Postdoc Seminar

Title: A note on relative Vaserstein symbol

Abstract: In an unpublished work of Fasel-Rao-Swan the notion of the relative Witt group WE (R, I ) is defined. In this talk we will give the details of this construction. Then we study the injectivity of the relative Vaserstein symbol VR,I : Um3(R,I)/E3(R,I) → WE(R,I). We established injectivity of this symbol if R is an affine non-singular algebra of dimension 3 over a perfect C1-field and I is a local complete intersection ideal of R. It is believed that for a 3-dimensional affine algebra non-singularity is not necessary for establishing injectivity of the Vaserstein symbol . At the end of the lecture we will give an example of a singular 3-dimensional algebra over a perfect C1-field for which the Vaserstein symbol is injective.

Title: Classification, reduction and stability of toric principal bundles

Abstract: In this talk, we describe a classification of torus equivariant principal G-bundles over a complex nonsingular toric variety where G is a complex linear algebraic group. When G is connected and reductive, we characterize their equivariant automorphisms and relate this description to the equivariant reduction of structure group. As an application, we show the principal bundle analogue of Kaneyama’s theorem on the existence of equivariant splitting of any torus equivariant vector bundle of rank r < n over a projective space of dimension n. We also show that for nonsingular projective toric varieties, equivariant stability of a principal bundle implies that the bundle is stable. This talk is based on a joint work with Indranil Biswas, Arijit Dey, Bivas Khan and Mainak Poddar.

Title: Square-reflexivity in relation to the Hasse-Minkowski Theorem

Abstract: The famous Hasse-Minkowski Theorem states that a quadratic form over a global field (number field or the function field of a curve over a finite field) has a non-trivial zero if and only if it has a non-trivial zero over all completions with respect to places. Such results are referred to local-global principles.

The theorem applies in particular to the case of a rational function field E(X) over a finite field E.

An analysis of the condition to have a local-global principle on E(X) over a field E of characteristic different from 2 sheds a new light on the special properties of finite fields. This property is given in a special way how square-classes can be realized in finite-dimensional etale / separable algebras over E. This property, called square-reflexivity, characterises the presence of the local-global principle over E(X) in dimension 4 and the fact that 5-dimensional forms always have a non-trivial zero, as is the case when E is finite. I will explain this by means of examples. This is a joint work with K.J. Becher.

Title: Free group generated by two parabolic maps in PU(2,1)

Abstract: In this talk, we will discuss the groups generated by two Heisenberg translations of PU(2,1) and determine when they are free. We used Klein’s combination theorem for proving our results. This is ongoing work with Prof.John R. Parker.

Title: Integration in Finite Terms: Polylogarithmic Integrals

Abstract: The talk concerns the problem of integration in finite terms with special functions. Our main result extends the classical theorem of Liouville in the context of elementary functions to special functions: dilogarithmic and trilogarithmic integrals. The results are important since they provide a necessary and sufficient condition for an element of differential field to have an antiderivative in a field extension generated by transcendental elementary functions and special functions. Our results simplify and generalize a theorem of Baddoura on integration with dilogarithmic integrals, and also prove a generalized version of his conjecture on integration with trilogarithmic integrals. Our results can be naturally extended to include polylogarithmic integrals and to this end, a conjecture will be stated for integration in finite terms with transcendental elementary functions and polylogarithmic integrals.

This is a joint work with Dr. Varadharaj R Srinivasan.

Title: Seshadri constants on Bott towers

Abstract: Seshadri constants measure local positivity of line bundles on algebraic varieties. In this talk, we consider a class of nonsingular projective toric varieties, namely Bott towers. We compute Seshadri constants of a nef line bundle on Bott tower. This is a joint work with I. Biswas, J. Dasgupta and K. Hanumanthu.

Title: A joint distribution theorem with applications to extremal primes

Abstract: An extremal prime $p$ for an elliptic curve $E$ is one for which $|a_p(E)| =[2\sqrt{p}]$ , i.e., $a_p(E)$ is maximal or minimal in view of the Hasse bound. In this talk, assuming GRH, we present a joint distribution result involving the Chebotarev Density Theorem. As a consequence, we obtain an upper bound for the number of primes satisfying $a_p(E) = [2\sqrt{p}] mod \ell$ for a sufficiently large prime $\ell$. This is joint work with Amita Malik.

Title: On the monotonicity property of the generalized eigenvalue for weakly-coupled cooperative elliptic systems.

Abstract: In this talk, we consider general linear non-degenerate weakly-coupled cooperative elliptic systems and study certain monotonicity properties of the generalized principal eigenvalue in $\mathbb{R}^d$ with respect to the potential. We show that monotonicity on the right is equivalent to the recurrence property of the twisted operator which is, in turn, equivalent to the minimal growth property at infinity of the principal eigenfunctions. We then show that the strict monotonicity property of the principal eigenvalue is equivalent with the exponential stability of the twisted operators. An equivalence between the monotonicity property on the right and the stochastic representation of the principal eigenfunction is also established. This talk is based on joint work with Ari Arapostathis and Anup Biswas.

Title: Reduction of Galois representations for slope 2

Abstract: Let $p$ be an odd prime and $E$ be a finite extension of $\mathbb{Q}_p$. Given an integer $k \geq 2$ and $a_p \in E$ with $v(a_p)> 0$, there exists an irreducible two-dimensional crystalline representation $V_{k,a_p}$ of $\mathrm{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$ over $E$ of weight $k$ and slope $v(a_p)$.

We discuss results about the mod $p$ reduction of $V_{k,a_p}$ when $v(a_p) = 2$. This builds up on the work of Ghate, Bhattacharya, Rozensztajn. The proof is an application of the compatibility between $p$-adic and mod $p$ local Langlands correspondence. This is an ongoing work with Ravitheja Vangala.

Title: Special classes of morphisms for monoid schemes and blue schemes

Abstract: In the last twenty years, there have been several approaches towards the absolute algebraic geometry or $\mathbb{F}_1$-geometry. One of the simplest minimalistic approaches (since it is contained in every other theory and therefore forms the core of $\mathbb{F}_1$-geometry) is via the theory of monoid schemes, originally developed by Kato, Deitmar, Weibel and Connes-Consani. Lorscheid developed a notion of $\mathbb{F}_1$-geometry based on the notion of blue schemes which enhances tropical varieties with a schematic structure and serve as a common language for the different approaches to tropicalizations. In this talk, we will discuss some important classes of morphisms for monoid schemes and blue schemes. This is a work in progress with Oliver Lorscheid.