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.

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.

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.

In this lecture I will outline a proof of the fact that the analytic index of an elliptic differential operator depends only on the K-theoretic symbol of the operator. Next I will discuss the Thom Isomorphism in K-theory and show that the index theorem for some special operators implies the theorem in general.

We shall discuss the homological invariants like depth and regularity of the edge ideals and their powers of bipartite graphs. We shall mostly concentrate to the cases where the edge ideal is unmixed or Cohen-Macaulay. Some recent progress and direction for future reseach shall be discussed.

We study some properties of Fatou and Julia sets for a family of holomorphic endomorphisms of C^k, k >1. In particular, the family we consider is a semigroup generated by various classes of holomorphic endomorphisms of C^k. Here we generalize a result from the dynamics of the iterates single holomorphic function in C^k, which was proved by Fornaess-Sibony(1998). Also we define recurrent Fatou components in this setup and give a classification result.