Tue, October 22, 2019
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11:00am [11:00am] Samarpita Ray: IISc Bengaluru
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