Mini Winter School: Geometry and Physics

Abstract

There are profound interactions between a broad spectrum of modern mathematics and mathematical physics, including complex geometry, differential geometry, noncommutative geometry, K-theory, index theory, gauge theory, quantum physics and string theory. We have organized a series of expository lectures, for students in both pure mathematics and mathematical physics, at the level of honours/beginning postgraduate students.

Titles and abstracts

Speaker: Peter Bouwknegt

Title: Introduction to String Theory. (3 lectures)

Abstract:
In these lectures I will give a concise introduction to String Theory. In particular I will give a discussion of T-duality for open and closed strings and introduce the concept of a D-brane.

Speaker: Nicholas Buchdahl

Title: Gauge theory and the topology of 4-manifolds. (3 lectures)

Abstract:
In these talks, I will discuss some results proved in the early 1980's by Simon Donaldson concerning the topology of smooth 4-dimensional manifolds. These results, which revolutionised modern differential topology, were remarkable not just because of their implications for the field, but also because of the way in which they were proved, namely by using methods and ideas from modern physics. In my talks, I shall describe some of the background physics and mathematics, and give an outline of how the theorem is proved. If there is time, I shall also attempt to describe some of the other quite remarkable results that followed on the heels of this fundamental work.

Speaker: Finnur Larusson

Title: Ellipticity and hyperbolicity in geometric complex analysis. (3 lectures)

Abstract:
In the first lecture, we shall review complex analysis in the complex plane, focusing on the several equivalent definitions of what it means for a function to be holomorphic and the basic properties of such functions. In the second and third talks we will introduce and explore important ideas from higher-dimensional complex analysis and complex geometry in the accessible setting of domains in the plane.

Speaker: Michael Murray

Title: Introduction to differential geometry and monopoles. (3 lectures)

Abstract:
The first two lectures will be an introduction to differential geometry, manifolds, vector bundles etc. This will be background for the talks by Mathai, Nick and Peter. The last lecture will be an introduction to Bogomolny, Prasad, Sommerfield monopoles

Speaker: Mathai Varghese

Title: On the Atiyah-Singer index theorem and applications. (3 lectures)

Abstract:
These lectures will be on the Atiyah-Singer index theorem, and its applications to mathematics and physics. The Atiyah-Singer index theorem is concerned with the existence and uniqueness of solutions to linear elliptic partial differential equations. The Fredholm index of an elliptic equation, which is the number of linearly independent solutions of the equation minus the number of linearly independent solutions of the adjoint equation, is a topological invariant. This means that continuous variations in the coefficients of an elliptic equation leave the Fredholm index unchanged. The Atiyah-Singer index theorem gives a striking calculation of this index. It continues to have a tremendous impact on mathematics and mathematical physics.

Schedule

Participant information

Venue

Benham Lecture Theatre (see C9 in map)

Registration

Registration closed on 10/6/2009.

Organisers