I will report on the ongoing project of classifying pointed Hopf algebras with finite Gelfand-Kirillov dimension.

A vast array of physical phenomena, ranging from the propagation of waves to the location of quantum particles, is dictated by the behavior of Laplace eigenfunctions. Because of this, it is crucial to understand how various measures of eigenfunction concentration respond to the background dynamics of the geodesic flow. In collaboration with J. Galkowski, we developed a framework to approach this problem that hinges on decomposing eigenfunctions into geodesic beams. In this talk, I will present these techniques and explain how to use them to obtain quantitative improvements on the standard estimates for the eigenfunction's pointwise behavior, Lp norms, and Weyl Laws. One consequence of this method is a quantitatively improved Weyl Law for the eigenvalue counting function on all product manifolds.

In the 1970’s Andrew Odlyzko proved good lower bounds for the discriminant of a number field. He also showed that his results could be sharpened by assuming the Generalized Riemann Hypothesis. Some years later Odlyzko suggested that it might be possible to do without GRH. I shall explain Odlyzko’s ideas and sketch how for number fields of reasonably small degree (say up to degree 11 or 12) one can indeed improve the lower known bounds by exploiting hypothetical violations of GRH. This is joint work with Karim Belabas, Francisco Diaz y Diaz and Salvador Reyes, extending unpublished results of Matías Atria.

Turbulence is a phenomenon of wide theoretical and practical interest and an area of research with intense current activity.

Mathematical modeling of turbulence relies, in an essential manner, on a thorough understanding of solutions of the Navier-Stokes equations at high Reynolds number.

A central question of the mathematical analysis of turbulent flows is whether inviscid dissipation actually occurs and, furthermore, whether there is dissipation of energy in the vanishing viscosity limit. Smooth solutions of the zero viscosity equations, also known as the Euler equations, conserve energy over time. However, according to turbulence theory, in turbulent regimes energy should be dissipated and, additionally, the energy dissipation rate should be non-positive in the vanishing viscosity limit. Clearly, the underlying flows cannot be smooth.

In 1949 Lars Onsager formulated what is now known as the Onsager Conjecture about the critical regularity which would allow for inviscid dissipation: solutions of the Euler equations with "more than a 1/3 derivative" (ie Holder continuous with exponent greater than 1/3) should be conservative. In contrast it should be possible to produce solutions with "less than 1/3 derivative" which dissipate energy.

During the last 10 to 12 years this area of research has seen a substantial amount of progress, culminating with the complete settling, in 2018, of the Onsager Conjecture. Complete? This is the subject of the talk

We shall study unique continuation properties (UCP) of solutions to some time evolution eq’s. We are interested in the following two questions:

(1) local : if \(u_1\), \(u_2\) are solutions of the eq. which agree in an open set Ω, do they agree in the whole domain?
(2) asymptotic at infinity : if \(u_1\), \(u_2\) are solutions of the eq. such that at two different times \(t_1\), \(t_2\)

$$ ||| u_1(·, t_j ) − u_2(·, t_j )||| < \infty,\quad j = 1, 2, $$ do they are equal in the whole domain? (\(|||\ ·\ |||\) represents an appropriate "norm") We shall concentrate on these questions for solutions of (i) the Kortewegde Vries eq., the Benjamin-Ono eq., (iii) the Intermediate Long Wave eq., (iv) the Camassa-Holm eq. and (v) related models. These are integrable models and the last three are non-local.

The family of classical spherical means \(A = \{A_t\}_t>0\) is given by: $$A_t f(x) = \int_{S^{d−1}}f(x-ty)d\sigma(y)$$ where \(d\sigma\) denotes the normalized surface measure on the unit sphere \(S^{d-1}\). E. M. Stein [1] (\(d\geq 3\)) and J. Bourgain [2] (\(d=2\)) proved that the spherical maximal function \(Sf(x):=\sup_{t>0}|A_{t}f(x)|\) defines a bounded operator on \(L^{p}(\mathbb{R}^{d})\) if and only if \(p>d/(d-1)\). Thus, for \(p\) in this range, we have \(\lim_{t\rightarrow 0}A_{t}f(x)=f(x)\) a.e. for all \(f\in L^{p}(\mathbb{R}^{d})\).

