $L(\Gamma)$. The classification of these group von Neumann algebras is a central theme in operator algebras. I will survey recent rigidity results which provide instances when various algebraic properties of groups, such as the existence or absence of a direct product decomposition, are remembered by their von Neumann algebras. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Jesse Peterson)

## Algebraic Methods in Numerical Computations

Lek-Heng Lim, University of Chicago

Location: Stevenson 5211

We show that in many instances, at the heart of a problem in numerical computation sits a special 3-tensor, the structure tensor of the problem that uniquely determines its underlying algebraic structure. For example, the Grothendieck constant, which plays an important role in unique games conjecture and SDP relaxations of NP-hard problems, arises as the spectral norm of such a structure tensor. In matrix computations, a decomposition of the structure tensor into rank-1 terms gives an explicit algorithm for solving the problem, its tensor rank gives the speed of the fastest possible algorithm, and its nuclear norm gives the numerical stability of the stablest algorithm. As an explicit example, we determine the fastest algorithms for the basic operation underlying Krylov subspace methods — the structured matrix-vector products for sparse, banded, triangular, symmetric, circulant, Toeplitz, Hankel, Toeplitz-plus-Hankel, BTTB matrices — by analyzing their structure tensors. Our method is a generalization of the Cohn–Umans method, allowing for arbitrary bilinear operations in place of matrix-matrix product, and arbitrary algebras (e.g., coordinate rings of schemes, cohomology rings of manifolds, PI algebras) in place of group algebras. The second part is joint work with Ke Ye. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Mike Neamtu)

## Vaught’s Conjecture in Computability Theory

Antonio Montalbán, University of California, Berkeley

Location: Stevenson 5211

We’ll describe Vaught’s conjecture, which is one of the most well-known and longest-standing open questions in mathematical logic. The conjecture essentially says that the continuum hypothesis holds when restricted to counting the number of models of a theory. We’ll mention the author’s result that this conjecture is equivalent to a computability-theoretic statement. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Costas Tsinakis)

## Connes’ Character Rigidity Conjecture for Lattices in Higher Rank Groups

Jesse Peterson, Vanderbilt University

Location: Stevenson 5211

A character on a group is a class function of positive type. For finite groups, the classification of characters is closely related to the representation theory of the group and plays a key role in the classification of finite simple groups. Based on the rigidity results of Mostow, Margulis, and Zimmer, it was conjectured by Connes that for lattices in higher rank simple Lie groups, the space of characters should be completely determined by their finite dimensional representations. In this talk, I will discuss the solution to this conjecture, and I will discuss its relationship to ergodic theory, invariant random subgroups, and von Neumann algebras. Tea at 3:33 pm in Stevenson 1425.

## Isogeometric Analysis

Thomas J.R. Hughes, University of Texas at Austin

Location: Stevenson 5211

The vision of Isogeometric Analysis was first presented in a paper published October 1, 2005 [1]. Since then it has become a focus of research within both the fields of Finite Element Analysis (FEA) and Computer Aided Design (CAD) and is rapidly becoming a mainstream analysis methodology and a new paradigm for geometric design [2]. The key concept utilized in the technical approach is the development of a new foundation for FEA, based on rich geometric descriptions originating in CAD, resulting in a single geometric model that serves as a basis for both design and analysis. In this overview, I will describe some areas in which progress has been made in developing improved Computational Mathematics methodologies to efficiently solve problems that have been at the very least difficult, if not impossible, within traditional FEA. I will also describe current areas of intense activity and areas where problems remain open, representing both challenges and opportunities for future research (see, e.g., [3,4]).

REFERENCES:[1] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs, Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement, Computer Methods in Applied Mechanics and Engineering, 194, (2005) 4135-4195.[2] J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, Chichester, U.K., 2009.[3] Special Issue on Isogeometric Analysis, (eds. T.J.R. Hughes, J.T. Oden and M. Papadrakakis), Computer Methods in Applied Mechanics and Engineering, 284, (1 February 2015), 1-1182.[4] Special Issue on Isogeometric Analysis: Progress and Challenges, (eds. T.J.R. Hughes, J.T. Oden and M. Papadrakakis), Computer Methods in Applied Mechanics and Engineering, 316, (1 April 2017), 1-1270.Tea at 3:33 pm in SC 1425. (Contact Person: Mike Neamtu)

## Word-Hyperbolic Surface Bundles

Christhoper Leininger, University of Urbana, Champaign

Location: Stevenson 5211

In the late 70’s and early 80’s, Thurston’s approach to studying 3-manifolds revolutionized the theory, showing that hyperbolic geometry provided a framework to more systematically study these manifolds. Specifically, he conjectured (and proved in many cases) that 3-manifolds could be canonically decomposed into geometric pieces, with hyperbolic geometry being the richest and most interesting geometric structure arising. Based on earlier work by Dehn, the key features of hyperbolic geometry were abstracted by Gromov to study more general spaces (most famously, finitely generated groups), and he has asked whether the analogue of the “hyperbolic parts” of Thurston’s geometrization hold in a more general setting. In this talk, I will describe a particular instance of Gromov’s “hyperbolization question”, motivated by Thurston’s approach, and explain some partial results in this direction. This is joint work with Bestvina, Bromberg, and Kent. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Mark Sapir)

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Kathryn Hess Bellwald, EPFL (École Polytechnique Fédérale de Lausanne)

Location: Stevenson 5211

Tea at 3:30 pm in Stevenson 1425. (Contact Person: Anna Marie Bohmann)

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Ruth Charney, Brandeis University

Location: Stevenson 5211

Tea at 3:30 pm in Stevenson 1425. (Contact Person: Spencer Dowdall)

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Bernhard G. Bodmann, University of Houston

Location: Stevenson 5211

Tea at 3:30 pm in Stevenson 1425. (Contact Person: Akram Aldroubi)

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Gigliola Staffilani, Massachusetts Institute of Technology

Location: Stevenson 5211

Tea at 3:30 pm in Stevenson 1425. (Contact Person: Giusy Mazzone)

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Lawrence Craig Evans, University of California, Berkeley

Location: Stevenson 5211

Tea at 3:30 pm in Stevenson 1425. (Contact Person: Vaughan Jones)

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Jeff Viaclovsky, University of California at Irvine

Location: Stevenson 5211

Tea at 3:30 pm in Stevenson 1425. (Contact Person: Ioana Suvaina)

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