Past

Let L be a positive definite lattice. There are only finitely many positive definite lattices

L' which are isomorphic to L modulo N for every N > 0: in fact, there is a formula for the number of such lattices, called the Siegel mass formula. In this talk, I'll review the Siegel mass formula and how it can be deduced from a conjecture of Weil on volumes of adelic points of algebraic groups. This conjecture was proven for number fields by Kottwitz, building on earlier work of Langlands and Lai. I will conclude by sketching joint work (in progress) with Dennis Gaitsgory, which uses topological ideas to attack Weil's conjecture in the case of function fields.

Numerical homogenization/upscaling for problems with multiple scales have attracted increasing attention in recent years. In particular, problems with non-separable scales pose a great challenge to mathematical analysis and simulation.

In this talk, we present some rigorous results on homogenization of divergence form scalar and vectorial elliptic equations with $L^\infty$ rough coefficients which allow for a continuum of scales. The first approach is based on a new type of compensation phenomena for scalar elliptic equations using the so-called ``harmonic coordinates''. The second approach, the so-called ``flux norm approach'' can be applied to finite dimensional homogenization approximations of both scalar and vectorial problems with non-separated scales. It can be shown that in the flux norm, the error associated with approximating the set of solutions of the PDEs with rough coefficients, in a properly defined finite-dimensional basis, is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard finite element space. We will also talk about the ongoing work on the localization of the basis in the flux norm approach.

This talk will follow the paper 0911.4708.

Witten showed that the Jones polynomial invariants of knots can be computed in terms of partition functions of a (2+1)-dimensional topological field theory, namely the SU(2) Chern-Simons theory. Reshetikhin and Turaev showed that this theory extends to a (1+1+1)-dimensional topological field theory---that is, there is a Chern-Simons-type invariant associated to 3-manifolds, 3-manifolds with boundary, and 3-manifolds with codimension-2 corners.

I will explain the notion of a local or (0+1+1+1)-dimensionaltopological field theory,  which has, in addition to the structure of a (1+1+1)-dimensional theory, invariants associated to 3-manifolds with codimension-3 corners.  I will describe a notion of 2-dimensional algebra that allows us to construct and investigate such local field theories.  Along the way I will discuss the geometric classification of local field theories, and explain the dichotomy between categorification and algebraification.

Simple groups of Lie type have a purely group theoretic characterization in terms of subgroup configurations. We here show that for certain Fraisse limits, the automorphism group is a simple group.

{\bf Keble Workshop on Partial Differential Equations

in Science and Engineering}

\\

\\Place: Roy Griffiths Room in the ARCO Building, Keble College

\\Time: 1:00pm-5:10pm, Thursday, June 10.

\\

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Program:\\

\\ 1:00-1:20pm: Coffee and Tea

\\

\\ 1:20-2:10pm: Prof. Walter Craig (Joint with OxPDE Lunchtime Seminar)

\\

\\ 2:20-2:40pm Prof. Mikhail Feldman

\\

\\ 2:50-3:10pm Prof. Paul Taylor

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\\ 3:20-3:40pm Coffee and Biscuits

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\\ 3:40-4:00pm: Prof. Sir John Ball

\\

\\ 4:10-4:30pm: Dr. Apala Majumdar

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\\ 4:40-5:00pm: Prof. Robert Pego

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\\ 5:10-6:00pm: Free Discussion

\\

\\{\bf Titles and Abstracts:}

\\

1.{\bf Title: On the singular set of the Navier-Stokes equations

\\ Speaker: Prof. Walter Craig, McMaster University, Canada}

\\ Abstract:\\

The Navier-Stokes equations are important in

fluid dynamics, and a famous mathematics problem is the

question as to whether solutions can form singularities.

I will describe these equations and this problem, present

three inequalities that have some implications as to the

question of singularity formation, and finally, give a

new result which is a lower bound on the size of the

singular set, if indeed singularities exist.

\\

\\{\bf 2. Title: Shock Analysis and Nonlinear Partial Differential Equations of Mixed Type.

\\ Speaker: Prof. Mikhail Feldman, University of Wisconsin-Madison, USA}

\\

\\ Abstract:\\ Shocks in gas or compressible fluid arise in various physical

situations, and often exhibit complex structures. One example is reflection

of shock by a wedge. The complexity of reflection-diffraction configurations

was first described by Ernst Mach in 1878. In later works, experimental and

computational studies and asymptotic analysis have shown that various patterns

of reflected shocks may occur, including regular and Mach reflection. However,

many fundamental issues related to shock reflection are not understood,

including transition between different reflection patterns. For this reason

it is important to establish mathematical theory of shock reflection,

in particular existence and stability of regular reflection solutions for PDEs

of gas dynamics. Some results in this direction were obtained recently.

