Seminars

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).

The min-max theory of the length/area functional in the space of hypersurfaces of a compact Riemannian manifold defines several numbers, called min-max widths, which are critical values of the functional. We will discuss situations where the knowledge of these numbers allow to recover, in full or in part, geometric properties of the manifold itself. This is joint work with F. Marques and A. Neves.

An old question steaming from Klein's Erlangen Program can be phrased
in modern terms as: Is a given geometric object uniquely determined by
its group of symmetries? The first part of this talk consists of an
introduction to the problem with some selected examples from outside
algebraic geometry.

In the second part of the talk we come to the setting of algebraic
geometry, where we show that, in general, the answer to the above
question is negative. After restricting the class of varieties we will
show an instance where the answer is affirmative. Indeed, we show that
complex affine toric surfaces are determined by the abstract group
structure of their regular automorphism groups in the category of
complex normal affine surfaces using properties of the Cremona group.

Let M be a properly embedded, connected, complete surface in R^3 with boundary a convex planar curve C, satisfying an elliptic equation H=f(H^2-K), where H and K are the mean and the Gauss curvature respectively – which we will refer to as Weingarten equation. In this talk, we discuss how the symmetries of C may induce symmetries of the whole M. When M is contained in one of the two halfspaces determined by C, we give sufficient conditions for M to inherit the symmetries of C. In particular, when M is vertically cylindrically bounded, we get that M is rotational if C is a circle. In the case in which the Weingarten equation is linear, we give a sufficient condition for such a surface to be contained in a halfspace. Both results are generalizations of results of Rosenberg and Sa Earp, for constant mean curvature surfaces, to the Weingarten setting. In particular, our results also recover and generalize the constant mean curvature case.

In this talk, we speak about our current results from [1] on the rank of Jacobian varieties of the family of algebraic curves of genus ≥ 1 defined by the affine equations $y^s=ax^r+b$  over a number field k, where r ≥ 2 and s ≥ 2 are fixed integers. Assuming the strong version of Lang’s conjecture on varieties of general type, we show that the Mordell-Weil rank of the Jacobian varieties of these curves is uniformly bounded. The proof proceeds by constructing a parameter space for curves in the family with a given number of rational points and analyzing the geometry of its fibers, which are shown to be complete intersection curves of increasing genus.

[1]: Sajad Salami, On the rank of Jacobian of the curves $y^s=ax^r+b$, accepted in J. Number Theory