The Dyson Brownian motion (DMB) is a system of infinitely many interacting Brownian motions with logarithmic interaction potential, which was introduced by Freeman Dyson '62 in relation to the random matrix theory. In this talk, we reveal that an infinite-dimensional differential structure induced by the DBM has a Bakry-Émery lower Ricci curvature bound. As an application, we show that the DBM can be realised as the unique Wasserstein-type gradient flow of the Boltzmann-Shannon entropy associated with $\operatorname{Sine}_\beta$ ensemble.
In the last two decades, Craig interpolation has emerged as an essential tool in formal verification, where first-order theories are used to express constraints on the system, such as on the datatypes manipulated by programs. Interpolants for such theories are largely exploited as an efficient method to approximate the reachable states of the system and for invariant synthesis. In this talk, we report recent results on a stronger form of interpolation, called uniform interpolation, and its connection with the notion of model completion from model-theoretic algebra. We discuss how uniform interpolants can be used in the context of formal verification of infinite-state systems to develop effective techniques for computing the reachable states in an exact way. Finally, we present some results about the transfer of uniform interpolants to theory combinations. We argue that methods based on uniform interpolation are particularly effective and computationally efficient when applied to verification of the so-called data-aware processes: these are systems where the control flow of a process can interact with a data storage.
Ricci flat metrics for Calabi-Yau threefolds are not known analytically. In this talk, I will discuss techniques from machine learning to deduce numerical flat metrics on Calabi-Yau two- and three-folds. In particular, I will focus on a particular type of approximation known as spectral neural networks. This type of network produces an exact Kähler metric. I will discuss the metric approximation for various examples, with particular focus on the Cefalú family of quartic two-folds, for which we study the corresponding characteristic forms. Furthermore, from the computation of the Euler characteristic, I will demonstrate that the numerical computations match the expectations, even in the case of singular geometries.
We will discuss some classification results for Hermitian non-Kähler gravitational instantons. There are three main results: (1) Non-existence of certain Hermitian non-Kähler ALE gravitational instantons. (2) Complete classification for Hermitian non-Kähler ALF/AF gravitational instantons. (3) Non-existence of Hermitian non-Kähler gravitational instantons under suitable curvature decay condition, when there is more collapsing at infinity (ALG, ALH, etc.). These are achieved by a thorough analysis of the collapsing geometry at infinity and compactifications.
In this talk we will discuss a collection of convolution inequalities for real valued functions on the hypercube, motivated by combinatorial applications.
In this talk, we propose a new approach to solving the Buckley-Leverett System, which is to consider a compressible approximation model characterized by a stiff pressure law. Passing to the incompressible limit, the compressible model gives rise to a Hele-Shaw type free boundary limit of Buckley-Leverett System, and it is shown the existence of a weak solution of it.
We study transitions from chaotic to integrable Hamiltonians in the double scaled SYK and $p$-spin systems. The dynamics of our models is described by chord diagrams with two species. We begin by developing a path integral formalism of coarse graining chord diagrams with a single species of chords, which has the same equations of motion as the bi-local Liouville action, yet appears otherwise to be different and in particular well defined. We then develop a similar formalism for two types of chords, allowing us to study different types of deformations of double scaled SYK and in particular a deformation by an integrable Hamiltonian. The system has two distinct thermodynamic phases: one is continuously connected to the chaotic SYK Hamiltonian, the other is continuously connected to the integrable Hamiltonian, separated at low temperature by a first order phase transition.
In recent years, there has been a growing number of applications of stable homotopy theory to condensed matter physics, many of which stem from a conjecture of Kitaev that gapped invertible phases of matter should be classified by the homotopy groups of a spectrum. This gives rise to a mathematical modeling question: how do we model quantum systems in such a way that this result can be better understood, perhaps even proved? In this talk, I will discuss some aspects of this modeling problem. This is based on joint work with Mike Hermele, Juan Moreno, Markus Pflaum, Marvin Qi and Daniel Spiegel, David Stephen, Xueda Wen.
I will present some models in ecology and epidemiology using a transport equation approach, so called structured models. The first models are of predator-prey type and include a variable hunger structure. They take the form of nonlocal transport equations coupled to ODEs. Then, we use a similar approach in an epidemiological model including disease awareness and variable susceptibility. We show well-posedness results, asymptotic behavior, and numerical simulations. This is joint work with C. Rebelo, A. Margheri, and P. Lafargeas.