Description: This course is a sequel to Math 930 (Riemannian geometry I). The course investigates various topics in comparison geometry. In particular, we will study Ricci curvature (including, the Cheeger-Gromoll splitting theorem, the Bishop volume comparison theorem and the Bochner technique) and then study sectional curvature (including the Toponogov comparison theorem, the sphere theorem and, perhaps, the soul theorem). We will then study other topics as time permits, for example, positive scalar curvature or Gromov-Cheeger compactness results. The text is: Riemannian Geometry (second edition) by Peter Petersen.
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: 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: This course is an introduction to geometric group theory. The central
theme is to study groups as geometric or topological objects via
their actions on nice spaces. The course will begin with a review of
the theory of covering spaces. Topics include Bass-Serre theory, non-
positive curvature, and boundaries of groups. The course will
conclude with a survey of the literature and fundamental open problems. Link to class website.
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.
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