Past

There is a well-behaving class of dense ordered abelian groups called "regularly dense ordered abelian groups". This first order property of ordered abelian groups is introduced by Robinson and Zakon as a generalization of being an archimedean ordered group. Every dense subgroup of the additive group of reals is regularly dense. In this talk we consider subgroups of the multiplicative group, S, of all complex numbers of modulus 1. Such groups are not ordered, however they have an "orientation" on them: this is a certain ternary relation on them that is invariant under multiplication. We have a natural correspondence between oriented abelian groups, on one side, and ordered abelian groups satisfying a cofinality condition with respect to a distinguished positive element 1, on the other side. This correspondence preserves model-theoretic relations like elementary equivalence. Then we shall introduce a first-order notion of "regularly dense" oriented abelian group; all infinite subgroups of S are regularly dense in their induced orientation. Finally we shall consider the model theoretic structure (R,Gamma), where R is the field of real numbers, and Gamma is dense subgroup of S satisfying the Mann property, interpreted as a subset of R^2. We shall determine the elementary theory of this structure.

We consider impulse control problems in finite horizon for diffusions with decision lag and execution delay. The new feature is that our general framework deals with the important case when several consecutive orders may be decided before the effective execution of the first one.

This is motivated by financial applications in the trading of illiquid assets such as hedge funds.

We show that the value functions for such control problems satisfy a suitable version of dynamic programming principle in finite dimension, which takes into account the past dependence of state process through the pending orders. The corresponding Bellman partial differential equations (PDE) system is derived, and exhibit some peculiarities on the coupled equations, domains and boundary conditions. We prove a unique characterization of the value functions to this nonstandard PDE system by means of viscosity solutions. We then provide an algorithm to find the value functions and the optimal control. This implementable algorithm involves backward and forward iterations on the domains and the value functions, which appear in turn as original arguments in the proofs for the boundary conditions and uniqueness results. Finally, we give several numerical experiments illustrating the impact of execution delay on trading strategies and on option pricing.

A well-known problem in protein modeling is the determination of the structure of a protein with a given set of inter-atomic or inter-residue distances obtained from either physical experiments or theoretical estimates. A general form of the problem is known as the distance geometry problem in mathematics, the graph embedding problem in computer science, and the multidimensional scaling problem in statistics. The problem has applications in many other scientific and engineering fields as well such as sensor network localization, image recognition, and protein classification. We describe the formulations and complexities of the problem in its various forms, and introduce a so-called geometric buildup approach to the problem. We present the general algorithm and discuss related computational issues including control of numerical errors, determination of rigid vs. unique structures, and tolerance of distance errors. The theoretical basis of the approach is established based on the theory of distance geometry. A group of necessary and sufficient conditions for the determination of a structure with a given set of distances using a geometric buildup algorithm are justified. The applications of the algorithm to model protein problems are demonstrated.

This talk will be about the mathematics and computer science behind my "Smoking Adjoints: fast Monte Carlo Greeks" article with Paul Glasserman in Risk magazine. At a high level, the adjoint approach is simply a very efficient way of implementing pathwise sensitivity analysis. At a low level, reverse mode automatic differentiation enables one to differentiate a "black-box" to get the sensitivity of a single output to multiple inputs at a cost no more than 4 times greater than the cost of evaluating the black-box, regardless of the number of inputs

The usual procedure to obtain uniqueness theorems for black hole space-times ("No Hair" Theorems) requires the construction of global coordinates for the domain of outer communications (intuitively: the region outside the black hole). Besides an heuristic argument by Carter and a few other failed attempts the existence of such a (global) coordinate system as been neglected, becoming a quite hairy hypothesis.

After a review of the basic aspects of causal theory and a brief discussion of the definition of black-hole we will show how to construct such coordinates focusing on the non-negativity of the "area function".

I shall explain two ways of embedding families of rescaled graphs into Cayley graphs of groups. The first one allows to construct finitely generated groups with continuously many non-homeomorphic asymptotic cones (joint work with M. Sapir). Note that by a result of Shelah, Kramer, Tent and Thomas, under the Continuum Hypothesis, a finitely generated group can have at most continuously many non-isometric asymptotic cones.

The second way is less general, but it works for instance for families of Cayley graphs of finite groups that are expanders. It allows to construct finitely generated groups with (uniformly convex Banach space)-compression taking any value in [0,1], and with asymptotic dimension 2. In particular it gives the first example of a group uniformly embeddable in a Hilbert space with (uniformly convex Banach space)-compression zero. This is a joint work with G. Arzhantseva and M.Sapir.

Let G be a connected reductive algebraic group over an

algebraically closed field of characteristic 0. A normal

irreducible G-variety X is called spherical if a Borel

subgroup of G has an open orbit on X. It was conjectured by F.

Knop that two smooth affine spherical G-varieties are

equivariantly isomorphic provided their algebras of regular

functions are isomorphic as G-modules. Knop proved that this

conjecture implies a uniqueness property for multiplicity free

Hamiltonian actions of compact groups on compact real manifolds

(the Delzant conjecture). In the talk I am going to outline my

recent proof of Knop's conjecture (arXiv:math/AG.0612561).

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