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.
Monday | Tuesday | Wednesday |
---|---|---|
9-10: Welcome tea | 9:30-10:30: Buchdahl | 9-10: Buchdahl |
10-11: Murray | 10:30-11: break | 10-11: Buchdahl |
11-12: Murray | 11-12: Larusson | 11-11:30: break |
12-1:30: Lunch | 12-1:30: Lunch | 11:30-12:30: Larusson |
1:30-2:30: Mathai | 1:30-2:30: Bouwknegt | 12:30-2:Lunch |
2:30-3:30: Larusson | 2:30-3:30: Bouwknegt | 2-3: Murray |
3:30-4: break | 3:30-4:break | 3-4: Mathai |
4-5: Bouwknegt | 4-5: Mathai | 4-5: Farewell Tea |
6-8: Winter school dinner (meant mainly for interstate guests) |