Past

Graph-theoretic ideas have become very useful in understanding modern large-scale data-mining techniques. We show in this talk that ideas from optimization are also quite useful to better understand the numerical behavior of the corresponding algorithms. We illustrate this claim by looking at two specific graph theoretic problems and their application in data-mining.

The first problem is that of reputation systems where the reputation of objects and voters on the web are estimated; the second problem is that of estimating the similarity of nodes of large graphs. These two problems are also illustrated using concrete applications in data-mining.

Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves on an arbitrary nonsingular toric

variety X. This combinatorial description can be used to construct moduli spaces of stable equivariant sheaves on X using Geometric Invariant Theory (analogous to techniques used in case of equivariant vector bundles on X by Payne and Perling). We study how the moduli spaces of stable equivariant sheaves on X can be used to explicitly compute the fixed point locus of the moduli space of all stable sheaves on X, i.e. the subscheme of invariant stable sheaves on X.

Cherednik algebras (always of type A in this talk) are an intriguing class of algebras that have been used to answer questions in a range of different areas, including integrable systems, combinatorics and the (non)existence of crepant resolutions. A couple of years ago Iain Gordon and I proved that they form a non-commutative deformation of the Hilbert scheme of points in the plane. This can be used to obtain detailed information about the representation theory of these algebras.

In the first part of the talk I will survey some of these results. In the second part of the talk I will discuss recent work with Gordon and Victor Ginzburg. This shows that the approach of Gordon and myself is closely related to Gan and Ginzburg's quantum Hamiltonian reduction. This again has applications to representation theory; for example it can be used to prove the equidimensionality of characteristic varieties.

A number of phylogenetic algorithms proceed by searching the space of all possible phylogenetic (leaf labeled) trees on a given set of taxa, using topological rearrangements and some optimality criterion. Recently, such an approach, called BSPR, has been applied to the balanced minimum evolution principle. Several computer studies have demonstrated the accuracy of BSPR in reconstructing the correct tree. It has been conjectured that BSPR is consistent, that is, when applied to an input distance that is a tree-metric, it will always converge to the (unique) tree corresponding to that metric. Here we prove that this is the case. Moreover, we show that even if the input distance matrix contains small errors relative to the tree-metric, then the BSPR algorithm will still return the corresponding tree.

A trader in finance is called an insider if she (or he) knows more about the prices in the market than can be obtained from the market history itself. This is the case if, for example, the trader knows something about the future price/value of a stock. We discuss the following question: What is the optimal portfolio of an insider who wants to maximize her expected profit at a given future time? The problem is that heavy trading by the insider will reveal parts of her inside price information to the market and thereby reduce her information advantage.

We will solve this problem by presenting a general anticipative stochastic calculus model for insider trading. Our results generalize equilibrium results due to Kyle (1985) and Back (1992).

The presentation is partly based on recent joint work with Knut Aase and Terje Bjuland, both at the Norwegian School of Economics and Business Administration (NHH).

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.

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.

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.