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ID de reunión: 838 5291 8034
Programa
Martes 31 de agosto de 2021
- Rodrigo Bañuelos
Universidad Purdue, EUA
10:00am (GTM -5)Conditional Expectation, Singular Integrals and Fourier Multipliers.
Resumen: In his 1983 research announcement "Some results in harmonic analysis in ℝn, for n → ∞" (Bulletin of the AMS), E.M. Stein announced new Lp estimates which do not depended on the dimension n for some classical operators in harmonic analysis. In this paper he also raised the following question and made a remarkably insightful statement:
"The results above raise the following question. Can one find an appropriate infinite-dimensional formulation of (that part of) harmonic analysis in ℝn, which displays in a natural way the above uniformity in n? A related question is to study the limit as n → ∞ of the above results, insofar as such limits may have a meaning. One might guess that a further understanding of these questions would involve, among other things, notions from probability theory: i.e. Brownian motion and possibly some variant of the central limit theorem."
This talk describes a general class of singular integrals and Fourier multipliers obtained from conditional expectations of transformations of stochastic integrals. From this, probabilistic techniques give norm bounds that are not only universal in terms of the geometry of the ambient space (including dimension) but in many important cases are sharp, or near sharp. To illustrate the techniques, we describe the solution of a 90+ year-old problem concerning the lp-norm of the discrete Hilbert transform and address similar problems for more general discrete Calderón-Zygmund operators.
- David Nualart
Universidad de Kansas, EUA
11:30am (GTM -5)
Spatial ergodicity and central limit theorems for the stochastic heat equation.
Resumen: Consider the multidimensional stochastic heat equation driven by a noise which is white in time and it has an homogeneous spatial covariance. In this talk we will review some recent results on the spatial ergodicity of the solution and on Gaussian fluctuations of spatial averages. We will show that, combining Malliavin calculus with Stein’s method for normal approximations, one can establish quantitative versions of the central limit theorem for spatial averages.
Miércoles 1 de septiembre de 2021
- Anton Wakolbinger
Universidad Goethe de Frankfurt, Alemania
10:00am (GTM -5)Evolving genealogies for branching populations under selection and competition.
Resumen: For a continuous state branching process with type-dependent selection and density-dependent competition, we present a "lookdown" description which admits the joint evolution of population size, type configurations and genealogies as the unique strong solution of a system of SDE’s, driven by Brownian and Poisson sources of randomness. A projection of this "fine model" gives the joint evolution of population size and symmetrized type configurations and genealogies, i.e. marked distance matrix distributions, which by Kurtz' Markov mapping theorem can be characterized in terms of a well-posed martingale problem.
The talk is based on joint work (https://arxiv.org/abs/2106.16201 ) with Airam Blancas (Mexico City), Stephan Gufler (Frankfurt), Sandra Kliem (Leipzig) and Viet Chi Tran (Paris).
- Víctor Pérez-Abreu
México
11:30am (GTM -5)Eigenvalues processes: An overview.
Resuemen: The Dyson-Brownian motion is the stochastic process of the eigenvalues of a symmetric or Hermitian matrix processes which entries are constructed from independent one-dimensional Brownian motions, and was first studied in 1962 by F. Dyson. Such a process is governed by a set of stochastic differential equations which diffusion coefficient is not smooth due to the important non-colliding property of the eigenvalues.
In this talk, we will review old and recent results of extensions to other matrix processes.
Jueves 2 de septiembre de 2021
- Georges Zaccour
HEC Montreal, Canadá
10:00am (GTM -5)Sustainability of Cooperation in Dynamic Games Played over Event Trees.
Resumen: Many problems in economics, engineering and management science have the following three features in common: (a) They involve a few agents (players) who have interdependent payoffs. (b) The agents cooperate or compete repeatedly over time, and the problem involves an accumulation process, e.g., production capacity, pollution stock. (c) Some of the parameter values are uncertain. A natural framework to deal with such problems is the theory of dynamic games played over event trees (DGPETs).
If the players decide to coordinate their strategies by signing a long-term contract, then they must ensure that all players follow their cooperative commitments as time goes by.
In this talk, I discuss different approaches to sustain cooperation over time in DGPETs.
- Alain Bensoussan
Universidad de Texas en Dallas, EUA
4:00 pm (GTM -5)New Mathematical Problems Of Management Science.
Resumen: Management Science has always been a source of innovation for applied mathematics. The type of problems is different from those originated from physics or mechanics, although some intersection exists. It does not mean it is easier mathematics.
In this talk, I discuss the evolution of stochastic control in the recent years, in relation with new problems in management science. This trend is different from that generated by Financial Engineering in the nineties. It is more diverse. An important aspect is the fact that an environment of numerous players, as in social networks, affects decisions. The relevant breakthrough in stochastic control is Mean Field Games, MFG. A similar theory, Mean Field Control develops the right mathematical apparatus to handle risk management. I will explain how MFG provides meaningful methods for capacity management.
If time permits, I will present new stochastic control problems in the context of management of corporations. These problems are linked to the commitment of managers or shareholders.
Machine Learning cannot be ignored. I will review the concepts and the connection with optimization and control theory. I will discuss how to make use of it to solve mathematical problems of management science.