IMPA - O Instituto de Matemática Pura e Aplicada

The Ramanujan vector field is a polynomial vector field defined on the affine space of dimension three. Its defining property is that it acts as the $\theta$ operator on quasimodular forms. In characteristic zero, it has only one invariant hypersurface and this fact is crucial in Nesterenko's proof that the transcendence degree of $\mathbb{Q}(q,E_2(q), E_4(q), E_6(q))$ is greater or equal than three, where $E_2,E_4,E_6$ are the normalized Eisenstein series. Following Nesterenko's result, Federico Pellarin gave a complete classification of all prime ideals invariant in characteristic zero. In this talk, we will talk about what is known about ideals invariant by the Ramanujan vector field in prime characteristic and, in particular, explore a method of Movasati of obtaining invariant ideals in this context.

The study of symplectic capacities has been central to symplectic topology since their discovery in the 80s, with important connections with systolic and convex geometry. We prove that all normalized capacities coincide for dynamically convex positive $S^1$-invariant domains in $\mathbb{C}^2$. We also give sufficient and necessary conditions for positive $S^1$-invariant domains to be dynamically convex. This is part of a joint work with V. Ramos and A. Vicente.  

 In this presentation, I will outline the Homological Mirror Symmetry conjecture with a focus on the category of matrix factorizations. Exploring the geometric implications on the mirror side, I will introduce the tensor product of matrix factorizations and demonstrate how it can be applied to construct new Landau-Ginzburg models for Fano manifolds. Finally, I will outline directions for future research.

Heavy-tailed distributions, known for producing observations that can be extremely large and distant from the mean, are often associated in machine learning and statistics with negative consequences such as outliers and numerical instability. Despite their daunting reputation, heavy-tailed behaviors are ubiquitous across many natural systems. The goal of this talk is to argue that heavy tails should not be conceived as 'surprising' or 'anomalous' phenomena; on the contrary, they may offer benefits for machine learning algorithms. This work brings together investigations conducted by my colleagues, students, postdocs, and myself on the emergence and implications of heavy-tailed phenomena in stochastic optimization, particularly in the context of deep learning and stochastic gradient descent (SGD). We first demonstrate that heavy tails can naturally arise in SGD, even when the underlying data is not heavy-tailed, highlighting the influence of algorithmic hyperparameters and multiplicative noise structures. Using continuous-time proxy models based on Lévy-driven stochastic differential equations, we develop generalization bounds that explicitly link heavy-tailed behavior to improved test performance. Further, we explore how heavy tails can create compressibility in neural network parameters and examine their potential in a differential privacy context. Our findings offer a nuanced perspective: while heavy tails can indeed be beneficial up to a certain point, 'excessive heavy' tails may eventually degrade optimization and generalization. 

We study long-waves propagation in two-dimensional waveguide channels with irregular geometries, focusing on curved and expanded junctions. These domains are mapped to simpler canonical channels through the Schwarz–Christoffel conformal transformation. Using modal decomposition in the conformal domain, we show that the averaged solution is governed by the fundamental mode, enabling an effective one-dimensional reduction. In this reduced model, junctions appear as delta-type perturbations, establishing a natural connection with quantum graph theory. We developed analytical and numerical approaches to quantify the effects of channel width, angle, smoothness, and wavelength on the wave dynamics.

The idea of simplifying a Lie group action by constructing a new action (a “resolution”) whose orbits all have the same dimension is not new. In this talk, I will describe some known results for proper actions, as well as a new construction for a subclass of proper actions called infinitesimal polar actions.

The resolution we construct has several desirable properties: for example, it preserves the orbit space and is independent of any choices. It allows us to show that, for such actions, (i) the orbit space carries a canonical orbifold structure, and (ii) there is a distinguished subgroup of the orbifold fundamental group, called the Weyl group, which is a Coxeter group. I will illustrate the theory with several explicit examples, including toric symplectic manifolds.

An important feature of our construction is that it is canonical and, unlike existing approaches, does not require any choice of Riemannian metric. Moreover, it extends from proper actions to a class of proper Lie groupoids called polar groupoids (or “polaroids”), and all notions and constructions are Morita invariant (so one can speak of polar stacks). I will focus mainly on group actions in order to keep the presentation accessible to those not familiar with Lie groupoids.

This talk is based on ongoing joint work with Marius Crainic (Utrecht) and David Martínez-Torres (Madrid).