Past

The space BD of functions of bounded deformation consists of all L1-functions whose distributional symmetrized derivative (defined by duality with the symmetrized gradient ($\nabla u + \nabla u^T)/2$) is representable as a finite Radon measure. Such functions play an important role in a variety of variational models involving (linear) elasto-plasticity. In this talk, I will present the first general lower semicontinuity theorem for symmetric-quasiconvex integral functionals with linear growth on the whole space BD. In particular we allow for non-vanishing Cantor-parts in the symmetrized derivative, which correspond to fractal phenomena. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to establish the lower semicontinuity result even without a BD-analogue of Alberti's Rank-One Theorem in BV, which is not available at present. A similar strategy also makes it possible to give a proof of the classical lower semicontinuity theorem in BV without invoking Alberti's Theorem.

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.