Oxford

I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to be a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a recent topological result on obstructions for this problem. I will also address the question of how a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.

Recently Frenkel and Hernandez introduced a kind of "Langlands duality" for characters of semisimple Lie algebras. We will discuss a representation-theoretic interpretation of their duality using quantum analogues of exceptional isogenies. Time permitting we will also discuss a branching rule and relations to Littelmann paths.

John Allen: The Bennett Pinch revisited

Abstract: The original derivation of the well-known Bennett relation is presented. Willard H. Bennett developed a theory, considering both electric and magnetic fields within a pinched column, which is completely different from that found in the textbooks. The latter theory is based on simple magnetohydrodynamics which ignores the electric field.

The discussion leads to the interesting question as to whether the possibility of purely electrostatic confinement should be seriously considered.

Angela Mihai: A mathematical model of coupled chemical and electrochemical processes arising in stress corrosion cracking

Abstract: A general mathematical model for the electrochemistry of corrosion in a long and narrow metal crack is constructed by extending classical kinetic models to also incorporate physically realistic kinematic conditions of metal erosion and surface film growth. In this model, the electrochemical processes are described by a system of transport equations coupled through an electric field, and the movement of the metal surface is caused, on the one hand, by the corrosion process, and on the other hand, by the undermining action of a hydroxide film, which forms by consuming the metal substrate. For the model problem, approximate solutions obtained via a combination of analytical and numerical methods indicate that, if the diffusivity of the metal ions across the film increases, a thick unprotective film forms, while if the rate at which the hydroxide produces is increased, a thin passivating film develops.

The Mathematical Institute invites you to attend the Inaugural Lecture of Professor Gui-Qiang G. Chen. Professor in the Analysis of Partial Differential Equations. Examination Schools, 75-81 High Street, Oxford, OX 4BG.

There is no charge to attend but registration is required. Please register your attendance by sending an email to @email specifying the number of people in your party. Admission will only be allowed with prior registration.

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ABSTRACT

While calculus is a mathematical theory concerned with change, differential equations are the mathematician's foremost aid for describing change. In the simplest case, a process depends on one variable alone, for example time. More complex phenomena depend on several variables – perhaps time and, in addition, one, two or three space variables. Such processes require the use of partial differential equations. The behaviour of every material object in nature, with timescales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modelled by nonlinear partial differential equations or by equations with similar features. The roles of partial differential equations within mathematics and in the other sciences become increasingly significant. The mathematical theory of partial differential equations has a long history. In the recent decades, the subject has experienced a vigorous growth, and research is marching on at a brisk pace.

In this lecture, Professor Gui-Qiang G. Chen will present several examples to illustrate the origins, developments, and roles of partial differential equations in our changing world.

This talk will be based on the article arXiv:1006.2788.

In this talk I will discuss "motivic" Donaldson-Thomas invariants, following the now not-so-recent paper of Kontsevich and Soibelman on this subject. I will, in particular, present some understanding of the mysterious notion of "orientation data," and present some recent work. I will of course do my best to make this talk "accessible," though if you don't know what a scheme or a category is it will probably make you cry.

This talk will begin with an introduction to calibrations and calibrated submanifolds. Calibrated geometry generalizes Wirtinger's inequality in Kahler geometry by considering k-forms which are analogous to the Kahler form. A famous one-line proof shows that calibrated submanifolds are volume minimizing in their homology class. Our examples of manifolds with a calibration will come from complex geometry and from manifolds with special holonomy.

We will then discuss the deformation theory of the calibrated submanifolds in each of our examples and see how they differ from the theory of complex submanifolds of Kahler manifolds.