Free probability theory, a species of non-commutative probability theory, is amazing for several reasons. Not only has it nice combinatorial features underlying it but also profound connections with other fields, in particular physics. Recently, we established a priori unexpected relations with some very prominent algebraic objects, in particular Hopf algebras. In this talk we will carefully introduce some of the basic features and give a glance at future directions.

http://www.city.ac.uk/engineering-maths/research/mathematics-centre/research-seminars/seminars-2011-2012

When an integrable quantum field theory in one space dimension has a Lie group symmetry, the Lie algebra is typically embedded in a larger algebra called a Yangian. When one adds a boundary which preserves integrability, this is extended to a (generalized) twisted Yangian. We explain the role of these algebras in physics, and in particular recent work by MacKay and Regelskis which uncovers their governing role in the scattering of worldsheet excitations off D-branes in the AdS/CFT correspondence.

We generalise a recent combinatorial description of the Verlinde or WZW fusion algebra of type A by defining cylindric Macdonald functions. The latter arise as weighted sums over non-intersecting paths on a square lattice with periodic boundary conditions. Expanding the cylindric Macdonald functions into Schur functions one obtains generalised Kostka-Foulkes polynomials. The latter contain ordinary Kostka-Foulkes polynomials, which appear in algebraic geometry, representation theory and combinatorics, as special case. We further motivate the cylindric Schur functions by showing that they are connected with a commutative Frobenius algebra which can be interpreted as a deformation of the Verlinde algebra: its structure constants are polynomials whose constant terms are the WZW fusion coefficients.

A Ricci soliton is a generalisation of an Einstein metric which evolves in a very simple way under the Ricci flow. We discuss ways of producing examples of Ricci solitons by looking for solutions with a high degree of symmetry

This talk will give a pedagogical account of the role played by integrability and instantons in N=2 SUSY gauge theories leading to recent understanding of what it means to quantize the integrable system.

Dimer models are typically studied in condesed matter physics and combinatorics. The correspondence between dimer models, toric Calabi-Yaus and quiver gauge theories on D-branes has had a profound impact in areas ranging from string phenomenology to mathematics. Today I will discuss a recently discovered correspondence between dimer models and integrable systems.

I will discuss scattering amplitudes in N=2,4,8 SYM in three-dimensions, concentrating on the N=8 case, with an emphasis on which properties of the N=4, D=4 SYM amplitudes survive under dimensional reduction. The on-shell supersymmetry algebra makes the SO(N) symmetry of the amplitudes manifest, while the Lagrangian displays only manifest SO(N-1) symmetry. I will also discuss the possibility of non-local Yangian-type symmetry, connections to BLG, and some perspectives on loop level results. Based on 1103.0786 / 1109.2792.

Abstract : Feynman Graph counting in Quantum Field Theory (QFT) can be formulated in terms of symmetric groups. This leads to expressions for graph counting and symmetry factors in terms of topological transition amplitudes for strings with a cylinder target, related to two dimensional topological field theory. The details of the interactions in the QFT are encoded in the boundary conditions which specify how the strings wind around circles. The QFTs discussed include scalar field theories and QED, where there is no large gauge group.

We shall talk generally about the state of the art in set theory and try to explain to what extent the independence results in set theory influence our understanding of mathematical foundations. The talk will start rather generally and will build up to describe some current research directions.

The importance of adequately modeling credit risk has once again been highlighted in the recent financial crisis. Defaults tend to cluster around times of economic stress due to poor macro-economic conditions, but also by directly triggering each other through contagion. Although credit default swaps have radically altered the dynamics of contagion for more than a decade, models quantifying their impact on systemic risk are still missing. Here, we examine contagion through credit default swaps in a stylized economic network of corporates and financial institutions. We analyse such a system using a stochastic setting, which allows us to exploit limit theorems to exactly solve the contagion dynamics for the entire system. Our analysis shows that CDS, when used to expand banks' loan books (arguing that CDS would offload the additional risks from banks' balance sheets), can actually lead to greater instability of the entire network in times of economic stress, by creating additional contagion channels. This can lead to considerably enhanced probabilities for the occurrence of very large losses and very high default rates in the system. Our approach adds a new dimension to research on credit contagion, and could feed into a rational underpinning of an improved regulatory framework for credit derivatives.

In algebraic geometry and string theory there has been a lot of recent work on so-called wall-crossing phenomena for Donaldson-Thomas invariants. In this talk we will study a baby example of wall-crossing, which already has some non-trivial consequences. I will not assume any previous knowledge of algebraic geometry, just some basic properties of the category of modules over a ring.