Past

This talk will discuss the notion of a Nahm transform in differential geometry, as a way of relating solutions to one differential equation on a manifold, to solutions of another differential equation on a different manifold. The guiding example is the correspondence between solutions to the Bogomolny equations on $\mathbb{R}^3$ and Nahm equations on $\mathbb{R}$. We extract the key features from this example to create a general framework.

Nematic elastomers are rubbery elastic solids made of cross-linked polymeric chains with embedded nematic mesogens. Their mechanical behaviour results from the interaction of electro-optical effects typical of nematic liquid crystals with the elasticity of a rubbery matrix. We show that the geometrically linear counterpart of some compressible models for these materials can be justified via Gamma-convergence. A similar analysis on other compressible models leads to the question whether linearised elasticity can be derived from finite elasticity via Gamma-convergence under weak conditions of growth (from below) of the energy density. We answer to this question for the case of single well energy densities.

We discuss Ogden-type extensions of the energy density currently used to model nematic elastomers, which provide a suitable framework to study the stiffening response at high imposed stretches.

Finally, we present some results concerning the attainment of minimal energy for both the geometrically linear and the nonlinear model.

The "de Rham-Witt complex" of Deligne and Illusie is a functorial complex of sheaves $W^*(X)$ on a smooth algebraic variety $X$ over a finite field, computing the cristalline cohomology of $X$. I am going to present a non-commutative generalization of this: even for a non-commutative ring $A$, one can define a functorial "Hochschild-Witt complex" with homology $WHH^*(A)$; if $A$ is commutative, then $WHH^i(A)=W^i(X)$, $X = Spec A$ (this is analogous to the isomorphism $HH^i(A)=H^i(X)$ discovered by Hochschild, Kostant and Rosenberg). Moreover, the construction of the Hochschild-Witt complex is actually simpler than the Deligne-Illusie construction, and it allows to clarify the structure of the de Rham-Witt complex.

I shall present a geometric property valid in many Hrushovski
amalgamation constructions, relative CM-triviality, and derive
consequences on definable groups: modulo their centre they are already
products of groups interpretable in the initial theories used for the
construction. For the bad field constructed in this way, I shall
moreover classify all interpretable groups up to isogeny.

The study of free groups and their automorphisms has a long pedigree, going back to the work of Nielsen and Dehn in the early 20th century, but in many ways the subject only truly reached maturity with the introduction of Outer Space by Culler and Vogtmann in the “Big Bang” of 1986. In this (non-expert) talk, I will walk us through the construction of Outer Space and some related complexes, and survey some group-theoretic applications.

We prove that the rank gradient vanishes for mapping class groups, Aut(Fn) for all n, Out(Fn), n > 2 and any Artin group whose underlying graph is connected. We compute the rank gradient and verify that it is equal to the first L2-Betti number for some classes of Coxeter groups.

We study the problem of covering R^d by overlapping translates of a convex polytope, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov and Minkowski, and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile R^d. By contrast, for k ≥ 2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational, centrally symmetric polytope, and if P has centrally symmetric facets, then P must k-tile R^d for some positive integer k.

I will explain an approach to study DT theory of toric CY 3-folds using $L_\infty$ algebras. Based on the construction of strong exceptional collection of line bundles on Fano toric stack of dimension two, we realize any bounded families of sheaves on local surfaces support on zero section as critical sets of the Chern-Simons functions. As a consequence of this construction, several interesting properties of DT invariants on local surfaces can be checked.

Abstract: We consider an implicit finite difference
scheme on uniform grids in time and space for the Cauchy problem for a second
order parabolic stochastic partial differential equation where the parabolicity
condition is allowed to degenerate. Such equations arise in the nonlinear
filtering theory of partially observable diffusion processes. We show that the
convergence of the spatial approximation can be accelerated to an arbitrarily
high order, under suitable regularity assumptions, by applying an extrapolation
technique.

