Georgia Institute of Technology College of Sciences

In the classical theory of optimal transport, Legendre duality arises naturally, as seen for example in Kantorovich’s duality theorem. Extending this idea to a general cost function naturally leads to a broader notion of functional cost-duality and the associated class of c-functions.

Similarly, in the setting of sets, taking polars provides an analogous notion of duality, mapping to the class of convex sets. In this talk, I will introduce cost dualities for sets and show that they correspond precisely to all order-reversing involutions on sets. Finally, I will explore the connections between c-duality and various geometric and functional inequalities.

We develop a theory of Hilbert-space valued stochastic integration with respect to cylindrical martingale-valued measures. As part of our construction, we expand the concept of quadratic variation, to the case of cylindrical martingale-valued measures that are allowed to have discontinuous paths; this is carried out within the context of separable Banach spaces. Our theory of stochastic integration is applied to address the existence and uniqueness of solutions to stochastic partial differential equations in Hilbert spaces. 

It is well known that  $H^2(\mathbb{D}^2)$ is a RKHS with the reproducing kernel $K( \lambda, z) = \frac{1}{(1-\overline{\lambda_1}z_1)(1 - \overline{\lambda_2}z_2)}$ and that for any submodule $M \subseteq H^2(\mathbb{D}^2)$ its reproducing kernel is $K^M( \lambda, z) = P_M K( \lambda, z)$ where $P_M$ is the orthogonal projection onto $M$. Associated with any submodule $M$ are the core function $G^M( \lambda, z) = \frac{K^M( \lambda, z)}{K( \lambda, z)}$ and the core operator $C_M$, an integral transform on $H^2(\mathbb{D}^2)$ with kernel function $G^M$. The utility of these constructions for better understanding the structure of a given submodule is evident from the various works in the past 20 years. In this talk, we will discuss the relationship between the rank, codimension, etc. of a given submodule and the properties of its core function and core operator. In particular, we will discuss the longstanding open question regarding whether we can characterize all submodules whose core function is bounded. This is a joint project with Rongwei Yang.