# Weekly bulletin

## Week starting Monday 27 November 2017

### Tuesday 28 November - Partial Differential Equations and Analysis Seminar

Start time: 1.30pm
Location:

John Dedman Building 27, Room G35

Presenter(s): Associate Professor Florence Lancien (Universite de Franche-Comte, Besancon, France)
Abstract:

Given $T$ a power bounded operator on a Banach space $X$ one can consider the discrete subordinated operator $S=\sum_{k=0}^\infty c_k T^k$ where $c_k\geq 0$ and $\sum_{k=0}^\infty c_k=1.$ N. Dungey gave conditions on $(c_k)$ for $S$ to be a Ritt operator i.e. $S$ powerbounded and $\sup_n n IIS^n-S^{n+1}II<\infty.$ With C. Le Merdy we concentrate on the following subordination situation : for $\nu$ a probability measure on a locally compact abelian group and $\pi: G\to B(X)$ a bounded representation we define the operator $S=\int_G \pi(t) d\vu(t).$ We show that under certain conditions on $\vu$ as well as on the geometry of the space $X$ the subordinated operator $S$ is a Ritt operator or admits a bounded $H^\infty$ functional calculus. This is joint work with Christian Le Merdy.

### Tuesday 28 November - Algebra and Topology Seminar

Start time: 3.30pm
Location:

John Dedman Building 27, Room G35

Presenter(s): Arun Ram (U. of Melbourne)
Abstract:

Recent work of Kato and Kato-Naito-Sagaki has provided an improved understanding of the combinatorics of integrable level 0 representations of the affine Lie algebra.  In particular, there is a connection to the Schubert calculus of the semi-infinite flag variety from which Kato-Naito-Sagaki prove an analogue of the  Pieri-Chevalley formula for the semi-infinite flag variety (line bundle multiplied with a Schubert class). The connection to crystals in this new formula (analogous to that in the Pieri-Chevalley formula of Pittie-Ram) provides a geometric interpretation of (a part of) the path model which was used by Ram-Yip to give a formula for Macdonald polynomials.

### Thursday 30 November - MSI Colloquium 2017

Start time: 4pm
Location:

John Dedman Building 27, Room G35

Presenter(s): Professor Gilles Lancien, Laboratoire de Mathématiques de Besançon
Abstract:

In the last fifteen years, the question of embedding metric spaces into “nice” Banach spaces has gathered researchers coming from very diverse origins, in particular from theoretical computer science, geometry of Banach spaces and geometric group theory. A very natural motivation is to aim at a better understanding of complicated metric spaces, such as graphs with a high number of vertices and edges, by embedding them into well understood Banach spaces, like Hilbert spaces.

In the so-called non linear geometry of Banach spaces, which is the subject of this talk, the approach is somewhat reversed. The starting point is to try to exhibit the linear properties of Banach spaces that are stable under some particular non linear maps. These non linear maps can be of very different nature: Lipschitz isomorphisms or embeddings, uniform homeomorphisms, uniform or coarse embeddings. Then, the next goal is to characterize these linear properties in purely metric terms. Usually, these characterizations are given by the (non)embeddability of special metric spaces, which are very often fundamental metric trees or graphs.

Finally, this can motivate the study of these properties, whose linear character can now be forgotten, in the larger setting of metric spaces. We will try to illustrate these ideas with a few results and open questions. We will explain what is known as the “Ribe Program”, which aims at characterizing local properties of Banach spaces (properties of their finite dimensional subspaces) in purely metric terms. This will be illustrated with famous results by J. Bourgain or A. Naor. If time allows, we will describe a recent line of research: looking for metric characterizations of “asymptotic properties” of Banach spaces. This subject could rightfully be named the “Kalton program”.

Updated:  22 November 2017/Responsible Officer:  Director/Page Contact:  School Manager