Georgia Institute of Technology College of Sciences

It is known for many years that various inequalities in convex geometry have information-theoretic analogues. The most well known example is the Entropy power inequality which corresponds to the Brunn-Minkowski inequality, but the theory of optimal transport allows to prove even better analogues. 

At the same time, in recent years there is a lot of interest in the role of symmetry in Brunn-Minkowski type inequalities. There are many open conjectures in this direction, but also a few proven theorems such as the Gaussian Dimensional Brunn-Minkowski inequality. In this talk we will discuss the natural question — do the known information-theoretic inequalities similarly improve in the presence of symmetry?  I will present some cases where the answer is positive together with some open problems. 

Based on joint work with Gautam Aishwarya. 

This seminar has beeb cancelled and will be rescheduled next year.  We discuss a kind of weak type inequality for the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operators that was first studied by Muckenhoupt and Wheeden and later by Sawyer. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior. Such inequalities arise in the generalization of weak-type spaces to the matrix weighted setting and find applications in scalar two-weight norm inequalities via interpolation with change of measures. In this talk, I will discuss quantitative estimates obtained for $A_p$ weights, $p > 1$, that generalize those results obtained by Cruz-Uribe, Isralowitz, Moen, Pott and Rivera-Ríos for $p = 1$. I will also discuss an endpoint result for the Riesz potentials.

The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\"{o}rmander asked if a more general class of operators (known as H\"{o}rmander type operators) all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\"{o}rmander's question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately H\"{o}rmander's question is known to fail in all dimensions $\geq 3$ by the work of Bourgain and many others. It continues to fail in all dimensions $\geq 3$ even if one adds a ``positive curvature'' assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension $3$ one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it ``Bourgain's condition''. We unify Fourier restriction and Bochner-Riesz by conjecturing that any H\"{o}rmander type operator satisfying Bourgain's condition should have the same $L^p$-boundedness as in those two conjectures. As evidence, we prove that the failure of Bourgain's condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about some history and some interesting components in our proof.

A meromorphic inner function is a bounded analytic function on the upper half plane with unit modulus almost everywhere on the real line and a meromorphic continuation to the complex plane. Meromorphic inner functions and equivalently meromorphic Herglotz functions play a central role in inverse spectral theory of differential operators. In this talk, I will discuss some uniqueness problems for meromorphic inner functions and their applications to inverse spectral theory of canonical Hamiltonian systems as Borg-Marchenko type results.