# Ohio University Analysis Seminar

This is the website for the Analysis Seminar of the Mathematics Department of Ohio University. Find below a list of past and upcoming talks. For further details, please contact Marcel Bischoff and/or Adam Fuller.

## Fall 2018 upcoming talk

### Thursday, November 29st, Marcel Bischoff (Ohio University), 2pm-3pm, Morton Hall 322

Title: Introduction to Subfactors

Abstract: A factor is a von Neumann algebra $M$ with trivial center and a subfactor is an unital inclusion of factors $N\subset M$. A theorem of Popa's says that a finite index finite depth subfactor of the hyperfine type II${}_1$ factor are completely classified by a certain commuting square of finite-dimensional C${}j^*$-algebras. I will introduce some of the concepts and sketch some ideas of the proof.

## Fall 2018

### Thursday, September 6th, Shezad Ahmed (Ohio University), 11am-12pm, Morton Hall 223

Title: Automorphisms of the Calkin Algebra

Abstract: Given a Hilbert space $H$, we let $\mathrm B(H)$ denote the space of all bounded operators on $H$ and $\mathrm K(H)$ denote the ideal of compact operators. The Calkin algebra is the quotient $\mathrm B(H)/\mathrm K(H)$. In this talk, we tackle the question of whether or not there are automorphisms of $\mathrm C(H)$ which are not simply "the obvious ones", i.e. those that are induced by unitary operators on $H$. Our focus will be on the case when $H$ is separable and infinite dimensional.

### Thursday, September 27th, Adam Fuller (Ohio University), 2pm-3pm, Morton Hall 322

Title: Dilations of Operators

Abstract: In 1953 Sz.-Nagy proved that given a contraction $T$ on a Hilbert space $H$, there is a unitary $U$ on a larger space $K$, such that $T^n = P_{H} U^n |_K$ for all $n \geq 0$. That is, every contraction is a compression of a unitary operator. This result and generalizations leads to, amongst other things, a simplified proof of von Neumann’s inequality, a contractive functional calculus, and has applications to interpolation theory. In this talk we will look at Sz.-Nagy’s original result, generalizations and impediments to generalizations.

### Thursday, October 4th, Vladimir Uspenskiy (Ohio University), 2pm-3pm, Morton Hall 322

Title: Topological groups and their representations on Banach spaces

Abstract: Does every topological group admit a faithful unitary representation? The answer is yes for locally compact groups and no in general. However, every topological group admits a faithful representation by linear isometries of a Banach space. Thus, surprisingly, certain Banach spaces have a "richer" isometry group than Hilbert spaces. We'll consider some of such Banach spaces, among them the space generated by the Urysohn universal metric space.

### Thursday, November 8th, Shezad Ahmed (Ohio University), 2pm-3pm, Morton Hall 322

Title: Souslin’s problem, trees, and their generalizations

Abstract: It is well know that one can characterize the real line (up to order isomorphism) as the unique linear order which:
1) Has no first or last element,
2) is connected in the order topology, and
3) is separable in the order topology.
In 1920, Souslin asked whether or not condition 3) can be weakened to asking that the countable chain condition is satisfied in the order topology. In this talk, we survey the history of this problem and trace a path to current research regarding the existence of certain types of trees.

### Thursday, November 29st, Marcel Bischoff (Ohio University), 2pm-3pm, Morton Hall 322

Title: Introduction to Subfactors

Abstract: A factor is a von Neumann algebra $M$ with trivial center and a subfactor is an unital inclusion of factors $N\subset M$. A theorem of Popa's says that a finite index finite depth subfactor of the hyperfine type II${}_1$ factor are completely classified by a certain commuting square of finite-dimensional C${}j^*$-algebras. I will introduce some of the concepts and sketch some ideas of the proof.

### Thursday, December 6th, Corey Jones (The Ohio State University), 2pm-3pm, Morton Hall 322

Title: Braided tensor categories associated to von Neumann algebras

Abstract: Given a separable von Neumann algebra, we will describe a braided W${}^\ast$-tensor category of bimodules, generalizing and categorifying Connes' chi invariant. The braiding on this category is strongly analogous to the braiding on the DHR category from algebraic quantum field theory. We will then discuss some examples coming from von Neumann algebras associated to finite depth subfactors. Based on joint work with Vaughan Jones.

## Former Seminars

Fall 2017 and Spring 2018

Last update: 2018-11-28 20:18:11.