Parallel sessions

Thursday & Friday

Algebra and Algebraic Geometry

at 7th floor of main building

Organizing committee: Ignacio López, Alvaro Ritatore

Scientific committee: Ignacio López, Pierre-Louis Montagard, Alvaro Ritatore

October 31st

Thursday

9:30 - 10:30 | Walter Ferrer (Universidad de la República)

Observable subgroups and generalizations

Motivated by the search of results on the linearization of Lie groups, the concept of observable subgroup H of an affine algebraic group G, appeared at the beginning of the 60s formulated initially as a theorem on the existence of a finite dimensional extension to G for every f.d. representation of H. It was observed that the concept of observability has a strong relation with many important results on geometric invariant theory such as the geometric structure of homogeneous spaces, the structure of invariant rational functions, the extension of characters, the finite generation of invariants, etc. In this talk I will describe some of the contributions of the integrants of the group of Algebra and algebraic geometry to the study of the concept of observability, introducing along the way new characterizations and generalizations in many directions. For example, we define the concept of observable actions on arbitrary affine varieties, and consider categorical formulations that allow the definition of what can be called categorical adjunctions.

10:30 - 11:00 Break

11:00 - 12:00 | Dalia Artenstein (Universidad de la República)

Nearly Frobenius Algebras

A characterization of Frobenius algebras is given by Abrams in [1] which states that A is a Frobenius algebra if and only if it has a coassociative comultiplication ∆ : A → A ⊗ A with counit, which is a map of A-bimodules. The notion of nearly Frobenius algebra is a weakening of this characterization, where we require a comultiplication ∆ : A → A ⊗ A that is a map of A-bimodules (coassociativity turns out to be a consequence of the above) without asking for a counit. This notion is motivated by the result proved in [2], which states that: the homology of the free loop space H∗(LM) has the structure of a Frobenius algebra without counit. The aim of the talk is to share some results obtained in the study of these algebras, including: families of examples, general constructions, relationships with the notion of separability and examples coming from Hochschild cohomology, among others.

[1] L. Abrams, Modules, Comodules, and Cotensor Products over Frobenius algebras. Journal of Algebra 219, pp. 201-213 (1999).

[2] R. Cohen and V. Godin. A Polarized View of String Topology. Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser. Vol. 308, Cambridge: Cambridge Univ. Press, pp. 127-154 (2004).

12:00 - 14:00 Lunch break

14:00 - 15:00 | Marco Pérez (Universidad de la República)

Families of cotorsion pairs induced by a Frobenius pair

Families of A Frobenius pair (X,w) is formed by two classes of objects X and w in an abelian category, with w ⊆ X, and such that:
-X is closed under direct summands, extensions and cokernels of monomorphisms,
- w is also closed under direct summands,
- the injective dimension of w relative to X is zero, and
- every object in X can be embedded into an object in w with cokernel in X.

The previous are sufficient conditions to construct right approximations for objects with finite X-resolution dimension by objects in X. Moreover, if (X,w) is strong (which means that the projective dimension of w relative to X is zero and that every object in X is the epimorphic image of an object in w with kernel in X) then we can also obtain an exact model structure whose homotopy category is equivalent to the stable category X / ~ (where two maps f and g with the same domain and codomain are equivalent if f-g factors through an object in w).

In this talk, we replace the strong condition in the previous definition by the possibility of approximating objects in the orthogonal complement of X by objects with finite w-resolution dimension. These new Frobenius pairs will be called w-complete. The idea is to explore which homological constructions and properties can be obtained from these Frobenius pairs, like for instance, relative cotorsion pairs and model structures formed by the classes of objects with finite X-resolution dimension and w-resolution dimension. Along the way, we will give examples in Gorenstein homological algebra, mention some open questions in the area of relative homological algebra, and propose alternative proofs of some well-known results about the completeness of certain cotorsion pairs.

