Universidad de Colima
March 13 – 17, 2017
Dyadic Harmonic Analysis, Ma. Cristina Pereyra
Abstract: In 2000 I gave a series of lectures in Cuernavaca on Dyadic Harmonic Analysis. The thesis was that there is a parallel dyadic world to classical harmonic analysis where often the objects and theorems are easier to grasp. Since then a lot of striking results have been produced including Hytönen’s proof of the A_2 conjecture (2012), Lacey ‘s (2014) et al proof of the two-weight boundedness of the Hilbert transform, and Lerner’s introduction in 2013 of the powerful idea, later refined by Hytönen, Lacey, and Lerner himself, that many operators can be controlled (pointwise!) by sparse positive dyadic operators. This is a very active area of research. These dyadic techniques are well-known by harmonic analysts but they have been stretched to cover settings as diverse as graphs, fractals, compact Lie groups, smooth manifolds with doubling measure, Carnot-Caratheodory spaces, etc. The link between all these seemingly disconnected setting is that they can be viewed as spaces of homogeneous type, and one can associate to such spaces multiple dyadic structures, Haar bases, and even wavelets with Hölder regularity. In these lectures I want to give you a glimpse into this world starting in the humble setting of the real line.
Coloquio de Física y Matemáticas
Averaging and (Harmonic) Analysis, Ma. Cristina Pereyra
Abstract: Averages are everywhere, they help us understand all sort of data. There are several possible meaningful averages, we will review some of these and how they compare to each other, first for finite sets of numbers then for functions. In doing so we will revisit some important and useful inequalities. Averaging or smoothing is at the heart of harmonic analysis, I will try to briefly describe a few classical and not so classical instances of this connection.
Dyadic Harmonic Analysis and Weighted Inequalities, Ma. Cristina Pereyra
Abstract: In this talk we I will give you a tour of one and two-weight inequalities with emphasis on the dyadic theory and how it influences the classical Calderón-Zygmund theory using the Hilbert transform as a toy model.
University of New Mexico
April 17 – 21, 2017
Harmonic Analysis on Fractals, Ricardo A. Sáenz
Abstract: In this course we study the basic concepts of the theory of Analysis on fractals, as developed by Jun Kigami in the early 1990s. We work out the details of the construction of a harmonic structure on the Sierpinski Gasket, its harmonic functions, and its Laplacian. We discuss the algorithm to construction harmonic functions, as well as the decimation algorithm to construction the eigenfunctions of the Laplacian.
Undergraduate Advanced Calculus 2 lectures
The Hausdorff dimension, Ricardo A. Sáenz
There exist subsets in the plane that lead to nonsense when we try to measure them, like curves with “infinite length” or sets with “zero area”. In order to avoid such situations, we have to find the “right exponent” to measure them. This exponent is called the Hausdorff dimension of the set, discovered by Felix Hausdorff at the start of the XX century. In these lectures we define the Hausdorff dimension of a subset of , and we review its basic properties. We calculate explicitly the Hausdorff dimension of some examples, and state and prove Hutchinson’s theorem on the dimension of self-similar sets under certain conditions.
Department of Mathematics and Statistics Colloquium
How does a fractal vibrate?, Ricardo A. Sáenz
The work of Jun Kigami defines a Laplacian on certain selfsimilar sets, called post-critically finite, as a limit of normalized differences. This allows us to calculate explicitly approximations to its eigenfunctions, and, in some cases, it is even posible to construct them by a recursive interpolation process called the decimation method. In this talk I discuss the Laplacian on post-critically finite sets, its spectrum, and some of its properties, comparing them to the spectrum of the classical Laplacian on Euclidean domains.
Poisson integrals and maximal functions, Ricardo A. Sáenz
Given a harmonic structure on an post-critically finite set K, and its induced Laplacian, we can define the Poisson kernel as the fundamental solution to the Poisson equation on , with either Dirichlet or Neumann boundary conditions. The corresponding Poisson integrals satisfy many of the properties of the classical integrals on , in particular, the existence of boundary limits, both vertical and nontangential. In this talk we discuss these properties, and we also discuss the maximal functions of Poisson integrals and the Hardy spaces determined by them.