Description
Speaker: Samarpita Ray.
Affiliation: IISc Bengaluru.
Date and Time: Tuesday 22 October, 11:00 am - 12:00 noon.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Some results on spectral spaces and spectral sequences.
Abstract: In this talk, I will present an overview of my research works
and further plans. As part of my thesis work, I have worked on two
different topics which straddle the fields of commutative algebra,
algebraic geometry and category theory. One of my problems is related to
the area of algebraic geometry over the "field with one element"
($\mathbb{F}_1$), several notions of which has been developed in the last
twenty years. It is in this context that monoids became topologically and
geometrically relevant objects of study. Spectral spaces, introduced by
Hochster, are topological spaces homeomorphic to the spectrum of a ring
and are widely studied in the literature. In our work, we present several
naturally occurring classes of spectral spaces using commutative algebra
on pointed monoids. For this purpose, our main tools are finite type
closure operations and continuous valuations on monoids which we introduce
and study in this work.
The other problem involves categorical generalization of certain Hopf
algebra results and a study of their cohomology using Grothendieck
spectral sequence.
It builds on B. Mitchel's famous "ring with several objects" viewpoint of
an arbitrary small preadditive category. In this respect, for a Hopf
algebra H, an H-category will denote an "H-module algebra with several
objects" and a co-H-category will denote an "H-comodule algebra with
several objects". Modules over such Hopf categories were first considered
by Cibils and Solotar. We present a study of cohomology in such module
categories using Grothendieck spectral sequences. I will briefly talk
about these thesis projects and also my further works in this direc