This presentation explores Mean Field Games (MFGs) through the lens of functional analysis, focusing on the role of monotonicity methods in understanding their properties. We begin by introducing MFGs as models of large populations of interacting rational agents and illustrate their derivation for deterministic problems. We then examine key questions regarding the existence and uniqueness of MFG solutions. Specifically, we present several new existence theorems obtained via $p$-Laplacian regularization. Finally, we discuss weak-strong uniqueness and establish conditions under which weak and strong solutions of MFGs coincide.
Quantum non-Gaussianity, i.e., the impossibility of a state being a mixture of Gaussian states, is a key resource in bosonic systems, especially in the context of continuous-variable quantum computation. Here we present a novel characterization of quantum non-Gaussianity in terms of the zeros of the Schrödinger wavefunction. Under a mild energy assumption, the wavefunction of a single mode admits a natural extension to a holomorphic function over the complex plane. This allows us to prove a Hudson-like theorem: a pure state is Gaussian if and only if its complex extension has no zeros, thereby making the presence of such zeros a faithful signature of non-Gaussianity. Exploiting the Gaussian dynamics of these complex zeros, we show that suitable phase shifts typically bring them to the real axis, where they become observable in some quadrature probability distribution. We then construct a certification protocol based on homodyne detection — a common and accessible measurement set-up — that allows us to witness quantum non-Gaussianity using data from only a single quadrature. Our work drastically simplifies the setup required to detect quantum non-Gaussianity in bosonic quantum states.
The motion of a charged particle in an electromagnetic field is governed by the Lorentz-force equation (LFE), a classical model independently introduced by Poincaré and Planck in the early twentieth century. Despite being an ordinary differential equation, the LFE presents important analytical challenges that have delayed a fully general mathematical treatment for over a century. First, the equation is vector-valued, rather than scalar. Moreover, the relativistic acceleration term becomes singular as the particle's velocity approaches the speed of light, leading to a non-smooth action functional. In addition, the action depends on the particle's velocity through a dot product, resulting in a sign-indefinite term which complicates the variational treatment of the equation. In this talk, I will survey recent variational methods developed to overcome these difficulties and to establish the existence of periodic solutions to the LFE. Some of these results are part of a joint work with Manuel Garzón (ICMAT, Madrid). I will conclude with a discussion of several open problems.
The one-dimensional Kardar-Parisi-Zhang (KPZ) universality class is a broad collection of models including one-dimensional random interface growth, directed polymers and particle systems. At its center lies the KPZ fixed point, a scaling invariant Markov process which governs the asymptotic fluctuations of all models in the class, and which contains all the rich fluctuation behavior seen in the class.
In these lectures, I will explain how one can derive explicit formulas for the transition probabilities of the KPZ fixed point, and how they reveal connections with random matrix theory and with certain classical integrable differential equations. As a starting point for the derivation, we will use the polynuclear growth model (PNG), a model for crystal growth in one dimension that is intimately connected to the classical longest increasing subsequence problem for a uniformly chosen random permutation. The solution will be obtained through a mix of probabilistic and integrable methods.
The lectures will begin with a general overview of the KPZ universality class and its conjectural scaling limits, and I will aim to keep prerequisite knowledge to a minimum.
The one-dimensional Kardar-Parisi-Zhang (KPZ) universality class is a broad collection of models including one-dimensional random interface growth, directed polymers and particle systems. At its center lies the KPZ fixed point, a scaling invariant Markov process which governs the asymptotic fluctuations of all models in the class, and which contains all the rich fluctuation behavior seen in the class.
In these lectures, I will explain how one can derive explicit formulas for the transition probabilities of the KPZ fixed point, and how they reveal connections with random matrix theory and with certain classical integrable differential equations. As a starting point for the derivation, we will use the polynuclear growth model (PNG), a model for crystal growth in one dimension that is intimately connected to the classical longest increasing subsequence problem for a uniformly chosen random permutation. The solution will be obtained through a mix of probabilistic and integrable methods.
