Past

I will describe how to build a noncommutative ring which dictates

the process of resolving certain two-dimensional quotient singularities.

Algebraically this corresponds to generalizing the preprojective algebra of

an extended Dynkin quiver to a larger class of geometrically useful

noncommutative rings. I will explain the representation theoretic properties

of these algebras, with motivation from the geometry.

For continuous maps $f: S^{2n-1} \to S^n$ one can define an integer-valued invariant, the so-called Hopf invariant. The problem of determining for which $n$ there are maps having Hopf invariant one can be related to many problems in topology and geometry, such as which spheres are parallelisable, which spheres are H-spaces (that is, have a product), and what are the division algebras over $\mathbb{R}$.

The best way to solve this problem is using complex K-theory and Adams operations. I will show how all the above problems are related, give an introduction to complex K-theory and it's operations, and show how to use it to solve this problem.

In these (three) lectures, I will discuss the following topics:

1. The theorems of Ax on the elementary theory of finite and pseudo-finite fields, including decidability and quantifier-elimination, variants due to Kiefe, and connection to Diophantine problems.

2. The theorems on Chatzidakis-van den Dries-Macintyre on definable sets over finite and pseudo-finite fields, including their estimate for the number of points of definable set over a finite field which generalizes the Lang-Weil estimates for the case of a variety.

3. Motivic and p-adic aspects.

This is the second of two talks, and probably will not be comprehensible unless you came to last week's talk.

A Kuranishi space is a topological space equipped with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on many moduli spaces in differential geometry, and in particular, in moduli spaces of stable $J$-holomorphic curves in symplectic geometry.

Let $Y$ be an orbifold, and $R$ a commutative ring. We define four topological invariants of $Y$: two kinds of Kuranishi bordism ring $KB_*(Y;R)$, and two kinds of Kuranishi homology ring $KH_*(Y;R)$. Roughly speaking, they are spanned over $R$ by isomorphism classes $[X,f]$ with various choices of relations, where $X$ is a compact oriented Kuranishi space, which is without boundary for bordism and with boundary and corners for homology, and $f:X\rightarrow Y$ is a strong submersion. These theories are powerful tools in symplectic geometry.

Today we discuss the definition of Kuranishi homology, and the proof that weak Kuranishi homology is isomorphic to the singular homology.

Abstract: The Maximum Induced Planar Subgraph problem asks

for the largest set of vertices in a given input graph G

that induces a planar subgraph of G. Equivalently, we may

ask for the smallest set of vertices in G whose removal

leaves behind a planar subgraph. This problem has been

linked by Edwards and Farr to the problem of _fragmentability_

of graphs, where we seek the smallest proportion of vertices

in a graph whose removal breaks the graph into small (bounded

size) pieces. This talk describes some algorithms

developed for this problem, together with theoretical and

experimental results on their performance. The material

presented is joint work either with Keith Edwards (Dundee)

or Kerri Morgan (Monash).

A brief overview of consonance by way of continued fractions and modular arithmetic.

This talk gives a survey on a series of work which I and co-authors have been doing for 10 years. I will start from the Feynman-Kac type formula for Dirichlet forms. Then a necessary and sufficient condition is given to characterize the killing transform of Markov processes. Lastly we shall discuss the regular subspaces of linear transform and answer some problems related to the Feynman-Kac formula

Counting paths, or walks, is an important ingredient in the classical representation theory of compact groups. Using Brownian paths gives a new flexible and intuitive approach, which allows to extend some of this theory to the non- cristallographic case. This is joint work with P. Biane and N. O'Connell

Abstract: For a large class of compactifications of interest in string phenomenology, the task of finding vacua of the four dimensional effective theories can be rewritten as a simple problem in algebraic geometry. Using recent developments in computer algebra, the task can then be rapidly dealt with in a completely algorithmic fashion. I shall review the main points of hep-th/0606122 and hep-th/0703249 in which this approach to finding vacua was set out, before moving on to a description of the Mathematica package STRINGVACUA (as described in arXiv:0801.1508 [hep-th]). This package uses the power of the computer algebra system Singular and provides a user-friendly implementation of our methods, intended for use by physicists, within the comfortable working environment of Mathematica.

