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.

Hilbert Function of a graded artin algebra is said to be unimodal if it increases (not necessarily strictly) from zero monotonically till it reaches its maximum value and then decreases ( again not necessarily strictly) till it reaches zero. The notion of unmorality can be imagined because the Gorenstein Artin algebras have symmetric Hilbert functions. However, it is known that unmodality is not always there even for Gorenstein algebras starting at codimension five. In this talk we consider this problem for low codimension Gorenstein and level algebras and prove it in many of the instances.

Let X= (X,O_X) be a projective scheme over a field and with twisting sheaf O_X(1). Let F be a coherent sheaf of O_X-modules. The cohomology table of F is defined as the family h F:= (hi(X,F(n)))(i,n)?N0×Z. We give a survey on results about the set of cohomology tables hC={hF|(X,F)?C},for certain classes C of pairs (X,F). We particularly look at the following questions: (1) Under which conditions is the set hC finite, if C is the class of all pairs (X,F) for which F has a given dimension? (2) Under which conditions is the set hC finite, if C is the class of all pairs (X,F) in which X = P‘r is a given projective space and F is an algebraic vector bundle over X? (3) What can be said if X runs throught all smooth complex projective surfaces and F=OX is the structure sheaf of X? Our results are related to the theory of Hilbert functions, Hilbert polynomials and Hilbert schemes, but also to Castelnuov-Mumford regularity and to vanishing results for cohomology.

One way to describe a class of graphs closed under taking induced subgraphs is by listing the set of all forbidden graphs, graphs that are not in the class but whose every proper induced subgraph is in the class. Such characterizations are known for some important classes. The class of perfect graphs is a prime example. I will mention such a theorem that we discovered recently for a generalization of threshold graphs. Then I will discuss ongoing work on finding such a characterization for quasi-triangulated graphs, a class halfway between chordal and weakly chordal graphs. The difficult question of whether anything can be said for a general hereditary class will be pointed out. This is joint work with Richard Behr and Thomas Zaslavsky.

Edge ideals of graphs and the relation between their algebraic properties and the graph invariants is receiving a lot of attention in the recent years. In this talk we present improved lower bound for the regularity of the binomial edge ideals of trees. We then prove an upper bound for the regularity of the binomial edge ideals proposed by Saeedi Madani and Kiani for a subclass of block graphs, which in particular contain lobsters. We further show that except for trees containing what we call a jewel as a subgraph, the regularity is one more than the number of internal vertices. This talk is based on a joint work with my colleagues A V Jayanthan and B V Raghavendra Rao.

An affine domain A over a field k is called a sub-principle plane if it satisfies the following: 1. A=k[p,q] \subset k[u,v] where k[u,v] is a polynomial ring in two variables over k. 2. There is a polynomial g \in k[u,v] such that k[p,q,g] = k[x,y]. We will discuss the problem of identifying properties of p,q which ensure the condition of A being a sub-principle plane. The problem is clearly important in order to determine if the polynomial F(X,Y,Z) defining the kernel of the homomorphism k[X,Y,Z] \mapsto k[u,v] is an abstract plane. A detailed description of A is hoped to help with the solution of the three dimensional epimorphism Problem.

In this talk, we shall first see some examples of a minimal graded free resolution of a finitely generated graded module $M$ over a commutative ring $R$. Given a field $K$, a positively graded $K$-algebras $R$ with $R_0=K$ is called "Koszul" if the field $K$ has an $R$-linear free resolution when viewed as an $R$-module via the identification $K=R/R_{+}$. We shall review the classical invariant Castelnouvo-Mumford regularity of a module and define Koszul algebras in terms of regularity. We shall also discuss several other characterizations of Koszul algebras. Then I will present some results on Koszul property of diagonal subalgebras of bigraded algebras; in particular, Koszul property of diagonal subalgebras of Rees algebras for a complete intersection ideal generated by homogeneous forms of equal degrees. At the end, I will present several problems concerning Koszul algebras.

In the first part of the talk we will review some basic results about reductive groups, their actions on affine varieties, rings of invariants, etc. In the second part I will mention many results I have proved in this area. In the last part I will state some results about reductive group actions on local analytic rings. Making use of these recent proofs of two conjectures I had made in 1990 will be mentioned.