October 28th
At the Large Auditorium of Edificio polifuncional José Luis Massera
14:00 - 14:15
Welcome
14:15 - 16:00
14:15 Brief history of mathematics in Uruguay - Roberto Markarian
14:45 Trajectory of the IFUMI - Claude Cibils
15:15 Testimonies and messages from Chichi León, María Simon, Gregory Randall and Walter Ferrer
15:45 Transatlantic messages
16:00 - 17:00
Mathematics through the eyes of children, Mathias Bourel and Mariana Haim with the participation of the Headmistress of the Lycée français Supervielle: Christelle Riom
17:00 - 18:00
17:00 Directeur du Bureau CNRS Amérique du Sud - Liviu Petru Nicu
17:10 Directeur scientifique adjoint de l’INSMI-CNRS - Frédéric Hérau
17:20 Director del Pedeciba - David Gonzalez
17:30 Vicerrector de la Udelar - Álvaro Mombrú
17:40 Ambassadeur de France en Uruguay - Jean-Paul Seytre
17:50 Ministro de Education y Cultura - Pablo da Silveira
With the participation of Laurence Arnoux (conseillière coopération et actions culturelles de l’Ambassade de France), Pablo Ezzati (decano de la Facultad de Ingenieria), Mónica Marin (Decana de la Facultad de Ciencias) y Maria Simon
Video Lab Del Plata Con Subtitulos.mp4Video by French colleagues
19:00
October 29th
Room 101 of the IMERL - Instituto de Matemática y Estadística Rafael Laguardia
9:00 - 9:30
9:00 Presentation of the IRL Jean-Christophe Yoccoz at Rio de Janeiro, Claudio Landim
9:15 Presentation of the IRL Laboratorio internacional Solomon Lefchetz at México City, Geronimo Uribe-Bravo
9:30 - 11:40
9:30 (40'+5) Hélène Eynard-Bontemps (France - Chili, Université Grenoble Alpes)
By « smooth », some people mean $C^1$, others $C^\infty$… In frenchman calculus, Newtonian mechanics or Riemannian geometry, $C^2$ seems more natural. Dynamical systems overflow with instances of problems and techniques that are highly sensitive to the regularity one considers, often for deep and interesting reasons. I will illustrate this claim with concrete examples drawn from one-dimensional dynamics, from the widely known Denjoy Theorem to the less known jewel that is Nancy Kopell’s Lemma.
10:20 - 10:50 - Coffee Break
10:50 (40'+5) Marina Gardella (Uruguay - France - Brazil, IMPA)
Images serve as potent information vectors, conveying a wealth of data and insights through visual representations. Their importance in various domains cannot be overstated, as they offer unique advantages for communication, understanding, and documentation. In an era characterized by the pervasive influence of digital imagery, image forensics represents a vital discipline that addresses the pressing need to uphold the veracity and trustworthiness of digital visual content.
Images are naturally endowed with a fingerprint, embedded during the image formation process. Indeed, the creation of a digital image, spanning from its acquisition at the camera sensor to its final storage, imprints distinct artifacts that serve as a unique signature. Such cues allow forgery detection. Indeed, though nowadays manipulations have the capability to achieve a high degree of visual fidelity, they concurrently introduce alterations to the intrinsic structure of the image. Such disruptions in the inherent fingerprint are exploited by most forgery detection methods to spot tampered regions.
In this talk, we will go through the main steps of the image formation pipeline and show the traces that are left by each of them. We will then see how those can be used as forensic evidence for image forgery detection and discuss some classic methods built upon them. Finally, we shall present real applications.
