Postdoc Seminar

Title : Some facts on Bass’s nil groups

Abstract: In 1966-67, A Bak introduced the concept of “form rings” and “form parame- ter” to give a uniform definition of classical groups. This group is known as Bak’s Unitary group or general quadratic group. In this talk we recall the definition of Bak’s group and its elementary subgroups. After recalling the notion of Bak’s Uni- tary group, we have deduced the graded Local-Global principle for this group. A relative Local-Global principle for this group is also established. The kernel of the group homomorphism K1GQλ(R[X], Λ[X]) → K1GQλ(R, Λ) induced from the form ring homomorphism (R[X],Λ[X]) → (R,Λ) : X 􏰀→ 0 is defined by NK1Qλ(R,Λ). We often say it as Bass’s nilpotent unitary K1-group of R. We have proved that Bass’s nil group has no k-torsion when kR = R. Using graded Local-Global principle of Unitary group, we also deduce the analog result for the graded rings.

This is a joint work with R. Basu.

Title: Equivariant principal bundles on nonsingular toric varieties

Abstract: Let G be a complex linear algebraic group. In this talk we will give an overview of torus equivariant principal G-bundles on nonsingular toric varieties and discuss some recent results. This is a work in progress with Bivas Khan and Mainak Poddar.

Title: Linkage of quaternion algebras

Abstract: A quaternion algebra over a field is given by a pair of nonzero param- eters from the field, which we call its slots. We discuss the property for a field F , called strong linkage, that any finite number of quaternion algebras over F have a common slot. The study of this field property is motivated by its relation to quadratic forms and further by the examples of global fields, where it can be shown using Class Field Theory.

In the particular case of the rational function field over a finite field, strong linkage can be obtained alternatively by using some crucial proper- ties of finite fields. We study the strong linkage property for the rational function field over quasi-finite fields, that is perfect fields having the same absolute Galois group as of a finite field. In this talk I will discuss relation of the linkage problem to quadratic form theory and different tools which are useful to show strong linkage.

Title: Local coordinates of loxodromic pairs in rank one.

Abstract: We classify conjugation orbits of generic pairs of loxodromic elements in G = SU(n,1). Such pairs, called ‘non-singular’, were introduced by Gongopadhyay and Parsad for SU(3,1). We extend this notion and classify SU(n,1)-conjugation orbits of such elements in arbitrary dimension. For n = 3, they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group; i.e.SU(3,1).

Title: Stability of tangent bundles over nonsingular toric varieties

Abstract: Let X be a nonsingular projective complex toric variety. In this talk, we give a combinatorial criterion of (semi)stability of equivariant torsion free sheaves on X with respect to a given polarization. As an application we obtain a complete answer to the question of (semi)stability of tangent bundle of X with Picard number 2. We also determine (semi)stability of tangent bundle of all toric Fano 4-folds with Picard number ≤ 3 which are classified by Batyrev. Our key tool is the combinatorial description of equivariant sheaves given by Perling. This is a joint work with Jyoti Dasgupta and Arijit Dey.

Title: On the policy improvement algorithm for ergodic risk-sensitive control.

Abstract: In this talk, we consider the ergodic risk-sensitive control problem for a large class of multidimensional controlled diffusions on the whole space. We study the minimization and maximization problems under either a blanket stability hypothesis or a near-monotone assumption on the running cost. We establish the convergence of the policy improvement algorithm for these models. If time permits, we also present a more general result concerning the region of attraction of the equilibrium of the algorithm. This is joint work with Ari Arapostathis and Anup Biswas.

Title: Completed cohomology of Shimura curves

Abstract: We will define the notion of completed cohomology given by Emerton. In the classical case, p-adic Local langlands Correspondence (for GL_2(Q_p)) can be realised inside the completed cohomlogy of modular curves. Using Emerton's local-global compatibility Chojecki proved that if \rho is a 2-dimensional p-adic Galois representation of G_Q such that \rho is absolutely irreducible mod p, then the p-adic Local Langlands Correspondence for GL_2(Q_p) appears in the etale cohomology of Lubin-Tate tower at infinity. This talk is about proving an analogous statement about the 2-dimensional p-adic representation of G_F for a totally real number field F. This is a work in progress with Debargha Banerjee.

Title: Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings

Abstract: As commutative rings provide an algebraic foundation of algebraic geometry, commutative idempotent semirings provide an algebraic foundation of tropical scheme theory, a new branch of tropical geometry. Spectral spaces, as introduced by Hochster in 1969, are topological spaces homeomorphic to the prime spectra of commutative rings. In this work, we prove that a space is spectral if and only if it is homeomorphic to the saturated prime spectrum of an idempotent semiring. We further provide examples of spectral spaces arising from sets of congruence relations on idempotent semirings. In particular, we prove that the space of valuations and the space of prime congruences on an idempotent semiring are spectral, and there is a natural bijection of sets between the two. We then introduce and study closure operations on idempotent semirings and spectral spaces arising from them. In particular, we introduce an integral closure operation and a Frobenius closure operation for idempotent semirings. This is a recently \u000Cfinished work in collaboration with Jaiung Jun and Jeffery Tolliver.