This is a joint work in progress with Víctor Becerril (CCM-UNAM, Morelia) cotorsion pairs induced by a Frobenius pair

15:00 - 15:30 Break

15:30 - 16:30 | Gustavo Rama (Universidad de la República)

Orthogonal modular forms for O(5), paramodular forms, and congruences

We introduce orthogonal modular forms, particularly modular forms for the group O(5). Along with Dummigam, Pacetti, and Tornaría, combined with a work by Rösner and Weissauer, we proved that certain of these forms correspond to Siegel modular forms invariant under the paramodular group. We also proved several examples of Harder's conjecture, which relates Hecke eigenvalues of classical modular forms to Hecke eigenvalues of paramodular forms. Additionally, we proved a conjectured congruence by Buzzard and Golyshev between a modular form of weight 2 and level 61 and the non-lift paramodular form of weight 3 and level 61.

Together with Assaf, Ladd, Tornaría, and Voight, we computed databases of paramodular forms for weights (k, j) = (3, 0), (4, 0), (3, 2) and levels less than 1000. We search for new congruences of Harder type and of Buzzard, Golyshev type. We will show new examples of these congruences and provide proofs for some of them.

November 1st

Friday

9:30 - 10:30 | Eugenia Ellis (Universidad de la República)

Homotopy structures realizing algebraic $kk$-theory

Algebraic $kk$-theory, introduced by Cortinas-Thom, is a bivariant K-theory defined on the category of algebras over a commutative unital ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor $j:\mathrm{Alg}\to kk$ that is the universal excisive, homotopy invariant and matrix-stable homology theory. Moreover, we can recover Weibel's homotopy K-theory from $kk$ since we have $kk(\ell,A) =3D KH(A)$ for any algebra $A$. In this talk we will see that the category of algebras with fibrations the split surjections and weak equivalences the $kk$-equivalences is a category of fibrant objects, whose homotopy category is $kk$. Using this, we are able to construct a stable infinity category whose homotopy category is $kk$. This is a work in progress, joint with Emanuel Rodr\'iguez Cirone.

10:30 - 11:00 Break

11:00 - 12:00 | Adrien Dubouloz (Université de Bourgogne - Université de Poitiers)

Unipotent structures on projective varieties

A unipotent structure on an algebraic variety is a faithful action of a unipotent group with an open orbit. For projective varieties and abelian unipotent groups (that is, products of the additive group), many classification results are known, in particular via the so-called Hasset-Tschinkel correspondence between conjugacy classes of abelian unipotent structures on projective spaces and isomorphsim classes of certain artinian unital commutative algebras. In this talk, I will give an overview of new methods for tackling the non necessarily abelian case, with particular attention by way of illustration to the case of smooth Fano threefolds endowed with a unipotent structure for the Heisenberg group.

Image Processing, Probability and Statistics

at 7th floor of main building

Organizing committee: Mathias Bourel, Nicolas Frevenza, Pablo Musé

Scientific committee: Andrés Almansa, Jean-Marc Azaïs, Paola Bermolen, Pablo Musé

October 31st

Thursday

9:30 - 10:30 | Alain Rouault (Université Paris-Saclay)

Wschebor's Theorem and its descendants

In his seminal article of 1992, Mario Wschebor proved an almost sure convergence for the small increments of Brownian motion. Since then, numerous extensions were considered. In a first part I will present some of them, including fluctuations and large deviations. In a second part, I will propose a version for matrix-valued processes and free probability.

Joint work with José León and Catherine Donati-Martin.

10:30 - 11:00 Break

11:00 - 12:00 | Rafael Grompone (Université Paris-Saclay)

Wild Speculations about Associative Memory and Artificial Intelligence

This talk will consider some possible explanations for the current success of neural network methods. It will be speculated that their ability to implement associative memory may be the key factor. It will be argued that associative memory is a basic building block of natural intelligence, and as a consequence, current neural networks are indeed a legitimate part of artificial intelligence. However, full intelligence is likely to require additional capabilities, so further developments may be needed in the future to achieve mature artificial intelligence.