Suppose we have a collection of blocks, which gradually split apart as time goes on. Each block waits an exponential amount
of time with parameter given by its size to some power alpha, independently of the other blocks. Every block then splits randomly,but according to the same distribution. In this talk, I will focus on the case where alpha is negative, which
means that smaller blocks split faster than larger ones. This gives rise to the phenomenon of loss of mass, whereby the smaller blocks split faster and faster until they are reduced to ``dust''. Indeed, it turns out that the whole state is reduced to dust in a finite time, almost surely (we call this the extinction time). A natural question is then: how do the block sizes behave as the process approaches its extinction time? The answer turns out to involve a somewhat unusual ``spine'' decomposition for the fragmentation, and Markov renewal theory.

This is joint work with Bénédicte Haas (Paris-Dauphine).

The melting of ice in Greenland and Antarctica would, of course, be by far the major contributor any possible sea level rise. Thus, to make science-based predictions about sea-level rise, it is crucial that the ice sheets covering those land masses be accurately mathematically modeled and computationally simulated. In fact, the 2007 IPCC report on the state of the climate did not include predictions about sea level rise because it was concluded there that the science of ice sheets was not developed to a sufficient degree. As a result, predictions could not be rationally and

confidently made. In recent years, there has been much activity in trying to improve the state-of-the-art of ice sheet modeling and simulation. In

this lecture, we review a hierarchy of mathematical models for the flow of ice, pointing out the relative merits and demerits of each, showing how

they are coupled to other climate system components (ocean and atmosphere), and discussing where further modeling work is needed. We then discuss algorithmic approaches for the approximate solution of ice sheet flow models and present and compare results obtained from simulations using the different mathematical models.

Motivated by a conjecture of Alday, Gaiotto and Tachikawa (AGT

conjecture), we construct an action of

a suitable $W$-algebra on the equivariant cohomology of the moduli

space $M_r$ of rank r instantons on $A^2$ (i.e.

on the moduli space of rank $r$ torsion free sheaves on $P^2$,

trivialized at the line at infinity). We show that

the resulting $W$-module is identified with a Verma module, and the

characteristic class of $M_r$ is the Whittaker vector

of that Verma module. One of the main ingredients of our construction

is the so-called cohomological Hall algebra of the

commuting variety, which is a certain associative algebra structure on

the direct sum of equivariant cohomology spaces

of the commuting varieties of $gl(r)$, for all $r$. Joint work with E. Vasserot.

I will present an efficient algorithm for computing certain special values of Rankin triple product $p$-adic L-functions and give an application of this to the explicit construction of rational points on elliptic curves.

The $p$-adic Gross-Zagier formula for diagonal cycles and the $p$-adic Beilinson formulae described in the lectures of Rotger and Bertolini respectively suggest a connection between certain  {\em $p$-adic iterated integrals} attached to modular forms and rational points on elliptic curves. I will describe an ongoing project (in collaboration with Alan Lauder and Victor Rotger) whose goal is to explore these relationships numerically, with the goal of better understanding the notion of {\em Stark-Heegner points}. It is hoped that these experiments might suggest new perspectives on Stark-Heegner points based on suitable {\em $p$-adic deformations} of the global objects--diagonal cycles, Beilinson-Kato and Beilinson-Flach elements-- described in the lectures of Rotger, Bertolini, Dasgupta, and Loeffler, following  the influential approach to $p$-adic $L$-functions pioneered by Coates-Wiles, Kato, and Perrin-Riou.

We define an integral version of Sczech cocycle on ${\rm GL}_n(\mathbf{Z})$ by raising the level at a prime $\ell$.As a result, we obtain a new construction of the $p$-adic L-functions of Barsky/Cassou-Nogu\`es/Deligne-Ribet. This cohomological construction further allows for a study of the leading term of these L-functions at $s=0$:\\1) we obtain a new proof that the order of vanishing is at least the oneconjectured by Gross. This was already known as result of Wiles.\\2) we deduce an analog of the modular symbol algorithm for ${\rm GL}_n$ from the cocyclerelation and LLL. It enables for the efficient computation of the special values of these $p$-adic L-functions.\\When combined  with a refinement of the Gross-Stark conjecture, we obtain some examples of numerical construction of $\mathfrak p$-units in class fields of totally real (cubic) fields.This is joint work with Samit Dasgupta.