- titulo:Diversity of statistical behavior in dynamical systems
abstract:For chaotic dynamical systems, it is unfeasible to compute long-term orbits precisely. Nevertheless, we may be able to describe the statistics of orbits, that is, to compute how often an orbit will visit a prescribed region of the phase space. Different orbits may or may not follow different statistics. I will explain how to measure the statistical diversity of a dynamical system. This diversity is called emergence, is independent of the traditional notions of chaos. I will begin the talk by discussing classic problems of discretization of metric spaces and measures. Then I will apply these ideas to dynamics and define two forms of emergence. I will present several examples, culminating with new dynamical systems for which emergence is as large as we could possibly hope for. This talk is based on joint work with Pierre Berger (Paris).
- titulo:Herramientas de geometría algebraica en biología de sistemas
abstract:Motivaré e introduciré algunos métodos y conceptos de la geometría algebraica que se están utilizando en los últimos años para analizar modelos estándar en biología molecular. La teoría algebraica de los sistemas de reacciones químicas tiene como objetivo comprender su comportamiento dinámico aprovechando la estructura algebraica inherente en las ecuaciones cinéticas, y no necesita la determinación de los parámetros a priori, que puede ser teórica o prácticamente imposible. También señalaré algunos de los desafíos matemáticos que surgen de esta aplicación.
- titulo:Entropy methods and sharp convergence of Markov Chains.
abstract:We describe how entropy methods can be used to derive quantitative versions of various scaling limits of Markov chains. We will focus on the the description of non-equilibrium states of interacting particle systems.
- titulo:Measuring the complexity of countable objects
abstract:Computability theory is the sub-area of mathematical logic that studies ways to measure the complexity of objects, constructions, theorems, and mathematical proofs related to countably infinite objects. On one hand, the natural objects seem to be linearly ordered from simpler to more complex, while, on the other hand the general objects are ordered in a chaotic way. This dichotomy between natural objects and objects in general is hard to study mathematically, as we don't have a formal definition of "natural object." The objective of this talk is to introduce Martin's conjecture (open for more than 40 years) and see how it explains this dichotomy.
abstract: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.
- titulo:Unconditional discriminant lower bounds exploiting violations of the generalized riemann hypothesis
abstract:|
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.
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,
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 \(r>2\) if \(d/(d-1)<p\leq 2d\). 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.
- titulo:Characterizations of Nonnegative polynomials of some varieties
abstract:|
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).
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.
- titulo:Transfer operators and atomic decomposition
abstract: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).