We consider the variation operator, for all \(1\leq r<\infty\), defined by, $$V_{r}A:=\sup_{N\in \mathbb{N}}\sup_{t_{1}<\ldots \lt t_{N}, t_{j}\in (0,\infty)} \bigg(\sum_{j=1}^{N-1}|A_{t_{j+1}}f(x)-A_{t_{j}}f(x)|^{r}\bigg)^{1/r}$$ Variation norms have received considerable attention in analysis as they are used to strengthen pointwise convergence results for families of operators. Variation norm inequalities have important consequences in ergodic theory and harmonic analysis. R. L. Jones, A. Seeger, and J. Wright proved [3] that \(V_{r}A\) is bounded in \(L^{p}(\mathbb{R}^{d})\) for all \(r \gt 2\) if \(d/(d-1) \lt p\leq 2d\). The endpoint result \(r=p/d\) was left open.

We show an endpoint result for \(V_{p/d}A\) in three and higher dimensions, and \(L^{p}\rightarrow L^{q}\) estimates for local and global \(r\)-variation operators associated to the family of spherical means. These can be understood as a strengthening of \(L^{p}\)-improving estimates for the spherical maximal function. The results imply associated sparse domination and consequent weighted inequalities.

Joint work with David Beltran, Richard Oberlin, Andres Seeger, and Betsy Stovall.

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[1] E. M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2176–2177.
[2] J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69–85.
[3] R. L. Jones, A. Seeger, and J. Wright, Strong variational and jump inequalities in harmonic analysis, Trans. Amer. Math. Soc. 360 (2008), 6711–6742.

Causal graphical models are statistical models represented by a directed acyclic graph in which each vertex stands for a random variable and a structural equation that generates it which is a function of its parents in the graph and an independent error.

I will start with a brief introduction of causal graphical models and of their use in the determination of identifiability and optimal estimation of the so-called average treatment effect (ATE) of static and personalized treatments in the presence of confounding variables.

I will then consider the problem of determining the best set of potential confounding variables at the stage of the design of a planned observational study aimed at assessing the population average causal effect of a point exposure personalized, i.e. dynamic, or static treatment. Given a tentative non-parametric graphical causal model, possibly including unobservable variables, the goal is to select the "best" set of observable covariates in the sense that it suffices to control for confounding under the model and it yields a non-parametric estimator of ATE with smallest variance. For studies without unobservables aimed at assessing the effect of a static point exposure we show that graphical rules recently derived for identifying optimal covariate adjustment sets in linear causal graphical models and treatment effects estimated via ordinary least squares also apply in the non-parametric setting. We further extend these results to personalized treatments. Moreover, we show that, in graphs with unobservable variables, but with at least one adjustment set fully observable, there exist adjustment sets that are optimal minimal (minimum), yielding non-parametric estimators with the smallest variance among those that control for observable adjustment sets that are minimal (of minimum cardinality). In addition, although a globally optimal adjustment set among observable adjustment sets does not always exist, we provide a sufficient condition for its existence. We provide polynomial time algorithms to compute the observable globally optimal (when it exists), optimal minimal, and optimal minimum adjustment sets. This is joint work with Ezequiel Smucler and Facundo Sapienza.

Since the groundbreaking contributions of Ruelle, the study of transfer operators has been one of the main tools to understand the ergodic theory of expanding maps, that is, discrete dynamical systems that locally expand distances. Questions on the existence of interesting invariant measures, as well the statistical properties of such dynamics system, as exponential decay of correlations and Central Limit Theorem, can be answered studying the spectral properties of the action of these operators on suitable spaces of functions. Using the method of atomic decomposition, we consider new Banach spaces of functions (that in some cases coincides with Besov spaces) that have a remarkably simple definition and allows us to obtain very general results on the quasi-compactness of the transfer operator acting in these spaces, even when the underlying phase space and expanding map are very irregular. Joint work with Alexander Arbieto (UFRJ-Brazil).

I will describe some recent results on the characterization of those polynomials that are nonnegative on a variety \(X\) in \(R^n\). In the first part of the talk I will explain why this is an interesting problem it turns out to have a wealth of applications ranging from nonconvex optimization to stochastic control. In the second part of the talk I will explain how this problem can be approached on algebraic curves and surfaces, presenting ongoing joint work with G. Blekherman (GA Tech), R. Sinn (U. Lepizig) and G.G. Smith (Queen's U).