The lectures will begin with a general overview of the KPZ universality class and its conjectural scaling limits, and I will aim to keep prerequisite knowledge to a minimum.
A well-known folklore theorem classifies 2-dimensional topological quantum field theories (TQFTs) in terms of Frobenius algebras, providing a unifying link between topology, algebra, and physics. In this talk, we explore what happens when the usual cobordism category is replaced by a category of nested cobordisms, in which 2-dimensional surfaces are equipped with embedded 1-dimensional submanifolds. We study symmetric monoidal functors out of this category and the resulting algebraic structures they encode. This talk is based on joint work with R. Hoekzema, L. Murray, N. Pacheco-Tallaj, C. Rovi, and S. Sridhar.
The one-dimensional Kardar-Parisi-Zhang (KPZ) universality class is a broad collection of models including one-dimensional random interface growth, directed polymers and particle systems. At its center lies the KPZ fixed point, a scaling invariant Markov process which governs the asymptotic fluctuations of all models in the class, and which contains all the rich fluctuation behavior seen in the class.
In these lectures, I will explain how one can derive explicit formulas for the transition probabilities of the KPZ fixed point, and how they reveal connections with random matrix theory and with certain classical integrable differential equations. As a starting point for the derivation, we will use the polynuclear growth model (PNG), a model for crystal growth in one dimension that is intimately connected to the classical longest increasing subsequence problem for a uniformly chosen random permutation. The solution will be obtained through a mix of probabilistic and integrable methods.
The lectures will begin with a general overview of the KPZ universality class and its conjectural scaling limits, and I will aim to keep prerequisite knowledge to a minimum.
The one-dimensional Kardar-Parisi-Zhang (KPZ) universality class is a broad collection of models including one-dimensional random interface growth, directed polymers and particle systems. At its center lies the KPZ fixed point, a scaling invariant Markov process which governs the asymptotic fluctuations of all models in the class, and which contains all the rich fluctuation behavior seen in the class.
In these lectures, I will explain how one can derive explicit formulas for the transition probabilities of the KPZ fixed point, and how they reveal connections with random matrix theory and with certain classical integrable differential equations. As a starting point for the derivation, we will use the polynuclear growth model (PNG), a model for crystal growth in one dimension that is intimately connected to the classical longest increasing subsequence problem for a uniformly chosen random permutation. The solution will be obtained through a mix of probabilistic and integrable methods.
The lectures will begin with a general overview of the KPZ universality class and its conjectural scaling limits, and I will aim to keep prerequisite knowledge to a minimum.
In this talk I will discuss some results obtained in collaboration with Filipe C. Mena and former PhD student Vítor Bessa on the global dynamics of a minimally coupled scalar field interacting with a perfect-fluid through a friction-like term in spatially flat homogeneous and isotropic spacetimes. In particular, it is shown that the late time dynamics contain a rich variety of possible asymptotic states which in some cases are described by partially hyperbolic lines of equilibria, bands of periodic orbits or generalised Liénard systems.
Existing machine learning frameworks operate over the field of real numbers ($\mathbb{R}$ ) and learn representations in real (Euclidean or Hilbert) vector spaces (e.g., $\mathbb{R}^d$ ). Their underlying geometric properties align well with intuitive concepts such as linear separability, minimum enclosing balls, and subspace projection; and basic calculus provides a toolbox for learning through gradient-based optimization.
But is this the only possible choice? In this seminar, we study the suitability of a radically different field as an alternative to $\mathbb{R}$ — the ultrametric and non-archimedean space of $p$-adic numbers, $\mathbb{Q}_p$ . The hierarchical structure of the $p$ -adics and their interpretation as infinite strings make them an appealing tool for code theory and hierarchical representation learning. Our exploratory theoretical work establishes the building blocks for classification, regression, and representation learning with the $p$-adics, providing learning models and algorithms. We illustrate how simple Quillian semantic networks can be represented as a compact $p$ -adic linear network, a construction which is not possible with the field of reals. We finish by discussing open problems and opportunities for future research enabled by this new framework.
