# Analysis Seminar

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

### Fall Quarter, 2019

- October 1,
**Michal Wojciechowski** (Institute of Mathematics of Polish Academy of Sciences)

On the trace of Sobolev spaces on the Von Koch’s snowflake
**Abstract**: We show that the boundary trace operator on Sobolev space of functions with summable gradient on von Koch’s snowflake has right inverse. This contrasts with the case of domains with nice boundaries in which, according to Petree’s theorem, a right inverse does not exists. Our proof is based on the characterization of the trace space. As a by-product we give a very simple proof of Petree’s theorem. Joint work with Krystian Kazaniecki.

As a byproduct of this work we made a woolen carpet which was presented on the Bridges – Art and Mathematics exhibition this year in Linz!

- October 8,
**David Blecher** (University of Houston)

Real positivity and maps on operator algebras
**Abstract**: We begin by reviewing the theory of real positivity initiated by the speaker and Charles Read, and present many new general results about real positive maps. The key point is that real positivity is often the right replacement in a general algebra A for positivity in C*-algebras. We then apply this to contractive projections and isometries of operator algebras. For example we describe recent joint work with Matt Neal in which we generalize and find variants of certain classical results on positive projections on C*-algebras and JB algebras due to Choi, Effros, Stormer, Friedman and Russo, and others. In previous work we had done the `completely contractive’ case. We also give a new Banach-Stone type theorem for isometries between our algebras, and an application of this is given to the characterization of a class of projections. In the last part of the talk, joint with Louis Labuschagne, we focus on a special case of the projections considered above that we consider to be a good noncommutative generalization of the ‘characters’ (i.e. homomorphisms into the scalars) on an algebra. We consider and solve several problems that arise when generalizing classical function algebra results involving characters.

- October 15,
**Ilan Hirschberg** (Ben Gurion University of the Negev)

Simple nuclear C*-algebras not isomorphic to their opposites
**Abstract**: For any C*-algebra A, one can associate the opposite C*-algebra A^{op}, which has the same Banach space structure only with multiplication reversed. It is a long-standing question whether there exist C*-algebras which are nuclear, simple and separable but are not isomorphic to their opposites. Examples in which one drops nuclearity or simplicity are known for some time. In a joint paper with Ilijas Farah, we constructed a non-separable example (or rather, showed that the existence of such an example is relatively consistent with ZFC). I will provide some background, and will try to outline the idea of the proof. Reference: Ilijas Farah and Ilan Hirshberg, Simple nuclear C*-algebras not isomorphic to their opposites, Proc. Nat. Acad. Sci. USA 114 no. 24 (2017), 6244–6249.

- October 22,
**Polona Durcik** (Caltech)

Singular Brascamp-Lieb inequalities
**Abstract**: Brascamp-Lieb inequalities are estimates for certain multilinear forms on functions on Euclidean spaces. They generalize several classical inequalities, such as Hoelder’s inequality or Young’s convolution inequality. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions in the Brascamp-Lieb inequality is replaced by a singular integral kernel. Examples include multilinear singular integral forms such as paraproducts and the multilinear Hilbert transform. In this talk we will survey some results in the area. Time permitting we will discuss an application to quantitative norm convergence of bilinear ergodic averages.

### Spring Quarter, 2020

- April 21,
**Krystal Taylor** (Ohio State University)

### Summer, 2019

- September 10,
**Jinxia Li** (Guangzhou University, China)

Estimates of Pointwise Anisotropic Singular Integrals

**Previous years:**

2018-19

2017-18

2016-17

2015-16

2014-15

2013-14