Past

In the past decades the $G_{n,p}$ model of random graphs has led to numerous beautiful and deep theorems. A key feature that is used in basically all proofs is that edges in $G_{n,p}$ appear independently.

The independence of the edges allows, for example, to obtain extremely tight bounds on the number of edges of $G_{n,p}$ and its degree sequence by straightforward applications of Chernoff bounds. This situation changes dramatically if one considers graph classes with structural side constraints. In this talk we show how recent progress in the construction of so-called Boltzmann samplers by Duchon, Flajolet, Louchard, and Schaeffer can be used to reduce the study of degree sequences and subgraph counts to properties of sequences of independent and identically distributed random variables -- to which we can then again apply Chernoff bounds to obtain extremely tight results. As proof of concept we study properties of random graphs that are drawn uniformly at random from the class consisting of the dissections of large convex polygons. We obtain very sharp concentration results for the number of vertices of any given degree, and for the number of induced copies of a given fixed graph.

: I will review various constructions and properties of complete constant scalar curvature metrics. I will emphasize the role played by the so called "Fowler's solutions" which give rise to metrics with cylindrical ends. I will also draw the parallel between these constructions and similar constructions which surprisingly (or not) appear in a different context : constant mean curvature surfaces and more recently the Allen-Cahn equation and some equation in the biological theory of pattern formation.

The talk will discuss the modeling of multi-phase fluid membranes surrounded by a viscous fluid with a particular emphasis on the inner flow--the motion of the lipids within the membrane surface.

For this purpose, we obtain the equations of motion of a two-dimensional viscous fluid flowing on a curved surface that evolves in time. These equations are derived from the balance laws of continuum mechanics, and a geometric form of these equations is obtained. We apply these equations to the formation of a protruding bud in a fluid membrane, as a model problem for physiological processes on the cell wall. We discuss the time and length scales that set different regimes in which the outer or inner flow are the predominant dissipative mechanism, and curvature elasticity or line tension dominate as driving forces. We compare the resulting evolution equations for the shape of the vesicle when curvature energy and internal viscous drag are operative with other flows of the curvature energy considered in the literature, e.g. the $L_2$ flow of the Willmore energy. We show through a simple example (an area constrained spherical cap vesicle) that the time evolutions predicted by these two models are radically different.

Joint work with Antonio DeSimone, SISSA, Italy.

Taking the intersection form of a 4n-manifold defines a functor from a category of cobordisms to a symmetric monoidal category of quadratic forms. I will present the theory of the Maslov index and some higher-categorical constructions as variations on this theme.

We describe individual based continuous models of random evolutions and discuss some effects of competitions in these models. The range of applications includes models of spatial ecology, genetic mutation-selection models and particular socio-economic systems. The main aim of our presentation is to establish links between local characteristics of considered models and their macroscopic behaviour

In this talk we revisit the setting of Bouchard, Touzi, and Zeghal (2004).

For an incomplete market and a non-smooth utility function U defined on the whole real line we study the problem:

sup E [U(XTx,θ – B)]

θΘ(S)

Here B is a bounded contingent claim and Xx,θ represents the wealth process with initial capital x generated by portfolio θ. We study the case when the portfolios are constrained in a closed convex cone.

For the case without constraints and with a smooth utility function the solution method is to approximate the utility function and look at the same problem on a bounded negative domain. However, when one attempts to solve this bounded domain problem for a non-smooth utility function, the standard methods of proof cannot be applied. To circumvent this difficulty the idea of quadratic inf-convolution was introduced in Bouchard, Touzi, and Zeghal (2004). This method is mathematically appealing but leads to lengthy and technical proofs.

We will show that despite the presence of constraints, the dependence on quadratic inf-convolution can be removed. We will also show the existence of a constrained replicating portfolio for the optimal terminal wealth when the filtration is generated by a Brownian motion. This provides a natural generalisation of the results of Karatzas and Shreve (1998) to the whole real line.

Abstract: Quantum Hall systems are characterized by a spectacular set of observations (universal low-temperature conductivity, critical behaviour and semi-circle laws for transitions between Quantum Hall states) that are more robust than would be expected from the detailed theory of underlying electron dynamics. The talk starts with a summary of these observations, and their derivation from the assumption that the important charge carriers at the low energies relevant to conductivity measurements are weakly interacting particles or vortices. This implies a large emergent duality symmetry (a level two subgroup of SL(2,Z)), whose presence underlies the robustness of the observations in question. The newly-discovered and unusual Quantum Hall properties of graphene are discussed as providing a new test of this picture.

In this talk we will investigate the properties of stochastic volatility models, to discuss to what extent, and with regard to which models, properties of the classical exponential Brownian motion model carry over to a stochastic volatility setting.

The properties of the classical model of interest include the fact that the discounted stock price is positive for all $t$ but converges to zero almost surely, the fact that it is a martingale but not a uniformly integrable martingale, and the fact that European option prices (with convex payoff functions) are convex in the initial stock price and increasing in volatility. We give examples of stochastic volatility models where these properties continue to hold, and other examples where they fail.

The main tool is a construction of a time-homogeneous autonomous volatility model via a time change.

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.

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.