12:00 - 14:00 Lunch break

14:00 - 15:00 | Gabriele Facciolo (Université Paris-Saclay)

Low Latency Video Restoration: Enhancing MIMO Architectures

We begin by reviewing state-of-the-art video restoration networks and their tradeoffs in terms of PSNR and processing time, highlighting the challenges of achieving both high quality and low latency. MIMO (multiple input, multiple output) architectures have emerged as a promising solution, taking multiple input frames (a stack) generating multiple output frames per evaluation, and offering a favorable trade-off between quality and computational efficiency. However, in low-latency settings, where the number of future frames is limited, we identify two key issues: performance drops due to reduced temporal receptive field, especially at stack borders, and motion artifacts caused by stack transitions. To address these, we propose two enhancements: recurrence across MIMO stacks to boost the temporal receptive field and overlapping stacks to mitigate transition artifacts. These improvements are applicable to any MIMO architecture, resulting in better reconstruction accuracy and temporal consistency. Additionally, we introduce a benchmark with drone footage that exposes temporal consistency issues often missed by standard benchmarks.

15:00 - 15:30 Break

15:30 - 16:30 | Leonardo Moreno (Universidad de la República)

GROS: A General Robust Aggregation Strategy

A new, very general, robust procedure for combining estimators in metric spaces is introduced (GROS). The method is reminiscent of the well-known median of means, as described in (L. Devroye, M. Lerasle, G. Lugosi, and R. I. Oliveira, 2016). Initially, the sample is divided into K groups. Subsequently, an estimator is computed for each group. Finally, these K estimators are combined using a robust procedure. We prove that this estimator is sub-Gaussian and we get its break-down point, in the sense of Donoho. The robust procedure involves a minimization problem on a general metric space, but we show that the same (up to a constant) sub-Gaussianity is obtained if the minimization is taken over the sample, making GROS feasible in practice. The performance of GROS is evaluated through five simulation studies: the first one focuses on classification using k-means, the second one on the multi-armed bandit problem, the third one on the regression problem. The fourth one is the set estimation problem under a noisy model. We apply GROS to get a robust persistent diagram. Lastly, an application of robust estimation techniques to determine the home-range of Canis dingo in Australia is implemented.

This work is a collaboration with Emilien Joly and Alejandro Cholaquidis.

November 1st

Friday

9:30 - 10:30 | Valeria Goicochea (Universidad de la República)

Large deviations for the Peano phenomenon

The theory of large deviations is concerned with the study of the probabilities of rare events. In particular, in this talk we will discuss the study of large deviations for stochastic processes that result from perturbing a system of ordinary differential equations for which the Peano phenomenon occurs. These differential equations are characterised by a lack of uniqueness of solution. The study of large deviations for the stochastic processes resulting from such perturbation makes it possible to identify which of the infinite solutions are most ‘important’. In this work, which is not yet completed, we are generalising the results of [Bafico-Baldi, 1981; Gradinaru-Herrmann, 2001] for the case of autonomous differential equations where the function involved comes from a homogeneous potential.

From the Feynman-Kac equations, we will see that one can transform the study of large deviations for the stochastic processes of interest into the study of the exponential behaviour of the principal eigenfunction (ground state) of a Schrödinger operator. Moreover, we obtain exponential bounds for the ground state by means of probabilistic tools, finer than those obtained by [Carmona-Simon, 1981] for more general potentials.

10:30 - 11:00 Break

11:00 - 12:00 | Andrés Almansa (Unviersité Paris Cité)

Posterior Sampling in Imaging with learned priors: From Langevin to diffusion models

Inverse problems in imaging are most often ill-posed. Traditionally, the most common approach was to use optimization algorithms to maximize the posterior, i.e. select the most likely solution. More recently posterior sampling started to become more widespread in the imaging community, thus enabling to compute posterior means, uncertainty quantification, and optimal parameter selection.

In this talk, we explore recent techniques to perform posterior sampling when the likelihood is known explicitly, and the prior is only known implicitly via a denoising neural network that has been trained on a large collection of images. We show how to extend the Unadjusted Langevin Algorithm (ULA) to this particular setting leading to Plug & Play ULA. We explore the convergence properties of PnP-ULA, the crucial role of the stepsize, and its relationship with the smoothness of the prior and the likelihood. To relax stringent constraints on the stepsize, annealed Langevin algorithms have been proposed, which are tightly related to generative denoising diffusion probabilistic models (DDPM). The image prior that is implicit in these generative models can be adapted to perform posterior sampling, by a clever use of Gaussian approximations, with varying degrees of accuracy, like in Diffusion Posterior Sampling (DPS) and Pseudo-Inverse Guided Diffusion Models (PiGDM). We conclude with an application to blind deblurring, where DPS and PiGDM are used in combination with an Expectation Maximization algorithm to jointly estimate the unknown blur kernel, and sample sharp images from the posterior.