In the 70´s Coxeter considered the 4-dimensional regular polytopes and used the so-called Petrie Polygons to obtain quotients of the polytopes that, while having all possible rotational symmetry, lack reflectional symmetry. He called these objects Twisted Honeycombs. Nowadays, objects with such symmetry properties are often called chiral. In this mini-course I will review Coxeter´s twisted honeycombs and explore a natural way to extend Coxeter´s work to polytope-like structures, in particular, we shall see some properties of so-called chiral skeletal polyhedra as well as of chiral abstract polytopes.

This will be an introductory course about lattices in (semi-simple) Lie groups and some famous theorems of Furstenberg, Margulis and Mostow (among others) about the "rigidity" of these groups. Time allowing, I'll discuss some recent theorems about actions of lattices on manifolds.

A Markov chain is a random process whose "memory"~of its own trajectory consists of its present state. Chains on finite spaces may be analysed through their spectrum. Under a crucial ``reversibility"~property that is satisfied in many applications, the spectrum is real and the difference between the two largest eigenvalues is the so-called spectral gap. We will present a simple comparison idea that often gives us the best possible estimates for several chain parameters in terms of the spectral gap. In particular, we re-obtain and improve upon several known results on hitting, meeting, and intersection times; return probabilities; and concentration inequalities for time averages. We then specialize to the graph setting, and obtain sharp inequalities in that setting. This course contains new and old results and is based on joint work with Yuval Peres.

Mathematics can help Art Historians and Art Conservators in studying and understanding art works, their manufacture process and their state of conservation. The presentation will review several instances of such collaborations, explaining the role of mathematics in each instance, and illustrating the approach with extensive documentation of the art works.

Conferencia grabada
Comenzando por un inocente concurso de números grandes iremos en esta charla descubriendo números tan enormes que retan a nuestra imaginación. Números más allá de lo que hubiéramos podido pensar y que nos llevan directamente a los límites de lo que las computadoras pueden y podrán hacer en el futuro.

I will report some recent advances on the classification of pointed Hopf algebras with abelian coradical and finite Gelfand-Kirillov dimension, with special emphasis on the presentation of new examples of Nichols algebras over abelian groups with finite growth. I will also discuss some open questions that would allow us to address the final answer to this problem.

Holomorphic correspondences are polynomial relations P(z,w)=0, which can be regarded as multi-valued self-maps of the Riemann sphere, this is implicit mapssending z to w. The iteration of such a multi-valued map generates a dynamical system on the Riemann sphere: dynamical system which generalises rational maps and finitely generated Kleinian groups. We consider a specific 1-(complex)parameter family of (2:2) correspondences F_a (introduced by S. Bullett and C. Penrose in 1994), which we describe dynamically. In particular, we show that for every parameter in a subset of the parameter plane called "the connectedness locus" and denoted by M_{\Gamma}, this family behaves as rational maps on a subset of the Riemann sphere and as the modular group on the complement: in other words, these correspondences are mating between the modular group and rational maps (in the family Per_1(1)). Moreover, we develop for this family of correspondences a complete dynamical theory which parallels the Douady-Hubbard theory of quadratic polynomials, and we show that M_{\Gamma} is homeomorphic to the parabolic Mandelbrot set M_1. This is joint work with S. Bullett (QMUL).

El anillo de operadores diferenciales, a pesar de no ser conmutativo, ha jugado un papel importante en el desarrollo del álgebra conmutativa en los últimos años. En esta charla nos enfocaremos en su uso para el estudio de singularidades.

The problem of determining dynamical properties from static information will be discussed. The setting of partially hyperbolic dynamics enjoys several such results in the conservative setting, but I would like to discuss a non-conservative instance.

El contenido de esta ponencia presenta algunos resultados preliminares del proyecto de investigación Anillo PIASOC180025 “Mujeres y matemáticas en Chile: sociología de un campo científico desde una perspectiva interdisciplinar y de género”. Específicamente, discute aspectos relativos a la construcción de subjetividades generizadas al interior de este campo científico. En base al análisis de 11 grupos focales y el diálogo con diferentes académicos/as e investigadores/as adscritos al campo de la matemática se distinguen modos de concebir sus posiciones desde la extrañeza y la otredad. En todos ellos, la perspectiva de género contribuye un marco crítico para abordar la práctica de la matemática como un proceso social, es por ello que el análisis se centra en las experiencias y trayectorias de quiénes la desarrollan, al estar marcadas por las características del campo social en que se despliegan.