Past

NO WORKSHOP - 09:45 coffee in DH Common Room for those attending the OCIAM Meeting

Elimination of wild ramification is used in the structure theory of valued function fields, with applications in areas such as local uniformization (i.e., local resolution of singularities) and the model theory of valued fields. I will give a survey on the role that Artin-Schreier extensions play in the elimination of wild ramification, and corresponding main theorems on the structure of valued function fields. I will show what these results tell us about local uniformization. I have shown that local uniformization is always possible after a separable extension of the function field of the algebraic variety (separable "alteration"). This was extended to the arithmetic case in joint work with Hagen Knaf. Recently, Michael Temkin has proved local uniformization by purely inseparable alteration.

Further, I will describe a classification of Artin-Schreier extensions with non-trivial defect. It can be used to improve one of the above mentioned main theorems ("Henselian Rationality"). This could be a key for a purely valuation theoretical proof of Temkin's result. On the other hand, the classification shows that separable alteration and purely inseparable alteration are just two ways to eliminate the critical defects. So the existence of these two seamingly "orthogonal" local uniformization results does not necessarily indicate that local uniformization without alteration is possible.

Interior-point methods for linear and convex quadratic programming

require the solution of a sequence of symmetric indefinite linear

systems which are used to derive search directions. Safeguards are

typically required in order to handle free variables or rank-deficient

Jacobians. We propose a consistent framework and accompanying

theoretical justification for regularizing these linear systems. Our

approach is akin to the proximal method of multipliers and can be

interpreted as a simultaneous proximal-point regularization of the

primal and dual problems. The regularization is termed "exact" to

emphasize that, although the problems are regularized, the algorithm

recovers a solution of the original problem. Numerical results will be

presented. If time permits we will illustrate current research on a

matrix-free implementation.

This is joint work with Michael Friedlander, University of British Columbia, Canada

This talk I present a study of BSDEs with non-linear terms of quadratic growth by using Girsanov's theorem. In particular we are able to establish a non-linear version of the Cameron-Martin formula, which can be for example used to obtain gradient estimates for some non-linear parabolic equations.

We consider a 3-dimensional elastic continuum whose material points

can experience no displacements, only rotations. This framework is a

special case of the Cosserat theory of elasticity. Rotations of

material points of the continuum are described mathematically by

attaching to each geometric point an orthonormal basis which gives a

field of orthonormal bases called the coframe. As the dynamical

variables (unknowns) of our theory we choose the coframe and a

density.

In the first part of the talk we write down the general dynamic

variational functional of our problem. In doing this we follow the

logic of classical linear elasticity with displacements replaced by

rotations and strain replaced by torsion. The corresponding

Euler-Lagrange equations turn out to be nonlinear, with the source

of this nonlinearity being purely geometric: unlike displacements,

rotations in 3D do not commute.

In the second part of the talk we present a class of explicit

solutions of our Euler-Lagrange equations. We call these solutions

plane waves. We identify two types of plane waves and calculate

their velocities.

In the third part of the talk we consider a particular case of our

theory when only one of the three rotational elastic moduli, that

corresponding to axial torsion, is nonzero. We examine this case in

detail and seek solutions which oscillate harmonically in time but

depend on the space coordinates in an arbitrary manner (this is a

far more general setting than with plane waves). We show [1] that

our second order nonlinear Euler-Lagrange equations are equivalent

to a pair of linear first order massless Dirac equations. The

crucial element of the proof is the observation that our Lagrangian

admits a factorisation.

[1] Olga Chervova and Dmitri Vassiliev, "The stationary Weyl

equation and Cosserat elasticity", preprint http://arxiv.org/abs/1001.4726

The aim of this talk is to get a feel for the Ricci flow. The Ricci flow was introduced by Hamilton in 1982 and was later used by Perelman to prove the Poincaré conjecture. We will introduce the notions of Ricci flow and Ricci soliton, giving simple examples in low dimension. We will also discuss briefly other types of geometric flows one can consider.

tba

There are two competing notions for a normal subsystem of a (saturated) fusion system. A recent theorem of mine shows how the two notions are related. In this talk I will discuss normal subsystems and their properties, and give some ideas on why this might be useful or interesting.

