## Mathematical Physics Seminars

### Programme

All seminars are held in Room M/2.06 on Thursdays at 3:10pm unless otherwise stated. All are welcome.

Programme Organiser and Contact: Professor David E Evans

#### 8 October 2014 at 14:00

**Mathematical Colloquium**

**Speaker:** Florin Boca (Urbana).

**Title:** Irregularities in the distribution of Euclidean and hyperbolic lattice angles.

**Abstract:** Spacing statistics measure the randomness of uniformly distributed sequences, or more generally increasing sequences of finite sets of real numbers. A familiar example of a uniformly distributed sequence of sets isgiven by the directions of vectors joining a fixed point in the Euclidean plane, with all (or only visible) points of integer coordinates inside balls of fixed center and increasing radius. However, these directions are not randomly distributed, and even the study of their most popular spacing statistics, limiting gap distribution and pair correlation function, turn out to pose challenges.

This talk will discuss recent progress in the study of the spacing statistics for this type of geometric configuration, comparing the Euclidean and the hyperbolic situations.

#### 9 October 2014

**Speaker:** Sameer Murthy (King's College, London).

**Title: **K3 surfaces, Mathieu moonshine and string theory.

**Abstract:** I shall discuss a conjecture of Eguchi, Ooguri and Tachikawa from 2010 that relates the elliptic genus of K3 surfaces and representations of

M24, the largest Mathieu group, and its extensions called Umbral moonshine. The generating functions of these representations are mock theta functions. I shall then present an ongoing attempt to understand these moonshine phenomena as arising in string theory that suggests a construction of a non-trivial infinite-dimensional M24-module. This is based on joint work with Jeff Harvey.

#### 16 October 2014

**Speaker:** Karen Vogtmann (Warwick).

**Title: **Hairy graphs and automorphisms of free groups.

**Abstract:** The group Out(F_n) of outer automorphisms of a free group is a complicated group with connections to many different areas of mathematics. A profitable way of studying Out(F_n) is to study its action on a space of graphs known as Outer space. In this talk I will describe Outer

space and then show how breaking graphs into "hairy" pieces can help to find new algebraic invariants for Out(F_n). Some of these invariants are related to classical modular forms for SL(n,Z).

#### 23 October 2014

**Speaker:** Jan Spakula (Southampton).

**Title: **Operator theory and coarse geometry.

**Abstract:** (Some) operator theorists study Fredholmness of certain operators on l^2(Z^n) using the so-called operator spectrum. John Roe, in 2004, explained that the operators of interest are really just elements of the Translation C*-algebra (also called the uniform Roe algebra) of Z^n, the C*-algebra encoding the large scale (or coarse) structure of Z^n.

In this talk, I will explain how to generalise the limit operator theory framework not only to other discrete groups, but to general discrete metric spaces. Furthermore, I will show how to further exploit the inherent connections to coarse geometry to generalise a recent result of Lindner and Siedel, which significantly simplifies the Fredholmness criterion (they refer to the problem they solve as "The core issue on Limit Operators (on Zn)").

#### 30 October 2014

**Speaker:** Ian Leary (Southampton).

**Title: **Right-angled Coxeter groups as a source of examples.

**Abstract:** Coxeter groups are groups generated by reflections; right-angled Coxeter groups are the simplest ones in which any two reflection planes are either parallel or perpendicular. I shall explain some of the ways in which these groups give rise to interesting examples in a range of areas, following the seminal work of Mike Davis.

#### 13 November 2014

**Speaker:** John Hunton (Durham).

**Title: **Attractive Tilings.

**Abstract:** This talk outlines a close connection between the moduli spaces of aperiodic tilings and attractors of certain types of dynamical systems. It should be reasonably self-contained, and I will introduce the necessary elements of these topics, together with a bit of homological algebra and geometric group theory, on the way.

#### 20 November 2014

**Speaker:** Shahn Majid (Queen Mary, London).

**Title: **Semiquantisation functor and Poisson-Riemannian geometry.

**Abstract:** TBC.

#### 27 November 2014

**Speaker:** Andre Henriques (Oxford).

**Title: **Bott periodicity and beyond.

**Abstract:** I will review Bott's classical periodicity result about topological K-theory (with period 2 in the case of complex K-theory, and period 8 in the case of real K-theory), and provide an easy (sketch of) proof, based on the algebraic periodicity of Clifford algebras. I will then introduce the `higher real K-theory' of Hopkins and Miller, also known as TMF. I'll discuss its periodicity (with period 576), and present a conjecture about a corresponding algebraic periodicity of `higher Clifford algebras'.

#### 5 February 2015

**Speaker:** Simon Willerton (Sheffield)

**Title: ** Categorifying the magnitude of graphs.

**Abstract:** Magnitude is a measure of the size of a metric space introduced by Tom Leinster. Whilst its origins lie in category theory, it has a very concrete definition and turns out to have connections with various aspects of mathematics, such as biodiversity measurement, integral geometry, potential theory and Minkowski dimension. A graph gives rise to a metric space by taking the shortest-path metric, thus a graph can be assigned a magnitude, and this, it transpires, can be considered as a formal power series with integer coefficients. Just as Khovanov homology has the Jones polynomial as its Euler characteristic, so it turns out that there is a homology theory of graphs that has graph magnitude as its Euler characteristic. I will explain the background and some properties of this magnitude homology of graphs. This is joint work with Richard Hepworth.

#### 26 February 2015

**Speaker:** Raymond Vozzo (Adelaide).

**Title: **TBC.

**Abstract:** TBC.

#### 12 March 2015

**Speaker:** Tom Leinster (Edinburgh)

**Title: ** TBC.

**Abstract:** TBC.