Title: Herzog-Kuhl Equations and its Applications - I
Abstract: In these talks, we will see relations between Hilbert series of a module and its graded Betti numbers. This gives relations between the
graded Betti numbers of a modules which are known as Herzog-Kuhl equations. As an application, we show that the property of R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module of finite projective dimension.
Time:
10:30am-11:30am
Location:
Room 215, Department of Mathematics
Description:
Speaker: Jai Laxmi
Title: Tate Resolutions - I
Abstract: Let S be a Noetherian ring, and R = S/I. It is always possible to construct a differential graded algebra (DG-algebra) resolution of R over S due to a result of Tate. If R is the residue field of S, then
Gulliksen proved that such a DG-algebra resolution is minimal. We shall discuss the construction of the Tate resolution in our talk.
Time:
11:00am
Location:
Ramanujan Hall
Description:
Speaker: Venkitesh S.I. (IITB)
Title: The Szemeredi-Trotter Theorem
Abstract:
Given a finite set of points P in R^2 and a finite family of lines L
in R^2, an incidence is a pair (p,l), where p\in P, l\in L and p is a
point in l.
The Szemeredi-Trotter Theorem states that the number of incidences is
atmost a constant multiple of (|L||P|)^{2/3} + |L| + |P|. We give a
proof by Tao, which uses the method of cell partitions.
Time:
4:00pm-5:00pm
Location:
Ramanujan Hall
Description:
Speaker: Prof. Eknath Ghate (TIFR)
Title: Reductions of Galois Representations: Act 1.5
Abstract: We shall describe recent progress on the question of writing
down the reductions of certain local Galois representations. We shall
focus on the case of half integral slopes (especially slope 3/2)
where the behaviour of the reduction is both more complicated and
more interesting.
Our proof uses the mod p Local Langlands Correspondence to reduce the
problem to computing the reductions of certain locally algebraic
representations of GL_2 of the p-adics on certain functions on
the underlying tree.
Time:
3:30pm
Location:
Room 215, Department of Mathematics
Description:
Speaker: Prof. R. V. Gurjar
Complex Algebraic Surfaces III