Upcoming Seminar/Colloquium

Abstract: In this talk, we will discuss the high frequency stability estimates for the linearized inverse boundary value problems for the Schrodinger equation and biharmonic operator with constant attenuation in some bounded domains. First, we will start with the inverse boundary problem introduced by Calderon and then, we will move to various results related to stability estimates for Calderon-type problems. Afterwards, we will explain the linearized versions of the inverse boundary value problems for the Schrodinger and biharmonic cases. Finally, we give the sketch of the proofs for calculating the stability estimates for both equations with constant attenuation in high frequency.

Abstract: Nonvanishing of L-functions has many consequences in analytic number theory, for example, the nonvanishing of the Riemann zeta function is the key point in proving the prime number theorem. The generalized Riemann hypothesis states that all the zeros of L-functions associated with a Hecke eigenform of weight k can lie only on the critical line Re(s)=k/2. Another interesting problem is to study the equivalence of modular properties of an automorphic form and analytic properties of L-functions attached to it, i.e. to derive the transformation properties from the functional equation of L-functions and vice versa.

Jacobi forms are natural generalizations of modular forms and they appear as Fourier-Jacobi coefficients of Siegel modular forms. Jacobi forms play a crucial role in the proof of Saito-Kurokawa conjecture. 

In the first part of the talk, we shall discuss the basics of Jacobi forms and present our results on the nonvanishing of L-functions and the Poincare series for Jacobi forms of integer index and matrix index as well. In the second part, we discuss the analytic continuation of L-functions and a converse theorem for Jacobi forms of half-integral weight. Then, we present certain properties of Rankin-Cohen brackets and their relation with the Poincare series for Jacobi forms. Finally, we discuss constructions of Jacobi cusp forms involving special values of certain Dirichlet series as their Fourier coefficients.

Abstract:  We provide a novel and simple range characterization for the spherical mean transform of radial functions supported in the unit ball of an odd-dimensional Euclidean
space. The new characterization involves a symmetry relation between the values of a linear differential operator applied to the function in the range of the transform. The range description is then used to construct a counterexample proving that unique continuation type results cannot hold for the spherical mean transform in odd dimensional spaces.

Most lectures on the Probabilistic method (more like a paradigm now) start with simple topics and peter out roughly the same place. What the speaker will do instead is to start where most of these lectures would drop off and go on to talk about some recent breakthrough results as well. Here is the (tentative) set of topics that the speaker will lecture on, along with their duration.

1. The "Rodl Nibble": Proof of the Erdos-Hanani conjecture by Pippenger-Spencer. (1 lecture)
2. The Janson Inequalities and applications (1 lecture)
3. The algorithmic aspects of the Local lemma and Moser's entropy reduction idea. (1 lecture)
4. The recent resolution of the Kahn-Kalai conjecture by Park-Pham. (2 lectures)

Depending on how these go, the speaker might include more material.

It is a series of 5 lectures, one hour each, from December 11, 2023, to December 15, 2023, at 3:30 PM. All the talks will be accessible to graduate students. The requirements to the basic minimum - an acquaintance with the outline of the probabilistic method will be assumed.

The ε-regularity of a graph is an important property. It helps to study the graph by seeing its subgraphs with certain properties. Szemeredi's regularity lemma is another important result in graph theory for studying dense graphs. In this talk, we shalll explain ε- regularity of a pair of vertices (A, B). We will describe a few properties of ε -regularity and state Szemeredi’s regularity lemma. We will further discuss some important applications of this lemma.