Description: This course is an introduction
to Riemannian geometry. We begin with the notion of a Riemannian
metric on a smooth manifold, its associated connection, the
Levi-Civita connection, and the curvature of the connection.
The remainder of this course and Math 931 are explorations
of the curvature and its relation to the topology and geometry
of the manifold and analysis on the manifold. In Math 930
we exploit the geometry of geodesics to explore some of these
relationships. We hope to conclude with the classical and
fundamental Toponogov comparison theorem.
Description: This is a year long course
in Algebraic Topology. Topics to be covered may include: Cohomology,
universal coefficient theorem, Kunneth formula, Poincare duality.
deRham theory, homotopy theory, cellular and simplicial approximations,
Whitehead and Hurewicz theorems, fiber bundles, Postnikov
towers, obstruction theory, characteristic classes and spectral
sequences.
Description: This is a topic course in Kahler geometry. In the first part we will discuss the Levi-Civita connection, curvature and Chern classes of a Kahler manifold. The Hodge theory on Kahler manifold will also be outlined. In the second part we will discuss the celebrated work of Calabi, Aubin, Yau, Tian and others on the existence of Kahler-Einstein metrics. In the third part, we will discuss some more recent work on extremal metrics and constant scalar curvature metrics in a general Kahler class.
Description: The Mumford conjecture gives a complete description of the stable part of the rational homology of the moduli space of complex algebraic curves (equivalently, Riemann surfaces). Here the stable part is the part independent of genus for large genus. This conjecture was proved in a series of works by Tillmann, Madsen-Tillmann, and Madsen-Weiss. Tillmann and Madsen gave invited lectures about this work at the last two Congresses (Madsen gave a plenary talk at Madrid Congress in 2006).
The proof starts with a transfer of the problem in topology, replacing the moduli space of curves by the classifying space of surface bundles and introducing some characteristic classes of surface bundles. Actually, the classifying spaces are the topologists’s moduli spaces.
The goal of the course is to give a broad introduction into the circle of ideas leading to the proof of the Mumford conjecture, in particular, to the theory of classifying spaces, and to present its proof. Some technical details will be inevitably skipped. On the other hand, I plan to discuss some related topics from K-theory, and, time permits, other branches of topology. The course is a year-long sequence.
The students are expected to be familiar with the basic theory of manifolds and the basic algebraic topology. The background of the audience will determine what details will be presented (and how far we will get).
Description: We will cover the recent preprint 'Knots, sutures, and excision' by Kronheimer and Mrowka.
Description: Algebraic geometry is one
of the oldest and deepest areas of mathematics. It began as
the study of the zero sets of polynomials of two or more variables;
that study leads to many interesting relations between algebra
and geometry. This course starts with introduction to classical
algebraic geometry: We begin by introducing affine and projective
varieties, study their basic properties and eventually concentrate
on the study of algebraic curves. We then discuss methods
of modern algebraic geometry which are also important tools
for studying number theory: schemes, sheaves and sheaf cohomology.
|