\\

In this talk we start by discussing examples of shocks in supersonic and

transonic flows of gas. Then we introduce the basic equations of gas dynamics:

steady and self-similar compressible Euler system and potential flow equation.

These equations are of mixed elliptic-hyperbolic type. Subsonic and supersonic

regions in the flow correspond to elliptic and hyperbolic regions of solutions.

Shocks correspond to certain discontinuities in the solutions. We discuss some

results on existence and stability of steady and self-similar shock solutions,

in particular the recent work (joint with G.-Q. Chen) on global existence of

regular reflection solutions for potential flow. We also discuss open problems

in the area.

\\

\\{\bf 3. Title: Shallow water waves - a rich source of interesting solitary wave

solutions to PDEs

\\ Speaker: Prof. Paul H. Taylor, Keble College and Department of Engineering Science, Oxford}

\\

\\Abstract:\\ In shallow water, solitary waves are ubiquitous: even the wave crests

in a train of regular waves can be modelled as a succession of solitary waves.

When successive crests are of different size, they interact when the large ones

catch up with the smaller. Then what happens? John Scott Russell knew by experiment

in 1844, but answering this question mathematically took 120 years!

This talk will examine solitary wave interactions in a range of PDEs, starting

with the earliest from Korteweg and De Vries, then moving onto Peregrine's

regularized long wave equation and finally the recently introduced Camassa-Holm

equation, where solitary waves can be cartoon-like with sharp corners at the crests.

For each case the interactions can be described using the conserved quantities,

in two cases remarkably accurately and in the third exactly, without actually

solving any of the PDEs.

The methodology can be extended to other equations such as the various versions

of the Boussinesq equations popular with coastal engineers, and perhaps even

the full Euler equations.

\\

{\bf 4. Title: Austenite-Martensite interfaces

\\ Speaker: Prof. Sir John Ball, Queen's College and Mathematical Institute, Oxford}

\\

\\Abstract:\\ Many alloys undergo martensitic phase transformations

in which the underlying crystal lattice undergoes a change of shape

at a critical temperature. Usually the high temperature phase (austenite)

has higher symmetry than the low temperature phase (martensite).

In order to nucleate the martensite it has to somehow fit geometrically

to the austenite. The talk will describe different ways in which this

occurs and how they may be studied using nonlinear elasticity and

Young measures.

\\

\\{\bf 5. Title: Partial Differential Equations in Liquid Crystal Science and

Industrial Applications

\\ Speaker: Dr. Apala Majumdar, Keble College and Mathematical Institute, Oxford}

\\

\\Abstract:\\

Recent years have seen a growing demand for liquid crystals in modern

science, industry and nanotechnology. Liquid crystals are mesophases or

intermediate phases of matter between the solid and liquid phases of

matter, with very interesting physical and optical properties.

We briefly review the main mathematical theories for liquid crystals and

discuss their analogies with mathematical theories for other soft-matter

phases such as the Ginzburg-Landau theory for superconductors. The

governing equations for the static and dynamic behaviour are typically

given by systems of coupled elliptic and parabolic partial differential

equations. We then use this mathematical framework to model liquid crystal

devices and demonstrate how mathematical modelling can be used to make

qualitative and quantitative predictions for practical applications in

industry.

\\

\\{\bf 6. Title: Bubble bath, shock waves, and random walks --- Mathematical

models of clustering

\\Speaker: Prof. Robert Pego, Carnegie Mellon University, USA}

\\Abstract:\\ Mathematics is often about abstracting complicated phenomena into

simple models. This talk is about equations that model aggregation

or clustering phenomena --- think of how aerosols form soot particles

in the atmosphere, or how interplanetary dust forms comets, planets

and stars. Often in such complex systems one observes universal trend

toward self-similar growth. I'll describe an explanation for this

phenomenon in two simple models describing: (a) ``one-dimensional

bubble bath,'' and (b) the clustering of random shock waves.

A behavioral mean-variance portfolio selection problem in continuous time is formulated and studied. Based on the standard mean-variance portfolio selection problem, the cumulative distribution function of the cash flow is distorted by a probability distortion function. Then the problem is no longer a convex optimization problem. This feature distinguishes it from the conventional linear-quadratic (LQ) problems.

The stochastic optimal LQ control theory no longer applies. We take the quantile function of the terminal cash flow as the decision variable.

