We define and give elementary properties of triangulated categories. We also give an application of triangulated categories to linkage theory in commutative algebra.

Upper bounds on the size of 3-AP free sets over finite fields: We will discuss a recent result of Ellenberg and Gijswijt who showed that if F is a finite field with three elements, and S is a subset of of F^n such that S does not that does not contain three elements in an arithmetic progression, then |S| is upper bounded by c^n for a constant c < 3.

In this talk we will discuss about the group structure on orbit spaces of unimodular rows over smooth real affine algebras. With a few definition and some results to start, we will prove a structure theorem of elementary orbit spaces of unimodular rows over aforementioned ring with the help of similar kind results on Euler class group. As a consequences, we will prove that : Let $X=Spec(R)$ be a smooth real affine variety of even dimension $d > 1$, whose real points $X(R)$ constitute an orientable manifold. Then the set of isomorphism classes of (oriented) stably free $R$ of rank $d > 1$ is a free abelian group of rank equal to the number of compact connected components of $X(R)$. In contrast, if $d > 2$ is odd, then the set of isomorphism classes of stably free $R$-modules of rank $d$ is a $Z/2Z$-vector space (possibly trivial). We will end this talk by giving a structure theorem of Mennicke symbols. PS: Soumi Tikader is a post doctoral candidate.

Abstract Let L be a line bundle of degree d on an elliptic curve C and ϕ : C → P n is a morphism given by a sub-linear system of the complete linear system |L| of dimension n + 1. When d = 4, n = 2, we prove that ϕ ∗TPn is semi-stable if deg(ϕ(C)) > 1. Moreover, we prove that ϕ ∗TPn is isomorphic to direct sum of two isomorphic line bundles if and only if deg(ϕ(C)) = 2. Conversely, for any rank two semi-stable vector bundle E on an elliptic curve C of degree 4, there is a non-degenerate morphism ϕ :C → P n such that ϕ ∗TPn (−1) = E. More precisely, E is isomorphic to direct sum of two isomorphic line bundles if and only if deg(ϕ(C)) = 2. Further E is either indecomposable or direct sum of non-isomorphic line bundles if and only if deg(ϕ(C)) = 4. When d = 5, n = 3, we compute the Harder-Narasimhan filtration of ϕ ∗TPn .

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

Given a prime p, an integer N and a 2 dimensional pseudo-representation of G_{Q,Np} over a finite field of characteristic p, we will analyze how the structure of the universal pseudo-deformation ring changes after allowing ramification at a prime $\ell$ not dividing Np. This question has been studied by Boston and Bockle for deformation rings of absolutely irreducible representations and Borel representations, respectively. As a related question, we will also determine when a pseudo-representation arises from an actual representation. The talk will begin with a brief survey of the theory of pseudo-representations.

In 1929 Oppenheim conjectured that for a nondegenerate, indefinite, irrational quadratic form Q in n ≥ 5 variables, Q(Zn) is dense in R. It was later strengthened to n ≥ 3 by Davenport and proved in 1987 by Margulis based on Raghunathan’s conjecture on closures of unipotent orbits. Later, Dani and Margulis proved the simultaneous density at integer values for a pair of quadratic and linear form in 3 variables when certain conditions are satisfied. We prove an analogue of this for the case of an inhomogeneous quadratic form and a linear form.

An Artin representation is simply a finite dimensional complex representation of the Galois group of a finite extension of the rational number field. Despite their apparent simplicity, Artin representations are very complicated and much harder to study than apaprently more complicated representations such as those attached to elliptic curves, and much of the theory remains conjectural. In this talk I will survey an aspect of the theory where Artin representations are actually simpler and more concrete than other kinds of representations.