11:45 - 12:15
11:50 Presentation of the IRL Pacific Institute for Mathematical Sciences - Özgür Yilmaz
12:05 Presentation of the IRL Centre des Recherches Mathématiques at Montreal - Emmanuel Royer
12:20 - 14:00
Lunch Break
14:00 - 16:00
14:00 Presentation of the IRN: Instituto Franco-Uruguayo de Física at Montevideo, Javier Brum
14:15 General audience lecture (40'+5): Marcela Peláez (Universidad de la República)
The strong nuclear interaction is responsible for binding protons and neutrons to form atomic nuclei, playing a crucial role in the stability of nuclei and, consequently, matter as we know it. This interaction is highly complex because, at a fundamental level, it acts between particles known as quarks, which cannot be isolated. Two of the most significant open questions in theoretical physics relate to this interaction: why these particles remain confined in nature, and how the mass of protons is generated from such small quark masses. Understanding how this interaction works from first principles is the primary goal of our research group. In particular, we work with a model that includes a mass term for the mediators of the interaction. This massive behavior has been identified in numerical simulations and, theoretically, allows for analytical treatments in an energy regime where such results are typically scarce. These analytical approaches provide deeper insights into the characteristics of the infrared regime. In this talk, I will present our recent progress in this line of research, as well as the challenges we are currently addressing.
15:00 - 15:15 Break
15:15 General audience lecture (40'+5): Denisse Sciamarella (IRL Instituto Franco-Argentino de estudios sobre el clima y sus impactos at Buenos Aires)
Henri Poincaré pioneered the study of dynamical systems through their topological properties. In phase space—where all possible states of a system are represented—topology determines whether two dynamics are qualitatively identical and if a model accurately reflects observed or simulated time series. While knot theory has long been used to categorize chaotic behavior in three-dimensional systems, it falls short in higher dimensions, where knots become untangled. The templex is a construct designed to extract not only the topology of a structure in phase space beyond three dimensions but also the topologically distinct paths around that structure, following the direction of time. By decomposing a reduced templex, every dynamical process, regardless of its complexity, can be represented as a combination of elementary oscillating and switching units, just as every computational code is ultimately a sequence of zeros and ones.
October30th
Room 101 of the IMERL - Instituto de Matemática y Estadística Rafael Laguardia
9:00 - 11:15
9:00 (50'+5) Jean-Michel Morel (City University of Hong Kong) online
In this general-purpose talk, I'll first describe the general machine learning procedure as the association of four objects: a dataset of input-output pairs representing a ground truth, a network structure, a loss function and an optimizer. The result of the optimization procedure is what we properly call a neural network, namely a computational structure endowed with learned weights. This machine learning procedure raises many issues. Some of the main ones are:
- variability of the result depending on stochastic optimization and on stochastic initialization,
- overfitting (to the dataset),
- domain dependence of the result.
To discuss the first issue, I’ll compare the role of noise in several applications of neural networks with the role of noise in Shannon’s theory of communication. Using two recent general theorems on optimization based machine learning and several practical examples, I'll give a formalization of overfitting (and domain dependence). I’ll deduce that neural networks are fundamentally good at denoising data, and that, according to Shannon’s theory, their useful output might not be contained in their proper output, but in the difference between input and output.
9:55 - 10:20 Break
10:20 (50'+5) Pablo Ferrari (Universidad de Buenos Aires, Argentina)
A Poisson line process is a random set of straight lines contained in the plane, as the image of the map $(x,v)\mapsto (x+vt)_{t\in\mathbb{R}}$, for each point $(x,v)$ of a Poisson process in the space-velocity plane. By associating a step with each line of the process, a random surface is obtained. The diffusive rescaling of the surface converges to the Lévy-Chentsov field, a classical Gaussian field. The process is also called Brownian motion with several parameters because a cut of the surface with a perpendicular plane, reveals Brownian motion.
A hard rod is an interval contained in $\mathbb{R}$ that travels ballistically until it collides with another hard rod, at which point they interchange positions. By associating each line with the ballistic displacement of a hard rod and associating surface steps with hard rod jumps, we obtain the hydrodynamic limits of the hard rods in the Euler and diffusive scalings. The main tools are law of large numbers and central limit theorems for Poisson processes.