The topic of this talk is the representation theory of Hopf-Galois extensions. We consider the following questions.

Let H be a Hopf algebra, and A, B right H-comodule algebras. Assume that A and B are faithfully flat H-Galois extensions.

1. If A and B are Morita equivalent, does it follow that the subalgebras A^coH and B^coH of H-coinvariant elements are also Morita equivalent?

2. Conversely, if A^coH and B^coH are Morita equivalent, when does it follow that A and B are Morita equivalent?

As an application, we investigate H-Morita autoequivalences of the H-Galois extension A, introduce the concept of H-Picard group, and we establish an exact sequence linking the H-Picard group of A and

the Picard group of A^coH.(joint work with Stefaan Caenepeel)

Lagrangian submanifolds are an important class of objects in symplectic geometry. They arise in diverse settings: as vanishing cycles in complex algebraic geometry, as invariant sets in integrable systems, as Heegaard tori in Heegaard-Floer theory and of course as "branes" in the A-model of mirror symmetry. We ask the difficult question: when are two Lagrangian submanifolds isotopic? Restricting to the simplest case of Lagrangian spheres in rational surfaces we will give examples where this question has a complete answer. We will also give some very pictorial examples (due to Seidel) illustrating how two Lagrangians can fail to be isotopic.

The talk will begin with a brief account of the construction of string topology operations. I will point out some mysteries with the formulation of these operations, such as the role of (moduli) of surfaces, and pose some questions. The remainder of the talk will address these issues. In particular, I will sketch some ideas for a higher-dimensional version of string topology. For instance, (1) I will describe an E_{d+1} algebra structure on the (shifted) homology of the free mapping space H_*(Map(S^d,M^n)) and (2) I will outline how to obtain operations H_*(Map(P,M)) -> H_*(Map(Q,M)) indexes by a moduli space of zero-surgery data on a smooth d-manifold P with resulting surgered manifold Q.

Take a group G and split it as the fundamental group of a graph of groups, then take the vertex groups and split them as fundamental groups of graphs of groups etc. If at some point you end up with a collection of unsplittable groups, then you have a hierarchy. Haken showed that for any 3-manifold M with an incompressible surface S, one can cut M along S and and then find other incompressible surfaces in M\S and cut again, and repeating this process one eventually obtains a collection of balls. Analogously, Delzant and Potyagailo showed that for any finitely presented group without 2-torsion and a certain sensible class E of subgroups of G, G admits a hierarchy where the edge groups of the splittings lie in E. I really like their proof and I will present it.

In this talk we consider the Boltzmann equation arising in gas dynamics with long-range interactions. Mathematically, it involves bilinear singular integral operators known as collision operators with non-cutoff collision kernels. As for the associated Cauchy problem, we develop a theory of weak solutions and present some of its a priori estimates related with physical quantities including the energy and moments.

Shifted Laplace preconditioners have attracted considerable attention as

a technique to speed up convergence of iterative solution methods for the

Helmholtz equation. In the talk we present a comprehensive spectral

analysis of the discrete Helmholtz operator preconditioned with a shifted

Laplacian. Our analysis is valid under general conditions. The propagating

medium can be heterogeneous, and the analysis also holds for different types

of damping, including a radiation condition for the boundary of the computational

domain. By combining the results of the spectral analysis of the

preconditioned Helmholtz operator with an upper bound on the GMRES-residual

norm we are able to derive an optimal value for the shift, and to

explain the mesh-depency of the convergence of GMRES preconditioned

with a shifted Laplacian. We will illustrate our results with a seismic test

problem.

Joint work with: Yogi Erlangga (University of British Columbia) and Kees Vuik (TU Delft)

(IN: LADY MARGARET HALL)

As part of the Conference on Geometric Model Theory in honour of Professor Boris Zilber

( IN: LADY MARGARET HALL)

As part of the Conference on Geometric Model Theory in honour of Professor Boris Zilber