# Analysis Seminar 2017

The analysis seminar is held on Tuesdays at 2:00-2:50 in 210 Deady Hall unless otherwise noted.

### Spring Quarter, 2018

- April 10,
**Xin Ma** (Texas A & M)

Dynamics and classification of crossed product C*-algebras
**Abstract**: In this talk I will talk about some dynamical properties and their relation to classification program of crossed products. These properties include dimensions, almost finiteness, comparison, and the small boundary property. In addition, I will talk about some recent classification results for crossed products based on dynamical properties mentioned above.

- April 17,
**John Jasper** (South Dakota State U)

ETFs: from harmonic analysis to *-algebras and combinatorial designs
**Abstract**: Several applications in signal processing require lines through the origin of a finite-dimensional Hilbert space with the property that the smallest interior angle is as large as possible. Packings that achieve equality in the Welch bound are known as equiangular tight frames (ETFs). The central problem in the study of ETFs is the question of existence, that is, given integers n and d, does there exist an ETF with n lines in d-dimensional space? To tackle this problem the main tool has been to develop new constructions of ETFs. One of the oldest general constructions uses harmonic analysis on finite abelian groups, together with special subsets of the group called difference sets. In this talk we will discuss two recently discovered constructions and how they were directly inspired by the difference set construction. In the first we use a *-algebra in the place of the group algebra to generalize beyond abelian groups. In the second we find the underlying combinatorics behind several known types of difference sets.

- April 24,
**Huaxin Lin** (UO)

Closure of convex hull of normal elements in C*-algebras of tracial rank zero
- May 1,
**Stefan Steinerberger** (Yale)

Oscillations, Roots and Eigenfunctions
**Abstract**: It is not very difficult to see that sin(nx) has n-1 roots in (0, pi). This was generalized quite subtantially by Sturm and is now called the Sturm Oscillation Theorem. However, Sturm actually proved a much stronger result: in particular, a*sin(mx) + b*sin(nx) has always at least min{m,n} – 1 roots and at most max{m,n} -1 roots. I will discuss some elementary approaches to these questions, discuss how these types of results appear naturally in PDEs, discuss a new inverse result as well as some statements and problems in higher dimensions.

- May 8,
**Marcin Bownik** (UO)

On a conjecture of Akemann and Weaver
- May 22,
**Daniel Wang** (South Houston State U)

Differential Characterization of Anisotropic Hardy Spaces

### Winter Quarter, 2018

- February 6,
**Qingyun Wang** (UO)

Classification of Rokhlin actions on classifiable C*-algebras
- February 13,
**Yuan Xu** (UO)

Minimal cubature rules and common zeros of polynomials
- February 20,
**Chris Phillips** (UO)

Isometric inner automorphisms of operator algebras
**Abstract**: Consider an isometric automorphism of a unital Banach algebra. Suppose it is inner. Can the implementing invertible element be taken to be an isometry in the algebra?

For C*-algebras, the answer is yes, by polar decomposition. For nonselfadjoint Hilbert space operator algebras, in general the answer is no. We give several examples of operator algebras for which the answer is yes (in one case, modulo verification of some details). The results suggest that a positive answer is (weak) evidence for the algebra in question to be “C*-like”.

This is joint work with Andrey Blinov.

- March 6,
**Jianchao Wu** (Penn State)

Demystifying Rokhlin dimension
**Abstract**: The theory of Rokhlin dimension was introduced by Hirshberg, Winter and Zacharias as a tool to study the regularity properties of C*-algebras in relation with group actions. It was inspired by the classical Rokhlin lemma in ergodic theory. Since then, it has been greatly developed as well as simplified, and connections to other areas have been discovered. In this talk, I will present some newer perspectives to help us understand this concept. In particular, I will explain its relation to the Schwarz genus for principal bundles in the context of generalized Borsuk-Ulam theorems. Time permitting, I will also indicate how one can extend the theory beyond residually finite groups. This includes recent and ongoing joint projects with Gardella, Hajac, Hirshberg, Hamblin, Tobolski and Zacharias.