The corresponding optimal terminal cash flow can be recovered by the optimal quantile function. Then the efficient strategy is the hedging strategy of the optimal terminal cash flow.

On any Hermitian manifold there is a unique Hermitian connection, called the Bismut connection, which has torsion a three-form. One says that the triplet consisting of the Hermitian structure together with the Bismut connection specifies a Kähler-with-torsion structure, or briefly a KT structure. If the torsion three-form is closed, we have a strong KT structure. The first part of this talk will discuss these notions and also address the problem of classifying strong KT structures.

\paragraph{} Despite their name, KT manifolds are generally not Kähler. In particular the fundamental two-form is not closed. If the KT structure is strong, we have instead a closed three-form. Motivated by the usefulness of moment maps in geometries involving symplectic forms, one may ask whether it is possible to construct a similar type of map, when we replace the symplectic form by a closed three-form. The second part of the talk will explain the construction of such maps, which are called multi-moment maps.

Clustering phenomena occur in numerous areas of science. This series of lectures will discuss:

(i) basic kinetic models for clustering- Smoluchowski's coagulation equation, random shock clustering, ballistic aggregation, domain-wall merging;

(ii) Criteria for approach to self-similarity- role of regular variation;

(iii) The scaling attractor and its measure representation. A particular theme is the use of methods and insights from probability in tandem with dynamical systems theory. In particular there is a

close analogy of scaling dynamics with the stable laws of probability and infinite divisibility.

Homology counts components and cycles, K-theory counts vector bundles and bundles of Clifford algebra modules.  What about geometric models for other generalized cohomology theories?  There is a vision, introduced by Segal, Stolz, and Teichner, that certain cohomology theories should be expressible in terms of topological field theories.

I will describe how the 0-th K-theory group can be formulated in terms of equivalence classes of 1-dimensional topological field theories.  Then I will discuss what it means to twist a topological field theory, and explain that the n-th K-theory group comes from twisted 1-dimensional topological field theories.

The expectation is that 2-dimensional topological field theories should be analogously related to elliptic cohomology.  I will take an extended digression to explain what elliptic cohomology is and why it is interesting.  Then I will discuss 2-dimensional twisted field theory and explain how it leads us toward a notion of higher

("2-dimensional") algebra.  

We will introduce both the classical Hanna Neumann Conjecture and its strengthened version, discuss Stallings' reformulation in terms of immersions of graphs, and look at some partial results. If time allows we shall also look at the new approach of Joel Friedmann.

We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.

Ordinary homology is a geometrically defined invariant of spaces: the 0-th homology group counts the number of components; the n-th homology group counts n-cycles, which correspond to an intuitive notion of 'n-dimensional holes' in a space.  K-theory, or more specifically the 0-th K-theory group, is defined in terms of vector bundles, and so also has an immediate relationship to geometry.  By contrast, the n-th K-theory group is typically defined homotopy-theoretically using the black box of Bott periodicity.

I will describe a more geometric perspective on K-theory, using Z/2-graded vector bundles and bundles of modules for Clifford algebras.  Along the way I will explain Clifford algebras, 2-categories, and Morita equivalence, explicitly check the purely algebraic 8-fold periodicity of the Clifford algebras, and discuss how and why this periodicity implies Bott periodicity.

The talk will not presume any prior knowledge of K-theory, Clifford algebras, Bott periodicity, or the like.

Starting from a definition of the cohomology of a group, we will define the bounded cohomology of a group. We will then show how quasi-homomorphisms lead to cocycles in the second bounded cohomology group, and use this to look at the second bounded cohomology of some of our favourite groups. If time permits we will end with some applications.

Consider a configuration of points in $d$-dimensional Euclidean space

together with a set of constraints

which fix the direction or the distance between some pairs of points.

Basic questions are whether the constraints imply that the configuration

is unique or locally unique up to congruence, and whether it is bounded. I

will describe some solutions

and partial solutions to these questions.

Consider the space M of parabolas y=ax^2+bx+c, with (a, b, c) as coordinates on M. Two parabolas generically intersect at two (possibly complex) points, and we can define a conformal structure on M by declaring two points to be null separated iff the corresponding parabolas are tangent. A simple calculation of discriminant shows that this conformal structure is flat.

In this talk (based on joint works with Godlinski and Sokolov) I shall show how replacing parabolas by rational plane curves of higher degree allows constructing curved conformal structures in any odd dimension. In dimension seven one can use this "twistor" construction to find G_2 structures in a conformal class.