For the integers $a$ and $b$ let $P(a^b)$ be all partitions of the

set $N= {1,..., ab}$ into parts of size $a.$ Further, let

$\mathbb{C}P (a^b)$ be the corresponding permutation module for the

symmetric group acting on $N.$ A conjecture of Foulkes says

that $\mathbb{C}P (a^b)$ is isomorphic to a submodule of $\mathbb{C}P

(b^a)$ for all $a$ not larger than $b.$ The conjecture goes back to

the 1950's but has remained open. Nevertheless, for some values of

$b$ there has been progress. I will discuss some proofs and further

conjectures. There is a close correspondence between the

representations of the symmetric groups and those of the general

linear groups, via Schur-Weyl duality. Foulkes' conjecture therefore

has implications for $GL$-representations. There are interesting

connections to classical invariant theory which I hope to mention.

The Nielsen realisation problem asks when a collection of diffeomorphisms, which form a group up to isotopy, is isotopic to a collection of diffeomorphisms which form a group on the nose. For surfaces this problem is well-studied, I'll talk about this problem in the context of K3 surfaces.

In these (three) lectures, I will discuss the following topics:

1. The theorems of Ax on the elementary theory of finite and pseudo-finite fields, including decidability and quantifier-elimination, variants due to Kiefe, and connection to Diophantine problems.

2. The theorems on Chatzidakis-van den Dries-Macintyre on definable sets over finite and pseudo-finite fields, including their estimate for the number of points of definable set over a finite field which generalizes the Lang-Weil estimates for the case of a variety.

3. Motivic and p-adic aspects.

A Kuranishi space is a topological space equipped with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on many moduli spaces in differential geometry, and in particular, in moduli spaces of stable $J$-holomorphic curves in symplectic geometry.

Let $Y$ be an orbifold, and $R$ a commutative ring. We shall define four topological invariants of $Y$: two kinds of Kuranishi bordism ring $KB_*(Y;R)$, and two kinds of Kuranishi homology ring $KH_*(Y;R)$. Roughly speaking, they are spanned over $R$ by isomorphism classes $[X,f]$ with various choices of relations, where $X$ is a compact oriented Kuranishi space, which is without boundary for bordism and with boundary and corners for homology, and $f:X\rightarrow Y$ is a strong submersion. Our main result is that weak Kuranishi homology is isomorphic to the singular homology of $Y$.

These theories are powerful tools in symplectic geometry for several reasons. Firstly, using them eliminates the issues of virtual cycles and perturbation of moduli spaces, yielding technical simplifications. Secondly, as $KB_*,KH_*(Y;R)$ are very large, invariants defined in these groups contain more information than invariants in conventional homology. Thirdly, we can define Gromov-Witten type invariants in Kuranishi bordism or homology groups over $\mathbb Z$, not just $\mathbb Q$, so they can be used to study the integrality properties of Gromov-Witten invariants.

This is the first of two talks. Today we deal with motivation from symplectic geometry, and Kuranishi bordism. Next week's talk discusses Kuranishi homology.

Packings and coverings in graphs are related to two main problems of

graph theory, respectively error correcting codes and domination.

Given a set of words, an error correcting code is a subset such that

any two words in the subset are rather far apart, and can be

identified even if some errors occured during transmission. Error

correcting codes have been well studied already, and a famous example

of perfect error correcting codes are Hamming codes.

Domination is also a very old problem, initiated by some Chess problem

in the 1860's, yet Berge proposed the corresponding problem on graphs

only in the 1960's. In a graph, a subset of vertices dominates all the

graph if every vertex of the graph is neighbour of a vertex of the

subset. The domination number of a graph is the minimum number of

vertices in a dominating set. Many variants of domination have been

proposed since, leading to a very large literature.

During this talk, we will see how these two problems are related and

get into few results on these topics.

One knows, for example by proving well-posedness for an initial value problem with data at the singularity, that there exist many cosmological solutions of the Einstein equations with an initial curvature singularity but for which the conformal metric can be extended through the singularity. Here we consider a converse, a local extension problem for the conformal structure: given an incomplete causal curve terminating at a curvature singularity, when can one extend the conformal structure to a set containing a neighbourhood of a final segment of the curve?