- March 13,
**David Cruz-Uribe** (Alabama)

Poincare inequalities and Neumann problems for the p-Laplacian
**Abstract**: I will discuss my recent work with Scott Rodney on the following equivalence: the existence of solutions to a degenerate p-Laplacian equation and the existence of a weighted (p,p) Poincare inequality in a degenerate Sobolev space. Our results build upon the definition of degenerate Sobolev spaces given by Sawyer and Wheeden, where the degeneracy is defined in terms of a degenerate elliptic matrix and I will spend some time talking about this general context. If time permits, I will also discuss the application of similar ideas to show that a local, weak Sobolev inequality implies the existence of a global Sobolev inequality.

- March 15,
**Hannah Feng** (UO)

Chebyshev-Type Cubature Formulas for Doubling Weights
**Abstract**: In this talk, I will present the recent joint work with Feng Dai on the strict Chebyshev-type cubature formula (CF) (i.e., equal weighted CF) for doubling weights on the unit sphere equipped with the usual surface Lebesgue measure and geodesic distance. Our main interest is on the minimal number of nodes required in a strict Chebyshev-type CF. Precisely, given a normalized doubling weight on unit sphere, we will establish the sharp asymptotical estimates of the minimal number of distinct nodes which admits a strict Chebyshev-type CF. If, in addition, the weight function is essentially bounded, the nodes involved can be configured well-separately in some sense. The proofs of these results rely on constructing new convex partitions of the unit sphere that are regular with respect to the weight. The weighted results on the unit sphere also allow us to establish similar results on strict Chebyshev-type CFs on the unit ball and the standard simplex.

- March 20,
**Xuanlong Fu** (East China Normal University)

**Special Time:** 1pm

Tracial nuclear dimension and regularity properties of C*-algebras

### Fall Quarter, 2017

- October 3,
**Qingyun Wang** (UO)

Stability of the rotation relations of three unitaries
- October 10,
**Marcin Bownik** (UO)

Lyapunov’s theorem for continuous frames
- October 17,
**Saeid Jamali**

Tracially Z-absorbing C*-algebras
- October 24,
**Ning Zhang** (Berkeley)

**Special Time/Location:** 1pm in 303 Deady

Analysis : Generalizations of Grunbaum’s Inequality
- October 24,
**Itay Londner** (Tel Aviv University)

Interpolation sets and arithmetic progressions
**Abstract**: Given a set S of positive measure on the unit circle, a set of integers K is an interpolation set (IS) for S if for any data {c(k)} in l^2(K) there exists a function f in L^2(S) such that its Fourier coefficients satisfy f^(k)=c(k) for all k in K.

In the talk I will discuss the relationship between the concept of IS and the existence of arbitrarily long arithmetic progressions with specified lengths and step sizes in K.

Multidimensional analogue and recent developments will also be considered.

Based on joint work with A. Olevskii.

- October 31,
**Mark Rudelson** (University of Michigan and MSRI)

Density of a projection of a random vector
- November 7,
**Hyun Ho Lee** (University of Ulsan, South Korea)

An extension of Phillip’s theorem to inclusions of unital C*-algebras
**Abstract**: N.C. Phillips extended M. Izumi’s works on the Rokhlin property of a finite group action and established tracial analogs of many results. Among them, there is a beautiful result which says that a finite abelian group action has the tracial Rokhlin property if and only if its dual group action is tracially approximately representable. We can formulate such notions in the setting of inclusions of unital C*-algebras and provide a duality between them. We also provide an evidence why this notion is an extension of Phillip’s definition.

- November 14,
**Huaxin Lin** (UO)

Classification of stably projectionless simple C*-algebras
- November 21,
**Chris Phillips** (UO)

L^p operator algebras with contractive approximate identities
- November 28,
**Rafał Latała** (University of Warsaw & MSRI)

Dimension-free bounds for nonhomogenous random matrices
**Abstract**: What does the spectrum of a random matrix look like when the entries can have an arbitrary variance pattern? Such questions, which are of interest in several areas of pure and applied mathematics, are largely orthogonal to problems of classical random matrix theory. For example, one might ask the following basic question: when does an infinite matrix with independent Gaussian entries define a bounded operator on l_2? In this talk, I will describe recent work with Ramon Van Handel and Pierre Youssef in which we completely answer this question, settling an old conjecture of Latała. More generally, we provide optimal estimates on the Schatten norms of random matrices with independent Gaussian entries. These results not only answer some basic questions in this area, but also provide significant insight on what such matrices look like and how they behave.

**Previous years:**

2016-17

2015-16

2014-15

2013-14