Past

The subgroup growth of finitely generated groups was seen last term, in a lecture of Dan Segal. This time, we see representation growth, and how it is similar to, and different from, subgroup growth.

Khovanov homology is an invariant of knots in $S^3$. In its original form,

it is a "homological version of the Jones polynomial"; Khovanov and

Rozansky have generalized it to other knot polynomials, including the

HOMFLY polynomial.

In the second talk, I'll discuss how Khovanov homology and its generalizations lead to a relation between the HOMFLY polynomial and the topology of flag varieties.

Khovanov homology is an invariant of knots in $S^3$. In its original form,

it is a "homological version of the Jones polynomial"; Khovanov and

Rozansky have generalized it to other knot polynomials, including the

HOMFLY polynomial.

The first talk will be an introduction to Khovanov homology and its generalizations.

We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface

into $R^n$, $n = 3, 4$, if there is one conformal immersion with Willmore energy smaller than a certain bound $W_{n,p}$ depending on codimension and genus $p$ of the Riemann surface. For tori in codimension $1$, we know $W_{3,1} = 8\pi$ . Joint work with Enrst Kuwert.

Seminar also with N. El Karoui and Y. Jiao

Dynamic modelling of default time for one single credit has been largely studied in the literature. For the pricing and hedging purpose, it is important to describe the price dynamics of credit derivative products. To this end, one needs to characterize martingales in the various filtrations and calculate conditional expectations by taking into account of default information, often modelized by a filtration $\bf{ D}$ generated by the jump process related to the default time $\tau$.

A general principle is to work with some reference filtration $\bf F$ which is often generated by some given processes. The calculations are then achieved by a formal passage between the enlarged filtration and the reference one on the set $\{\tau>t\}$ and the models are developed on the filtration $\bf F$.

In this paper, we are interested in what happens after a default occurs, i.e., on the set $\{\tau\leq t\}$. The motivation is to study the impact of a default event on the market, which will be important in a multi-credits setting. To this end, we adopt a new approach which is based on the knowledge of conditional survival probabilities. Inspired by the enlargement of filtration theory, we assume that the conditional law of $\tau$ admits a density.

We also present how our computations can be used in a multi-default setting.

In 1993 in his paper "A new strongly minimal set" Hrushovski produced a family of counter examples to a conjecture by Zilber. Each one of these counter examples carry a pregeometry. We answer a question by Hrushovski about comparing these pregeometries and their localization to finite sets. We first analyse the pregeometries arising from different variations of the construction before the collapse. Then we compare the pregeometries of the family of new strongly minimal structures obtained after the collapse.

Friction-driven vibration occurs in a number of contexts, from the violin string to brake squeal and machine tool vibration. A review of some key phenomena and approaches will be given, then the talk will focus on a particular aspect, the "twitchiness" of squeal and its relatives. It is notoriously difficult to get repeatable measurements of brake squeal, and this has been regarded as a problem for model testing and validation. But this twitchiness is better regarded as an essential feature of the phenomenon, to be addressed by any model with pretensions to predictive power. Recent work examining sensitivity of friction-excited vibration in a system with a single-point frictional contact will be described. This involves theoretical prediction of nominal instabilities and their sensitivity to parameter uncertainty, compared with the results of a large-scale experimental test in which several thousand squeal initiations were caught and analysed in a laboratory system. Mention will also be made of a new test rig, which attempts to fill a gap in knowledge of frictional material properties by measuring a parameter which occurs naturally in any linearised stability analysis, but which has never previously been measured.

The aim of this talk is to render the power of (short-step) interior-point methods for linear programming (and by extension, convex programming) intuitively understandable to those who have a basic training in numerical methods for dynamical systems solving. The connection between the two areas is made by interpreting line-search methods in a forward Euler framework, and by analysing the algorithmic complexity in terms of the stiffness of the vector field of search directions. Our analysis cannot replicate the best complexity bounds, but due to its weak assumptions it also applies to inexactly computed search directions and has explanatory power for a wide class of algorithms.

Co-Author: Coralia Cartis, Edinburgh University School of Mathematics.