The next two talks in the Geometry-Topology seminar (on 04/03/16 and 11/03/16) will be on introduction to the basics of topological K-theory. Topological K-theory attaches to each topological space, a ring gotten using isomorphism classes of topological vector bundles. In the first lecture, I will prove some basic facts about complex vector bundles on compact, Hausdorff spaces and introduce the K-groups. The second lecture will be largely devoted to sketching a proof of Bott periodicity; a central theorem in this subject. Two classical (and related) applications of this theorem are in proving non-parallelizability of spheres other than those of dimension 1,3 and 7 and non-existence of finite-dimensional division algebras over R except in dimensions 1,2, 4 and 8.

We shall discuss ergodic and dynamical properties of flows on homogeneous spaces, and their applications to diophantine approximation and geometry, starting with a basic introduction to the topic and leading up to some recent developments, following especially the work of Marina Ratner on Raghunathan's conjecture.

A theorem of Bayer and Stillman asserts that if I is an ideal in a polynomial ring S over a field (in finitely many variables), then the projective dimension and regularlity of S/I are equal to those of S/Gin(I), where Gin(I) is the generic initial ideal of I in the reverse lexicographic order. In this series of talks, we will discuss the necessary background material, and prove the above theorem.

We shall discuss ergodic and dynamical properties of flows on homogeneous spaces, and their applications to diophantine approximation and geometry, starting with a basic introduction to the topic and leading up to some recent developments, following especially the work of Marina Ratner on Raghunathan's conjecture.

We shall discuss ergodic and dynamical properties of flows on homogeneous spaces, and their applications to diophantine approximation and geometry, starting with a basic introduction to the topic and leading up to some recent developments, following especially the work of Marina Ratner on Raghunathan's conjecture.

A theorem of Bayer and Stillman asserts that if I is an ideal in a polynomial ring S over a field (in finitely many variables), then the projective dimension and regularlity of S/I are equal to those of S/Gin(I), where Gin(I) is the generic initial ideal of I in the reverse lexicographic order. In this series of talks, we will discuss the necessary background material, and prove the above theorem.

We shall discuss ergodic and dynamical properties of flows on homogeneous spaces, and their applications to diophantine approximation and geometry, starting with a basic introduction to the topic and leading up to some recent developments, following especially the work of Marina Ratner on Raghunathan's conjecture.

In this lecture we will first revisit Hodge theory from the point of view of the Heat equation. In particular, we will see how to prove the Hodge theorem by assuming the existence of the Heat Kernel. We will then look at the asymptotic expansion of the Heat Kernel and see how it leads to the signature theorem. In particular we will see how both the Hodge theorem and the signature theorem are special cases of the general statement "Analytical Index = Topological Index" (which is basically the statement of the Atiyah Singer Index Theorem).

A theorem of Bayer and Stillman asserts that if I is an ideal in a polynomial ring S over a field (in finitely many variables), then the projective dimension and regularlity of S/I are equal to those of S/Gin(I), where Gin(I) is the generic initial ideal of I in the reverse lexicographic order. In this series of talks, we will discuss the necessary background material, and prove the above theorem.