Past

In recent joint work with Maarten Solleveld we could give a complete classification of the set the irreducible discrete series characters of affine Hecke algebras (including the non simply-laced cases). The results can be formulated in terms of the K-theory of the Schwartz completion of the Hecke algebra. We discuss these results and some related conjectures on formal dimensions and on elliptic characters.

We discuss a model of finite strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. Based on the multiplicative decomposition a thermodynamically admissible flow rule for F_p is described involving as plastic gradient Curl F_p. The formulation is covariant w.r.t. superposed rigid rotations of the reference, intermediate and spatial configuration but the model is not spin-free due to the nonlocal dislocation interaction and cannot be reduced to a dependenceon the plastic metric C_p=F_p^T F_p.
The linearization leads to a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the non-symmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied.
Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: d_tp.τ=0 on the microscopically hard boundary Γ_D⊂∂Ω and [Curlp].τ=0 on the microscopically free boundary ∂Ω\Γ_D, where τ are the tangential vectors at the boundary ∂Ω. Moreover, I show that a weak reformulation of the infinitesimal model allows for a global in-time solution of the corresponding rate-independent initial boundary value problem. The method of choice are a formulation as a quasivariational inequality with symmetric and coercive bilinear form, following the abstract framework proposed by Reddy. Use is made of new Hilbert-space suitable for dislocation density dependent plasticity.

Now that the "classical" philosophies that have danced attendance upon intuitionistic mathematics (Brouwer's subjectivism, Heyting's eclecticism, and contemporary anti-realism) are recognized as failures, it is encumbent upon intuitionists to develop new foundations for their mathematics. In this talk, we assay such efforts, in particular, investigations into the various mathematical grounds on the basis of which the law of the excluded third might be proven invalid. It will also be necessary, along the way, to explode certain mistaken ideas about intuitionism, among them the notion that the logical signs of the intuitionists bear meanings different from those attached to the corresponding signs in conventional mathematics.

Please let Bruno Whittle (@email) know if you would like to go out to dinner with the speaker after the seminar.

We consider a linearly edge-reinforced random walk

on a class of two-dimensional graphs with constant

initial weights. The graphs are obtained

from Z^2 by replacing every edge by a sufficiently large, but fixed

number of edges in series.

We prove that a linearly edge-reinforced random walk on these graphs

is recurrent. Furthermore, we derive bounds for the probability that

the edge-reinforced random walk hits the boundary of a large box

before returning to its starting point.

Part I will also include an overview on the history of the model.

In part II, some more details about the proofs will be explained.

The Borel conjecture asserts that aspherical manifolds are topologically rigid, i.e., every homotopy equivalence between such manifolds is homotopic to a homeomorphism. This conjecture is strongly related to the Farrell-Jones conjectures in algebraic K- and L-theory. We will give an introduction to these conjectures and discuss the proof of the Borel conjecture for high-dimensional aspherical manifolds with word-hyperbolic fundamental groups.

We consider a linearly edge-reinforced random walk

on a class of two-dimensional graphs with constant

initial weights. The graphs are obtained

from Z^2 by replacing every edge by a sufficiently large, but fixed

number of edges in series.

We prove that a linearly edge-reinforced random walk on these graphs

is recurrent. Furthermore, we derive bounds for the probability that

the edge-reinforced random walk hits the boundary of a large box

before returning to its starting point.

Part I will also include an overview on the history of the model.

In part II, some more details about the proofs will be explained.

Abstract: We discuss the string-inspired approach to gauge theory amplitudes prompted by the work of Alday and Maldacena, in particular its application to weak coupling.

We shall present work in progress in collaboration with E. Hrushovski on the geometry of spaces of stably dominated types in connection with non archimedean geometry \`a la Berkovich

When liquidating large portfolios of securities one faces a trade off between adverse market impact of sell orders and the impatience to generate proceeds. We present a Black-Scholes model with an impact factor describing the market's distress arising from previous transactions and show how to solve the ensuing optimization problem via classical calculus of variations. (Joint work with Dirk Becherer, Humboldt Universität zu

Berlin)

Meshfree methods become more and more important for the numerical simulation of complex real-world processes. Compared to classical, mesh-based methods they have the advantage of being more flexible, in particular for higher dimensional problems and for problems, where the underlying geometry is changing. However, often, they are also combined with classical methods to form hybrid methods.

In this talk, I will discuss meshfree, kernel based methods. After a short introduction along the lines of optimal recovery, I will concentrate on results concerning convergence orders and stability. After that I will address efficient numerical algorithms. Finally, I will present some examples, including one from fluid-structure-interaction, which will demonstrate why these methods are currently becoming Airbus's preferred solution in Aeroelasticity.

The partition function of the random cluster model on a graph $G$ is also known as its Potts model partition function. (Only the points at which it is evaluated differ in the two models.) This is a multivariate generalization of the Tutte polynomial of $G$, and encodes a wealth of enumerative information about spanning trees and forests, connected spanning subgraphs, electrical properties, and so on.

An elementary property of electrical networks translates into the statement that any two distinct edges are negatively correlated if one picks a spanning tree uniformly at random. Grimmett and Winkler have conjectured the analogous correlation inequalities for random forests or random connected spanning subgraphs. I'll survey some recent related work, partial results, and more specific conjectures, without going into all the gory details.

We outline a method to solve the stationary Einstein equations with source a body in rigid rotation consisting of elastic matter.

This is work in progress by R.B., B.G.Schmidt, and L.Andersson

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).