Dynamical Systems and Geometry

Room 101 of Instituto de Matemática Rafael Laguardia (IMERL)

Organizing committee: Sébastien Alvarez, Matilde Martinez
Scientific committee: Sébastien Alvarez, Matilde Martinez, Isabelle Liousse, Maxime Wolff

October 31st

Thursday

9:30 - 10:30 | Thierry Barbot (Avignon Université)

On transverse pairs of foliations

In 2001, S. Matsumoto and T. Tsuboi exhibited examples of pair of transverse foliations in the unit tangent bundle of a closed surface, each isotopic to the weak stable/unstable foliations of the geodesic flow, but such that the intersection foliation is not isotopic to the geodesic flow. In this talk, I will give some results for other pairs of Anosov foliations.This is a joint work with S. Fenley and R. Potrie.

10:30 - 11:00 | Break

11:00 - 12:00 | Juliana Xavier (Universidad de la República)

Indecomposability of Julia sets from a topological viewpoint

It is a longstanding question in one-dimensional complex dynamics if there exists a rational function whose Julia set is an indecomposable continuum. We explore this problem from a topological viewpoint: is it even possible for a branched covering of the sphere of degree d>1? This problem relates to another old conjecture about C^1 maps of the sphere S^2 of degree d>1, stating that the number of fixed points of the iterates $f^n$ should have an exponential growth rate.

The aim of this talk is to introduce the two problems, show how they interact and tell the little bits of truth that I know of.

12:00 - 14:00 | Lunch break

14:00 - 15:00 | Nancy Guelman (Universidad de la República)

Reversibility and Interval Exchange Transformations

An element f in a group G is reversible if it is conjugated in G to its own inverse. We will show that an interval exchange transformation that is reversible can be reversed by an finite (even) order element.

15:00 - 15:30 | Break

15:30 - 16:30 | Peter Haïssinsky (Université d'Aix Marseille)

Characterizations of Kleinian groups

In low dimension, it is expected that topological properties determine a natural geometry. In this spirit, several characterisations are conjectured for Kleinian groups, i.e., discrete subgroups of PSL(2,C). We will survey different methods that lead to their topological and dynamical characterisations, and point out their limits and the difficulties encountered in obtaining a complete answer.

November 1st

Friday

9:30 - 10:30 | Pierre-Antoine Guihéneuf (Sorbonne Université)

Rotation sets in genus at least 2

Rotation theory for dynamics on hyperbolic surfaces regained attention in the last few years thanks to the so-called "forcing theory" that was developed by Le Calvez and Tal about 10 years ago.

After defining the ergodic rotation set for homeomorphisms of closed surfaces, I will present a decomposition theorem for these sets in the case of surfaces of genus at least 2. This statement will be accompanied by a family album of examples.

Joint work with A. Garcia and P. Lessa.

10:30 - 11:00 | Break

11:00 - 12:00 | Jean-François Quint (Université de Montpellier)

Limit theorems for conditioned random matrix products

Let $g_1,\ldots,g_n,\ldots$ be iid random matrices in the group ${\rm GL}_d(\mathbb R)$. Suppose the first Lyapunov exponent is zero, that is, the averaged norm $\|g_n\cdots g_1\|^{\frac{1}{n}}$ almost surely goes to $1$ as $n$ goes to $\infty$. Under this assumption (as well as moment and irreducibility conditions), given a non zero vector $v$ in $\mathbb R^d$, we will study the first time when the random trajectory $g_1v,g_2g_1v,\ldots, g_n\cdots g_1 v, \ldots$ leaves the unit ball of $\mathbb R^d$. We will show that, if $v$ is small enough, the probability for this exit time to take the value $n$ is equivalent to $C(v)n^{-\frac{3}{2}}$, for some $C(v)>0$. This is a joint work with Ion Grama and Hui Xiao.