: In his seminal paper in 2001, Henri Darmon proposed a systematic construction of p-adic points on elliptic curves over the rational numbers, viz. Stark–Heegner points. In this talk, I will report on the construction of p-adic cohomology classes/cycles in the Harris–Soudry–Taylor representation associated to a Bianchi cusp form, building on the ideas of Henri Darmon and Rotger–Seveso. These local cohomology classes are conjectured to be the restriction of global cohomology classes in an appropriate Bloch–Kato Selmer group and have consequences towards the Bloch–Kato–Beilinson conjecture as well as Gross–Zagier type results. This is based on a joint work with Chris Williams (Imperial College London).

Inequalities play an important role in the analysis of partial differential equations. The best constants involved in these equations and the case equality in these inequalities are of particular interest as they are connected with many interesting phenomenon in various problems. In this talk we will discuss some of these inequalities and related problems.

In this article, we investigate the existence and uniqueness of local mild solutions for the one-dimensional generalized stochastic Burgers equation (GSBE) containing a non-linearity of polynomial type and perturbed by α-regular cylindrical Volterra process and having Dirichlet boundary conditions. The Banach fixed point theorem (or contraction mapping principle) is used to obtain the local solvability results. The L∞- estimate on both time and space for the stochastic convolution involving the α-regular cylindrical Volterra process is obtained. Further, the existence and uniqueness of global mild solution of GSBE up to third order nonlinearity is shown. 2010 Mathematics Subject Classification. Primary: 60H15, 60G22; Secondary: 35Q35, 35R60. Key-words: Stochastic Burgers equation, Volterra process, γ-Radonifying operator, Stopping time.

We will consider mainly the following situation. Let R,S be complete normal local domains over an alg. closed field k of char. 0 such that S is integral over R. Our aim is to describe three ideals in S; I_N, I_D, I_K (Noether, Dedekind, Kahler differents resp.) each of which capture the ramified prime ideals in S over R. In general these three ideals are not equal. An important special case when all are equal is when S is flat over R. We will prove many of these statements. The case when there is a finite group G of k-automorphisms of S such that R is the ring of invariants is already very interesting. Then many nice results are proved. These include works of Auslander-Buchsbaum, Chevalley-Shephard-Todd, Balwant Singh, L. Avramov, P. Roberts, P. Griffith, P. Samuel.... I will try to discuss all these results. I believe that these results and ideas involved in them will be very valuable to students and faculty both. Prerequisites. Basic knowledge of Commutative Algebra and language of Algebraic Geometry (no sheaf theory!). I will "throw in" topological proofs from time to time to make the results intuitively more clear.

In this talk we will describe connections between second order partial differential equations and Markov processes associated with them. This connection had been an active area of research for several decades. The talk is aimed at Analysts and does not assume familiarity with probability theory.

In this talk we will describe connections between second order partial differential equations and Markov processes associated with them. This connection had been an active area of research for several decades. The talk is aimed at Analysts and does not assume familiarity with probability theory.

By a pure number field we mean an algebraic number field of the type Q( √n a) where the polynomial x n − a with integer coefficients is irreducible over the field Q of rationals. In this talk our aim is to provide a formula for the discriminant of pure number fields K = Q( √n a) where for each prime p dividing n, p does not divide the gcd of a and vp(a); vp(a) stands for the highest power of p dividing a. We also describe explicitly an integral basis of such fields. This takes care of all pure fields K = Q( √n a), where either a, n are coprime or a is squarefree.

We will consider mainly the following situation. Let R,S be complete normal local domains over an alg. closed field k of char. 0 such that S is integral over R. Our aim is to describe three ideals in S; I_N, I_D, I_K (Noether, Dedekind, Kahler differents resp.) each of which capture the ramified prime ideals in S over R. In general these three ideals are not equal. An important special case when all are equal is when S is flat over R. We will prove many of these statements. The case when there is a finite group G of k-automorphisms of S such that R is the ring of invariants is already very interesting. Then many nice results are proved. These include works of Auslander-Buchsbaum, Chevalley-Shephard-Todd, Balwant Singh, L. Avramov, P. Roberts, P. Griffith, P. Samuel,.... I will try to discuss all these results. I believe that these results and ideas involved in them will be very valuable to students and faculty both. Prerequisites. Basic knowledge of Commutative Algebra and language of Algebraic Geometry (no sheaf theory!). I will

We prove an explicit central value formula for a family of complex L-series of degree 6 for GL2 × GL3 which arise as factors of certain Garret--Rankin triple product L-series associated with modular forms. Our result generalizes a previous formula of Ichino involving Saito--Kurokawa lifts, and as an application, we prove Deligne's conjecture about the algebraicity of the central values of the considered L-series up to the relevant periods. I would also include some other arithmetic applications towards subconvexity problem, construction of associated p-adic L function etc. This is joint work with Carlos de Vera Piquero.

The problem of constructing flat representations of spherical surfaces arises naturally in geography and astronomy while making maps. We look at a mathematical formulation of this problem using the notion of conformal mapping, and discuss its relation with complex analysis. After reviewing the contributions of Gauss, Riemann, and Poincaré to this problem, we end with some glimpses of 20th century developments. This will be an expository talk accessible to undergraduate and postgraduate students.