Past

Azema martingales arise naturally in the study of the chaotic representation property; they also provide classical interpretations of quantum stochastic calculus. The talk will not insist on these aspects, but only define these processes and address the problem of their classification. This raises algebraic questions concerning tensors. Everyone knows that matrices can be diagonalized in some common orthonormal basis if and only if they are symmetric and commute with each other; we shall see an analogous statement for tensors with more
than two indices. This, and other theorems in the same vein, make it possible to associate to any multidimensional Azema martingale an orthogonal decomposition of the state space into one- and two-dimensional subspaces; the behaviour of the process becomes simpler when split into its components in these sub-spaces.

We shall discuss the moduli problem for irreducible holomorphic symplectic manifolds. If these manifolds are equipped with a polarization (an ample line bundle), then they are parametrized by (coarse) moduli spaces. We shall relate these moduli spaces to arithmetic quotients of type IV domains and discuss when they are rational or not. This is joint work with V.Gritsenko and G.K.Sankaran.

 "Scattering amplitudes in gauge theories and gravity have extraordinary properties that are completely invisible in the textbook formulation of quantum field theory using Feynman diagrams. In this usual approach, space-time locality and quantum-mechanical unitarity are made manifest at the cost of introducing huge gauge redundancies in our description of physics. As a consequence, apart from the very simplest processes, Feynman diagram calculations are enormously complicated, while the final results turn out to be amazingly simple, exhibiting hidden infinite-dimensional symmetries. This strongly suggests the existence of a new formulation of quantum field theory where locality and unitarity are derived concepts, while other physical principles are made more manifest. The past few years have seen rapid advances towards uncovering this new picture, especially for the maximally supersymmetric gauge theory in four dimensions.

These developments have interwoven and exposed connections between a remarkable collection of ideas from string theory, twistor theory and integrable systems, as well as a number of new mathematical structures in algebraic geometry. In this talk I will review the current state of this subject and describe a number of ongoing directions of research."

We show how to solve optimization problems in the presence of proportional transaction costs by determining a shadow price, which is a solution to the dual problem. Put differently, this is a fictitious frictionless market evolving within the bid-ask spread, that leads to the same optimization problem as in the original market with transaction costs. In addition, we also discuss how to obtain asymptotic expansions of arbitrary order for small transaction costs. This is joint work with Stefan Gerhold, Paolo Guasoni, and Walter Schachermayer.

The timing of developmental milestones such as egg hatch or bud break

can be important predictors of population success and survival. Many

insect species rely directly on temperature as a cue for their

developmental timing. With environments constantly under presure to

change, developmental timing has become highly adaptive in order to

maintain seasonal synchrony. However, climatic change is threatening

this synchrony.

Our model couples existing models of developmental timing to a

quatitative genetics framework which descibes the evolution of

developmental parameters. We use this approach to examine the ability of a

population to adapt to an enviroment that it is highly maladapted to.

Through a combination of numerical and analtyical approaches we explore

the dynamics of the infinite dimensional system of

integrodifference equations. The model indicates that developmental timing

is surprisingly robust in its ability to maitain synchrony even under

climatic change which works constantly to maintain maladaptivity.

The mechanics of thin elastic or viscous objects has applications in e.g. the buckling of engineering structures, the spinning of polymer fibers, or the crumpling of plates and shells. During the past decade the mathematics, mechanics and physics communities have witnessed an upsurge of interest in those issues. A general question is to how patterns are formed in thin structures. In this talk I consider two illustrative problems: the shapes of an elastic knot, and the stitching patterns laid down by a viscous thread falling on a moving belt. These intriguing phenomena can be understood by using a combination of approaches, ranging from numerical to analytical, and based on exact equations or low-dimensional models.

In this talk, we focus on the analytical properties of a decentralized auction, namely the PAUSE Auction Procedure. We prove that the revenue of the auctioneer from PAUSE is greater than or equal to the profit from the well-known VCG auction when there are only two bidders and provide lower bounds on the profit for arbitrary number of bidders. Based on these bounds and observations from auctions with few items, we propose a modification of the procedure that increases the profit. We believe that this study, which is still in progress, will be a milestone in designing better decentralized auctions since it is the first analytical study on such auctions with promising results.

Much mathematical work has gone into the creation of time-consistent nonlinear expectations. When we think of implementing these, various problems arise and destroy the beautiful consistency properties we have worked so hard to create. One of these problems is to do with horizon dependence, in particular, where a portfolio's value is considered at a time t+m, where t is the present time and m is a fixed horizon.

In this talk we shall discuss various notions of time consistency and the corresponding solution concepts. In particular, we shall focus on notions which pay attention to the space of available policies, allowing for commitment devices and non-markovian restrictions. We shall see that, for any time-consistent nonlinear expectation, there is a notion of time consistency which is satisfied by the moving horizon problem.

