After recalling some motivation for studying highly-connected graphs in the context of operator algebras and large-scale geometry, we will introduce the notion of "asymptotic expansion" recently defined by Li, Nowak, Spakula and Zhang. We will explore some applications of this definition, hopefully culminating in joint work with Li, Vigolo and Zhang.

In 1997 Pimsner described how to construct two universal C*-algebras associated with an injective C*-correspondence, now known as the Toeplitz--Pimsner and Cuntz--Pimsner algebras. In this talk I will recall their construction, focusing for simplicity on the case of a finitely generated projective correspondence. I will describe the associated six-term exact sequence in K(K)-theory and explain how these can be used in practice for computational purposes. Finally, I will describe how, in the case of a self-Morita equivalence, these exact sequences can be interpreted as an operator algebraic version of the classical Gysin sequence for circle bundles.

In this talk, I would like to advertise the strict equality between two objects from very different areas of mathematical physics:

- Kahane's Gaussian Multiplicative Chaos (GMC), which uses a log-correlated field as input and plays an important role in certain conformal field theories.

- A reference model in random matrices called the Circular Beta Ensemble (CBE).

The goal is to give a precise theorem whose loose form is GMC = CBE. Although it was known that random matrices exhibit log-correlated features, such an exact correspondence is quite a surprise. 

Ambitwistor strings are a class of holomorphic worldsheet models that directly describe massless quantum field theories, such as supergravity and super Yang-Mills. Their correlators give remarkably compact amplitude representations, known as the CHY formulas: characteristic worldsheet integrals that are fully localized on a set of polynomial constraints known as the scattering equations. Moreover, the ambitwistor string models provide a natural way of extending these formulas to loop level, where the constraints can be used to simplify the formulas (originally on higher genus curves) to 'forward limit-like' constructions on nodal spheres. After reviewing these developments, I will discuss one of the peculiar features of this approach: the worldsheet formulas on nodal spheres result in a non-standard integrand representation that makes it difficult to e.g. apply established integration techniques. While several approaches for addressing this look feasible or have been put forward in the literature, they only work for the simplest toy models. Taking inspiration from these attempts, I want to discuss a novel strategy to overcome this difficulty, and formulate compact worldsheet formulas with standard Feynman propagators.

The heart of modern machine learning (ML) is the approximation of high dimensional functions. Traditional approaches, such as approximation by piecewise polynomials, wavelets, or other linear combinations of fixed basis functions, suffer from the curse of dimensionality (CoD). We will present a mathematical perspective of ML, focusing on the issue of CoD. We will discuss three major issues: approximation theory and error analysis of modern ML models, dynamics and qualitative behavior of gradient descent algorithms, and ML from a continuous viewpoint. We will see that at the continuous level, ML can be formulated as a series of reasonably nice variational and PDE-like problems. Modern ML models/algorithms, such as the random feature and two-layer and residual neural network models, can all be viewed as special discretizations of such continuous problems. We will also present a framework that is suited for analyzing ML models and algorithms in high dimension, and present results that are free of CoD. Finally, we will discuss the fundamental reasons that are responsible for the success of modern ML, as well as the subtleties and mysteries that still remain to be understood.

Most of the basic results on martingale problems extend to the setting in which the generator depends on a control.  The “control” could represent a random environment, or the generator could specify a classical stochastic control problem.  The equivalence between the martingale problem and forward equation (obtained by taking expectations of the martingales) provides the tools for extending linear programming methods introduced by Manne in the context of controlled finite Markov chains to general Markov stochastic control problems.  The controlled martingale problem can also be applied to the study of constrained Markov processes (e.g., reflecting diffusions), the boundary process being treated as a control.  The talk includes joint work with Richard Stockbridge and with Cristina Costantini. 

In this talk, I will present different PDE models involving surface tension where it may be efficient to consider augmented versions.