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.

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.

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.

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.

Let (A,m) be a Noetherian local ring with depth(A) > 1, I an m-primary ideal, M a finitely generated A-module of dimension r, and G_n, the associated graded module of M with respect to I^n. We will discuss a necessary and sufficient condition for depth (G_n) > 1 for all sufficiently large. This talk is based on a paper by Tony Joseph Puthenpurakal (Ratliff-Rush filtration, regularity and depth of higher associated graded modules: Part I )

In this talk algebraic parabolic bundles on smooth projective curves over algebraically closed field of positive characteristic is defined. We will show that the category of algebraic parabolic bundles is equivalent to the category of orbifold bundles defined in. Tensor, dual, pullback and pushforward operations are also defined for parabolic Bundles.