Joint work with Chiara Franceschini, Dante Grevino, Herbert Spohn, and Stefano Olla.
https://www.doi.org/10.21711/217504322023/em387
https://arxiv.org/abs/2305.13037
11:15 - 11:30
11:15 Presentation of the IRL: Centro de Modelamiento Matematico at Santiago de Chile, Hector Ramirez
11:30 - 14:00
Lunch Break
14:00 - 17:10
14:00 (50'+5) Yves de Cornulier (CNRS - Université de Nantes, France)
Although groups are defined as abstract algebraic structures, historically, they were initially introduced as acting groups (finite permutation groups). Indeed, many specific groups are introduced as acting groups, such as automorphism groups of various structures or subgroups therein. However, some groups are defined roughly as acting groups, but not exactly, since group elements act "almost everywhere". In algebraic geometry, one main example is given by the Cremona group of birational self-transformations of the plane. In dynamics, there are groups that are defined as acting "piecewise" (e.g., piecewise isometries with polygonal pieces) without specifying how they act on the boundary of the pieces. How can we obtain genuine actions of these groups that still capture these "almost" actions? The talk will discuss all this in more detail, mentioning recent results along the lines of the above question
15:00 (50'+5) Claude Cibils (Université de Montpellier, France)
We will begin by considering examples of algebras over the complex numbers: matrix algebras, group algebras, quiver algebras and quotients of them. Then we will have an overview of a powerful tool, Hochschild cohomology of algebras. Several examples of computations will be given. The global dimension of an algebra is a measure of how intricate is an algebra. Dieter Happel proved that if the global dimension is finite, then the same happens with the Hochschild cohomology. The converse is false unfortunately. Then the question raised by Happel has a negative answer. With Marcelo Lanzilotta, Eduardo Marcos and Andrea Solotar, we manage a new cohomological tool, using the Auslander-Reiten translation and tao tilting theory. This tao Hochschild cohomology is totally adapted for algebras with infinite+ global dimension. For a quiver with less than two vertices all infinite global dimension algebras are infinite+ global dimension. We suspect that this is always true, answering positively to the tau analog oh Happel's question.
16:00 - 16:15 Break
16:15 (50'+5) Alejandro Passeggi (Universidad de la República, Uruguay)
Chaos theory has emerged from certain differential equations in physics and engineering, with two of the most fundamental examples being the three-body problem in the Hamiltonian framework and the Van der Pol equation in the dissipative setting.
Despite significant conceptual advances and the development of clear mathematical models, the theory is still far from being readily applicable to specific systems in applications. Rigorous proofs of chaos are extremely rare in such systems, and when they do exist, they require highly detailed analysis of the particular system and restrict the parameters to very narrow conditions. On the other hand, there are countless studies suggesting chaos through numerical simulations, and from a theoretical perspective, we have a “generic” theory for the existence of chaos, but many families of systems arising from applications tend to escape this generic framework.
Poincaré recognized early on that a two-dimensional topological theory of chaos was necessary to describe these differential equations. Since then, a theory has evolved and is now known as Topological Surface Dynamics.
In this talk, we will summarize this theory and present recent results that intend to transform the previously established topological conditions—implying annular chaos—into more testable concepts for specific systems. We will show how these conditions allow computer-assisted proofs of chaos for well-known analytical maps, such as the conservative and dissipative standard maps, along with their variations. Finally, we will discuss how these criteria could be applied to the three-body problem and more general Hamiltonian systems.
Organizing committee: Ignacio López, Alvaro Ritatore
Scientific committee: Ignacio López, Pierre-Louis Montagard, Alvaro Ritatore
October 31st
9:30 - 10:30 | Walter Ferrer (Universidad de la República)
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)
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 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)
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
9:30 - 10:30 | Eugenia Ellis (Universidad de la República)
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)
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.
Organizing committee: Mathias Bourel, Nicolas Frevenza, Pablo Musé
Scientific committee: Andrés Almansa, Jean-Marc Azaïs, Paola Bermolen, Pablo Musé
October 31st
9:30 - 10:30 | Alain Rouault (Université Paris-Saclay)
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)
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)
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)
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
9:30 - 10:30 | Valeria Goicochea (Universidad de la República)
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)
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.
October 31st
9:30 - 10:30 | Thierry Barbot (Avignon Université)
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)
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)
15:00 - 15:30 | Break
15:30 - 16:30 | Peter Haïssinsky (Université d'Aix Marseille)
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
9:30 - 10:30 | Pierre-Antoine Guihéneuf (Sorbonne Université)
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)
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.
November 1st
12:00 - Closing Parrillada
Universidad de la República
Julio Herrera y Reissig 565
