The usual language of algebraic geometry is not appropriate for Arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace rings by the collection of “vectors” or by bi-operads and another based on “matrices” or props. These are the two languages of [Har17], but we omit the involutions which brings considerable simplifications. Once one understands the delicate commutativity condition one can proceed following Grothendieck footsteps exactly. The square matrices, when viewed up to conjugation, give us new commutative rings with Frobenius endomorphisms.

Data-driven reduced-order modeling aims at constructing models describing the underlying dynamics of unknown systems from measurements. This has become an increasingly preeminent discipline in the last few years. It is an essential tool in situations when explicit models in the form of state space formulations are not available, yet abundant input/output data are, motivating the need for data-driven modeling. Depending on the underlying physics, dynamical systems can inherit differential structures leading to specific physical interpretations. In this work, we concentrate on systems that are described by differential equations and possess linear dynamics. Extensions to more complicated, nonlinear dynamics are also possible and will be briefly covered here if time permits.

The methods developed in our study use rational approximation based on Loewner matrices. Starting with the approach by Antoulas and Anderson in '86, and moving forward to the one by Mayo and Antoulas in '07, the Loewner framework (LF) has become an established methodology in the model reduction and reduced-order modeling community. It is a data-driven approach in the sense that what is needed to compute the reduced models is solely data, i.e., samples of the system's transfer function. As opposed to conventional intrusive methods that require an actual large-scale model to reduce (described by many differential equations), the LF only needs measurements in compressed format. In the former category of approaches, we mention balanced truncation (BT), arguably one of the most prevalent model reduction methods. Introduced in the early 80s, this method constructs reduced-order models (ROMs) by using balancing and truncating steps (with respect to classical system theory concepts such as controllability and observability). We show that BT can be reinterpreted as a data-driven approach, by using again the Loewner matrix as a central ingredient. By making use of quadrature approximations of certain system theoretical quantities (infinite Gramian matrices), a novel method called QuadBT (quadrature-based BT) is introduced by G., Gugercin, and Beattie in '22. We show parallels with the LF and, if time permits, certain recent extensions of QuadBT. Finally, all theoretical considerations are validated on various numerical test cases.

Recently there is a rising interest in the research of mean-field optimization, in particular because of its role in analyzing the training of neural networks. In this talk, by adding the Fisher Information (in other word, the Schrodinger kinetic energy) as the regularizer, we relate the mean-field optimization problem with a so-called mean field Schrodinger (MFS) dynamics. We develop a free energy method to show that the marginal distributions of the MFS dynamics converge exponentially quickly towards the unique minimizer of the regularized optimization problem. We shall see that the MFS is a gradient flow on the probability measure space with respect to the relative entropy. Finally we propose a Monte Carlo method to sample the marginal distributions of the MFS dynamics. This is a joint work with Giovanni Conforti, Zhenjie Ren and Songbo Wang.

The Anderson operator is a perburbation of the Laplace-Beltrami operator by a space white noise potential. I will explain how to get a short self-contained functional analysis construction of the operator and how a sharp description of its heat kernel leads to useful quantitative estimates on its eigenvalues and eigenfunctions. One can associate to Anderson operator a (doubly) random field called the Anderson Gaussian free field. The law of its (random) partition function turns out to characterize the law of the spectrum of the operator. The square of the Anderson Gaussian free field turns out to be related to a probability measure on paths built from the operator, called the polymer measure.