Over the last decade, Venkatesh has formulated vast conjectures. The first significant instance has been considered by Harris and Venkatesh. It asserts the proportionality of two collections of numbers (a_q) and (b_q) where q runs through the prime numbers, and a_q, b_q belong to the finite field F_q with q elements. But a_q and b_q arise from different worlds : modular forms on one side and representations of the Galois group on the other side. We will explain the conjecture other prerequisites than basic algebraic number theory. We will review the progress accomplished on the conjecture. (The new part of this talk is joint work with Emmanuel Lecouturier.)

The type SL2 bosonic/symmetric Macdonald polynomials P_m are special cases of Askey-Wilson polynomials, also known by the names q-ultraspherical polynomials, Rogers polynomials. These polynomials are two parameter q,t-generalization of characters of SL2 representations. The electronic/nonsymmetric Macdonald polynomials E_m are closely related to the symmetric polynomials. In this talk we discuss the product rules for computing E_k*P_m and P_k*P_m. Following ideas of Martha Yip, we use techniques from double affine Hecke algebra, but execute a compression to reduce the sum from 2*3^{m-1} signed terms to 2m positive terms. We obtain two universal formulas inside the double affine Hecke algebra that capture the product E_k*P_m and P_k*P_m for all m simultaneously. This is based on joint work with Arun Ram.

In this talk, we will study different views of "entropy" of a random variable. The speaker "expects" that the audience do not "gain" much "information" about the talk from the abstract!!

The classical Berger-Coburn-Lebow (BCL) theorem gives a unique factorization of the discrete semigroup $\{M_z^n \times I | n>=0\}$ in vector-valued Hardy space $H^2_D(E)$ in terms of contractive holomorphic semigroups. Here we study the (continuou) 1-parameter version of this and get a similar factorization with the associated infinitesmal, holomorphic generators giving rise to operator-valued Mobius transformations $D \times E$.