Seminários de até

I will construct an example of a bounded planar domain with one single hole for which the nodal line of a second Dirichlet eigenfunction is closed and does not touch the boundary. This shows that Payne's nodal line conjecture can at most hold for simply-connected domains in the plane.

In this talk, I would like to present some recent results on the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the stationary nonlinear Schrödinger equation with Sobolev critical power nonlinearity. It is well known that, in the free case, the associated energy functional has a mountain pass geometry on the $L^2$-sphere. This boils down, in higher dimensions, to the existence of a mountain pass solution which is (a suitable scaling of) the Aubin-Talenti function. In this talk, we consider the problem in bounded domains, in the presence of weakly attractive potentials, or under trapping potentials, and investigate the following questions:

1. whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and
2. if the existence of a mountain-pass solution persists.

We provide positive answers under suitable assumptions. This is based on joint work with Dario Pierotti and Gianmaria Verzini (Politecnico di Milano).

I will discuss certain relations between 3-dimensional topological gauge theories with continuous and finite gauge groups, commonly known as Chern–Simons and Dijkgraaf–Witten theories respectively. The relations of this form appear when the continuous and finite gauge groups are the same algebraic group considered over the complex/real numbers and a finite field, respectively. In this talk, I will focus on the SU(2) example and consider the relationship on the level of the corresponding invariants of closed 3-manifolds: Witten–Reshetikhin–Turaev and Dijkgraaf–Witten invariants.

In this talk, we will discuss how Yau's relative entropy method can be used to establish a quantitative law of large numbers for the particle density in systems with Glauber dynamics, as well as to determine the mixing times of these processes. The presentation will be illustrated with examples drawn from exclusion processes with reservoirs and reaction–diffusion models.

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.

The Hopfield Neural Network has played, ever since its introduction in 1982 by John Hopfield, a fundamental role in the inter-disciplinary study of storage and retrieval capabilities of neural networks, further highlighted by the recent 2024 Physics Nobel Prize.

From its strong link with biological pattern retrieval mechanisms to its high-capacity Dense Associative Memory variants and connections to generative models, the Hopfield Neural Network has found relevance both in Neuroscience, as well as the most modern of AI systems.

Much of our theoretical knowledge of these systems however, comes from a surprising and powerful link with Statistical Mechanics, first established and explored in seminal works of Amit, Gutfreund and Sompolinsky in the second half of the 1980s: the interpretation of associative memories as spin-glass systems.

In this talk, we will present this duality, as well as the mathematical techniques from spin-glass systems that allow us to accurately and rigorously predict the behavior of different types of associative memories, capable of undertaking various different tasks.