Past

I will describe a new family of groups exhibiting wild geometric and computational features in the context of their Conjugacy Problems. These features stem from manifestations of "Hercules versus the hydra battles."

This is joint work with Martin Bridson.

The convergence of Markov processes to stationary distributions is a basic topic of introductory courses in stochastic processes, and the theory has been thoroughly developed. What happens when we add killing to the process? The process as such will not converge in distribution, but the survivors may; that is, the distribution of the process, conditioned on survival up to time t, converges to a "quasistationary distribution" as t goes to infinity.

This talk presents recent work with Steve Evans, proving an analogue of the transience-recurrence dichotomy for killed one-dimensional diffusions. Under fairly general conditions, a killed one-dimensional diffusion conditioned to have survived up to time t either escapes to infinity almost surely (meaning that the probability of finding it in any bounded set goes to 0) or it converges to the quasistationary distribution, whose density is given by the top eigenfunction of the adjoint generator.

These theorems arose in solving part of a longstanding problem in biological theories of ageing, and then turned out to play a key role in a very different problem in population biology, the effect of unequal damage inheritance on population growth rates.

Let d be a fixed natural number. There is a theorem, due to Chvátal, Rodl,

Szemerédi and Trotter (CRST), saying that the Ramsey number of any graph G

with maximum degree d and n vertices is at most c(d)n, that is it grows

linearly with the size of n. The original proof of this theorem uses the

regularity lemma and the resulting dependence of c on d is of tower-type.

This bound has been improved over the years to the stage where we are now

grappling with proving the correct dependency, believed to be an

exponential in d. Our first main result is a proof that this is indeed the

case if we assume additionally that G is bipartite, that is, for a

bipartite graph G with n vertices and maximum degree d, we have r(G)

The classical maximum principle for optimal control of solutions of stochastic differential equations (developed by Pontryagin (deterministic case), Bismut, Bensoussan, Haussmann and others), assumes that the system is Markovian and that the controller has access to full, updated information about the system at all times. The classical solution method involves an adjoint process defined as the solution of a backward stochastic differential equation, which is often difficult to solve.

We apply Malliavin calculus for Lévy processes to obtain a generalized maximum principle valid for non-Markovian systems and with (possibly) only partial information available for the controller. The backward stochastic differential equation is replaced by expressions involving the Malliavin derivatives of the quantities of the system.

The results are illustrated by some applications to finance

Abstract: The alternative action for Yang-Mills theory, which Lionel Mason formulated in twistor space, explains some of the simplicities of gluon scattering amplitudes. We will review the derivation of the familiar CSW rules concerning tree-level scattering, show that the `missing' three-point amplitude can be correctly recovered and elucidate the connection with the canonical Lagrangian approach of Mansfied, Morris, et. al.

In this paper we derive a stochastic partial di¤erential equation whose solutions are processes relevant to the portfolio choice problem. The mar- ket is incomplete and asset prices are modelled as Ito processes. We provide solutions of the SPDE for various choices of its volatility coe¢ - cient. We also show how to imbed the classical Merton problem into our framework.

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.