We obtain necessary and sufficient conditions based on boundedness of tractor curvature components. (Based on work with Christian Luebbe: arXiv:0710.5552, arXiv:0710.5723.)

We study Onsager’s model of isotropic–nematic phase transition with orientation parameter on a circle and sphere. We show the axial symmetry and derive explicit formulae for all critical points. Using the information about critical points we investigate a theory of orientational order in nematic liquid crystals which interpolates between several distinct approaches based on the director field (Oseen and Frank), order parameter tensor (Landau and de Gennes), and orientation probability density function (Onsager). As in density-functional theories, the free energy is a functional of spatially-dependent orientation distribution, however, the spatial variation effects are taken into account via phenomenological elastic terms rather than by means of a direct pair-correlation function. As a particular example we consider a simplified model with orientation parameter on a circle and illustrate its relation to complex Ginzburg-Landau theory.

The arc complex is a combinatorial moduli space, very similar to the curve complex. Using the techniques of Masur and Minsky, as well as new ideas, I'll sketch the theorem of the title. (Joint work with Howard

Masur.) If time permits, I'll discuss an application to the cusp shapes of fibred hyperbolic three-manifolds. (Joint work with David Futer.)

We are planning to have dinner at Chiang Mai afterwards.

If anyone would like to join us, please can you let me know today, as I plan to make a booking this evening. (Chiang Mai can be very busy even on a Monday.)

The Bakry Emery criterion asserts that a probability measure with a strictly positive generalized curvature satisfies a logarithmic Sobolev inequality, and by results of Bakry and Ledoux an isoperimetric inequality of Gaussian type. These results were complemented by a theorem of Wang: if the curvature is bounded from below by a negative number, then under an additional Gaussian integrability assumption, the log-Sobolev inequality is still valid.

The goal of this joint work with A. Kolesnikov is to provide an extension of Wang's theorem to other integrability assumptions. Our results also encompass a theorem of Bobkov on log-concave measures on normed spaces and allows us to deal with non-convex potentials when the convexity defect is balanced by integrability conditions. The arguments rely on optimal transportation and its connection to the entropy functional

Some recent joint results with N. V. Krylov on the convergence of solutions of finite difference schemes are presented.

The finite difference schemes, considered in the talk correspond to discretizations (in the space variable) of second order parabolic and of second order elliptic (possibly degenerate) equations.

Space derivatives of the solutions to the finite difference schemes are estimated, and these estimates are applied to show that the convergence of finite difference approximations for equations in the whole space can be accelerated to any given rate. This result can be applied to stochastic PDEs, in particular to the Zakai equation of nonlinear filtering, when the signal and observation noises are independent.

Computational Mechanics (CM) has become

a scientific discipline in itself, being High Perfomance Computational

Mechanics (HPCM) a key sub-discipline. The effort for the most efficient use of

distributed memory machines provides a different perspective to CM scientists

relative to a wide range of topics, from the very physics of the problem to

solve to the numerical method used. Marenostrum supercomputer is the largest

facility in Europe and the 5th in the world (top500.org - Spring 2007). This

talk describes the research lines in the CASE Dpt. of the BSC applied to

Aerospace, Bio-mechanics, Geophysics or Environment, through the development of

Alya, the in-house HPCM code for complex coupled problems capable of running

efficiently in large distributed memory facilities.

Abstract: I show that the effective action of string compactifications has astructure that can naturally solve the supersymmetric flavour and CP problems. At leading order in the $g_s$ and $\alpha'$ expansions, the hidden sector factorises. The moduli space splits into two mirror parts that depend on K\"ahler and complex structure moduli. Holomorphy implies the flavour structure of the Yukawa couplings arises in only one part. In type IIA string theory flavour arises through the K\"ahler moduli sector and in type IIB flavour arises through the complex structure moduli sector. This factorisation gives a simple solution to the supersymmetric flavour and CP problems: flavour physics is generated in one sector while supersymmetry is broken in the mirror sector. This mechanism does not require the presence of gauge, gaugino or anomaly mediation and is explicitly realised by phenomenological models of IIB flux compactifications.