We'll survey some of the ways that hyperbolic groups have been studied

using analysis on their boundaries at infinity.

We analyze the question of the minimal index of a normal subgroup in a free group which does not contain a given element. Recent work by BouRabee-McReynolds and Rivin give estimates for the index. By using results on the length of shortest identities in finite simple groups we recover and improve polynomial upper and lower bounds for the order of the quotient. The bounds can be improved further if we assume that the element lies in the lower central series.

I describe a conjecture equating the two items appearing in the title.

In this talk we will first consider the Allen-Cahn action functional that controls the probability of rare events in an Allen-Cahn type equation with additive noise. Further we discuss a perturbation of the Allen-Cahn equation by a stochastic flow. Here we will present a tightness result in the sharp interface limit and discuss the relation to a version of stochastically perturbed mean curvature flow. (This is joint work with Luca Mugnai, Leipzig, and Hendrik Weber, Warwick.)

Coarse geometry provides a very useful organising point of view on the study
of geometry and analysis of discrete metric spaces, and has been very
successful in the context of geometric group theory and its applications. On
the other hand, the work of Carlsson, Ghrist and others on persistent
homology has paved the way for applications of topological methods to the
study of broadly understood data sets. This talk will provide an
introduction to this fascinating topic and will give an overview of possible
interactions between the two.

Topological spaces and manifolds are commonly used to model configuration
spaces of systems of various nature. However, many practical tasks, such as
those dealing with the modelling, control and design of large systems, lead
to topological problems having mixed topological-probabilistic character
when spaces and manifolds depend on many random parameters.
In my talk I will describe several models of stochastic algebraic topology.
I will also mention some open problems and results known so far.

"Rough paths of inhomogeneous degree of smoothness (Pi-rough paths) can be treated as p-rough paths (of homogeneous degree of
smoothness) for a sufficiently large p. The theory of integration with respect to p-rough paths can be applied to prove existence and uniqueness of solutions of differential equations driven by Pi-rough paths. However the required conditions on the one-form determining the differential equation are too strong and can be weakened. The talk proves the existence and uniqueness under weaker conditions and explores some applications of Pi-rough paths

The behaviour of the tail of the distribution of the first passage time over a fixed level has been known for many years, but until recently little was known about the behaviour of the probability mass function or density function. In this talk we describe recent results of Vatutin and Wachtel, Doney, and Doney and Rivero which give such information whenever the random walk or Levy process is asymptotically stable.

Many proteins cleave and reseal DNA molecules in precisely orchestrated
ways. Modelling these reactions has often relied on the axis of the DNA
double helix
being circular, so these cut-and-seal mechanisms can be
tracked by corresponding changes in the knot type of the DNA axis.
However, when the DNA molecule is linear, or the
protein action does not manifest itself as a change in knot type, or the
knots types are not 4-plats, these knot theoretic models are less germane.

We thus give a taxonomy of local DNA axis configurations. More precisely, we
characterise
all rational tangles obtained from a given rational tangle via a rational
subtangle
replacement (RSR). This builds on work of Berge and Gabai. 
We further determine the sites for these RSR of distance greater than 1.
Finally, we classify all knots in lens spaces whose exteriors are
generalised Seifert fibered spaces and their lens space surgeries, extending work of
Darcy-Sumners.

Biologically then, this classification is endowed with a distance that
determines how many protein reactions
of a particular type (corresponding to steps of a specified size) are
needed to proceed from one local conformation to another.
We conclude by discussing a variety of biological applications of this
categorisation.

Joint work with Ken Baker

The central axis of the famous DNA double helix can become knotted
or linked as a result of numerous biochemical processes, most notably
site-specific recombination. Site-specific recombinases are naturally
occurring enzymes that cleave and reseal DNA molecules in very precise ways.
As a by product of their main purpose, they manipulate cellular DNA in
topologically interesting and non-trivial ways. So if the axis of the DNA
double helix is circular, these cut-and-seal mechanisms can be tracked by
corresponding changes in the knot type of the DNA axis.  In this talk, I'll
explain several topological strategies to investigate these biological
situations. As a concrete example, I will disscuss my recent work, which
predics what types of DNA knots and links can arise from site-specific
recombination on DNA twist knots.

Persistent homology is a relatively new tool to analyse the topology of data sets.

We will give a brief introduction and tutorial as preparation for the third talk in the afternoon.

We present three examples of credit products whose valuation poses challenging modeling problems related to armageddon scenarios and extreme losses, analyzing their behaviour pre- and in-crisis.

The products are Credit Index Options (CIOs), Collateralized Debt Obligations (CDOs), and Credit Valuation Adjustment (CVA) related products. We show that poor mathematical treatment of possibly vanishing numeraires in CIOs and lack of modes in the tail of the loss distribution in CDOs may lead to inaccurate valuation, both pre- and especially in crisis. We also consider the limits of copula models in trying to represent systemic risk in credit intensity models. We finally enlarge the picture and comment on a number of common biases in the public perception of modeling in relationship with the crisis.

This will be on the topic of the CASE project Thales will be sponsoring from Oct '11.

It is known that the expansion of the real field by some quasianalytic algebras of functions are o-minimal and polynomially bounded. We prove that, for these structures, the preparation theorem for definable functions proved by L. van den Dries and P. Speissegger has an explicit form, from which it is easy to deduce a quantifier elimination result.

"Most drivers will recognize the scenario: you are making steady progress along the motorway when suddenly you come to a sudden halt at the tail end of a lengthy queue of traffic. When you move off again you look for the cause of the jam, but there isn't one. No accident damaged cars, no breakdown, no dead animal, and no debris strewn on the road. So what caused everyone to stop?" RAC news release (2005)

The (by now well-known) answer is that such "phantom traffic jams" exist as waves that propagate upstream (opposite to the driving direction) - so that the vast majority of individuals do not observe the instant at which the jam was created - yet what exactly goes on at that instant is still a matter of debate. In this talk I'll give an overview of empirical data and models to describe such spatiotemporal patterns. The key property we need is instability: and using the framework of car-following (CF) models, I'll show how different sorts of linear (convective and absolute) and nonlinear instability can be used to explain empirical patterns.

Polynomial Programs are ussually solved by using hierarchies of convex relaxations. This scheme rapidly becomes computationally expensive and is often tractable only for problems of small sizes. We propose an iterative scheme that improves an initial relaxation without incurring exponential growth in size. The key ingredient is a dynamic scheme for generating valid polynomial inequalities for general polynomial programs. These valid inequalities are then used to construct better approximations of the original problem. As a result, the proposed scheme is in principle scalable to large general combinatorial optimization problems.

Joint work with Bissan Ghaddar and Miguel Anjos

The topology of the moduli space of stable bundles (of coprime rank and degree) on a smooth curve can be understood from different points of view. Atiyah and Bott calculated the Betti numbers by gauge-theoretic methods (using equivariant Morse theory for the Yang-Mills functional), arriving at the same inductive formula which had been obtained previously by Harder and Narasimhan using arithmetic techniques. An intermediate interpretation (algebro-geometric in nature but dealing with infinite-dimensional parameter spaces as in the gauge theory picture) comes from thinking about vector bundles in terms of matrix divisors, generalising the Abel-Jacobi map to higher rank bundles.

I'll sketch these different approaches, emphasising their parallels, and in the end I'll speculate about how (some of) these methods could be made to work when the underlying curve acquires nodal singularities.

We consider an optimal stopping problem arising in connection with the exercise of an executive stock option by an agent with inside information.

The agent is assumed to have noisy information on the terminal value of the stock, does not trade the stock or outside securities, and maximises the expected discounted payoff over all stopping times with regard to an enlarged filtration which includes the inside information. This leads to a stopping problem governed by a time-inhomogeneous diffusion and a call-type reward. Using stochastic flow ideas we establish properties of the value function (monotonicity, convexity in the log-stock price), conditions under which the option value exhibits time decay, and derive the smooth fit condition for the solution to the free boundary problem governing the maximum expected reward. From this we derive the early exercise decomposition of the value function. The resulting integral equation for the unknown exercise boundary is solved numerically and this shows that the insider may exercise the option before maturity, in situations when an agent without the privileged information may not.

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.

This is joint work with Boris Andreianov and Nils Henrik Risebro.

I will give a brief introduction to the Steenrod squares and move on to show some applications of them in Topology and Geometry.

The representation theory of the symmetric groups is far more advanced than that of arbitrary finite groups. The blocks of symmetric groups with defect group of order pⁿ are classified, in the sense that there is a finite list of possible Morita equivalence types of blocks, and it is relatively straightforward to write down a representative from each class.

In this talk we will look at the case where n=2. Here the theory is fairly well understood. After introducing combinatorial wizardry such as cores, the abacus, and Scopes moves, we will see a new result, namely that the simple modules for any p-block of weight 2 "come from" (technically, have isomorphic sources to) simple modules for S_2p or the wreath product of S_p and C₂.

I will prove that generating sets of surface groups are either reducible or Nielsen equivalent to standard generating sets, improving upon a theorem of Zieschang. Equivalently, Aut(F_n) acts transitively on Epi(F_n,S) when S is a surface group.