Seminars and Colloquia by Series

Tuesday, July 18, 2017 - 16:00 , Location: Skiles 005 , Ishwari Kunwar , Georgia Tech , Organizer: Michael Lacey
In this thesis, we introduce multilinear dyadic paraproducts and Haar multipliers, and discuss boundedness properties of these operators and their commutators with locally integrable functions in various settings. We also present pointwise domination of these operators by multilinear sparse operators, which we use to prove multilinear Bloom’s inequality for the commutators of multilinear Haar multipliers. Along the way, we provide several characterizations of dyadic BMO functions. 
Tuesday, July 18, 2017 - 16:00 , Location: Skiles 005 , Ishwari Kunwar , Georgia Tech , Organizer: Michael Lacey
In this thesis, we introduce multilinear dyadic paraproducts and Haar multipliers, and discuss boundedness properties of these operators and their commutators with locally integrable functions in various settings. We also present pointwise domination of these operators by multilinear sparse operators, which we use to prove multilinear Bloom’s inequality for the commutators of multilinear Haar multipliers. Along the way, we provide several characterizations of dyadic BMO functions. 
Monday, May 15, 2017 - 10:05 , Location: Skiles 006 , Mikel Viana , Georgia Tech , Organizer:
We first discuss the construction of whiskered invariant tori for fibered holomorphic dynamics using a Nash-Moser iteration. The results are in a-posteriori form.  The iterative procedure we present  has numerical applications (it lends itself to efficient numerical implementations) since it is not based on transformation theory but rather in applying corrections which ameliorate notably the curse of dimensionality. Then we will discuss results on compensated domains in a Banach space.
Monday, May 15, 2017 - 10:05 , Location: Skiles 006 , Mikel Viana , Georgia Tech , Organizer:
We first discuss the construction of whiskered invariant tori for fibered holomorphic dynamics using a Nash-Moser iteration. The results are in a-posteriori form.  The iterative procedure we present  has numerical applications (it lends itself to efficient numerical implementations) since it is not based on transformation theory but rather in applying corrections which ameliorate notably the curse of dimensionality. Then we will discuss results on compensated domains in a Banach space.
Thursday, April 27, 2017 - 10:00 , Location: Skiles 005 , Lei Zhang , Georgia Institute of Technology , Organizer: Lei Zhang
We present two distinct problems in the field of dynamical systems.I the first part, we cosider an atomic model of deposition of materials over a quasi-periodic medium, that is, a quasi-periodic version of the well-known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show that there are always perturbative expansions. We also prove a KAM theorem in a-posteriori form.In the second part, we consider a simple model of chemical reaction and present a numerical method calculating the invariant manifolds and their stable/unstable bundles based on parameterization method. 
Thursday, April 27, 2017 - 10:00 , Location: Skiles 005 , Lei Zhang , Georgia Institute of Technology , Organizer: Lei Zhang
We present two distinct problems in the field of dynamical systems.I the first part, we cosider an atomic model of deposition of materials over a quasi-periodic medium, that is, a quasi-periodic version of the well-known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show that there are always perturbative expansions. We also prove a KAM theorem in a-posteriori form.In the second part, we consider a simple model of chemical reaction and present a numerical method calculating the invariant manifolds and their stable/unstable bundles based on parameterization method. 
Wednesday, April 26, 2017 - 09:00 , Location: Skiles 005 , Hagop Tossounian , Georgia Tech , hagopt87@gatech.edu , Organizer: Hagop Tossounian
Kinetic theory is the branch of mathematical physics that studies the motion of gas particles that undergo collisions. A central theme is the study of systems out of equilibrium and approach of equilibrium, especially in the context of Boltzmann's equation. In this talk I will present Mark Kac's stochastic N-particle model, briefly show its connection to Boltzmann's equation, and present known and new results about the rate of approach to equilibrium, and about a finite-reservoir realization of an ideal thermostat.
Wednesday, April 26, 2017 - 09:00 , Location: Skiles 005 , Hagop Tossounian , Georgia Tech , hagopt87@gatech.edu , Organizer: Hagop Tossounian
Kinetic theory is the branch of mathematical physics that studies the motion of gas particles that undergo collisions. A central theme is the study of systems out of equilibrium and approach of equilibrium, especially in the context of Boltzmann's equation. In this talk I will present Mark Kac's stochastic N-particle model, briefly show its connection to Boltzmann's equation, and present known and new results about the rate of approach to equilibrium, and about a finite-reservoir realization of an ideal thermostat.
Wednesday, April 12, 2017 - 14:30 , Location: Skiles 114 , Yanxi Hou , Georgia Institute of Technology , yhou44@gatech.edu , Organizer: Yanxi Hou
This thesis addresses asymptotic behaviors and statistical inference methods for several newly proposed risk measures, including relative risk and conditional value-at-risk. These risk metrics are intended to measure the tail risks and/or systemic risk in financial markets. We consider conditional Value-at-Risk based on a linear regression model. We extend the assumptions on predictors and errors of the model, which make the model more flexible for the financial data. We then consider a relative risk measure based on a benchmark variable. The relative risk measure is proposed as a monitoring index for systemic risk of financial system. We also propose a new tail dependence measure based on the limit of conditional Kendall’s tau. The new tail dependence can be used to distinguish between the asymptotic independence and dependence in extreme value theory. For asymptotic results of these measures, we derive both normal and Chi-squared approximations. These approximations are a basis for inference methods. For normal approximation, the asymptotic variances are too complicated to estimate due to the complex forms of risk measures. Quantifying uncertainty is a practical and important issue in risk management. We propose several empirical likelihood methods to construct interval estimation based on Chi-squared approximation.
Wednesday, April 12, 2017 - 14:30 , Location: Skiles 114 , Yanxi Hou , Georgia Institute of Technology , yhou44@gatech.edu , Organizer: Yanxi Hou
This thesis addresses asymptotic behaviors and statistical inference methods for several newly proposed risk measures, including relative risk and conditional value-at-risk. These risk metrics are intended to measure the tail risks and/or systemic risk in financial markets. We consider conditional Value-at-Risk based on a linear regression model. We extend the assumptions on predictors and errors of the model, which make the model more flexible for the financial data. We then consider a relative risk measure based on a benchmark variable. The relative risk measure is proposed as a monitoring index for systemic risk of financial system. We also propose a new tail dependence measure based on the limit of conditional Kendall’s tau. The new tail dependence can be used to distinguish between the asymptotic independence and dependence in extreme value theory. For asymptotic results of these measures, we derive both normal and Chi-squared approximations. These approximations are a basis for inference methods. For normal approximation, the asymptotic variances are too complicated to estimate due to the complex forms of risk measures. Quantifying uncertainty is a practical and important issue in risk management. We propose several empirical likelihood methods to construct interval estimation based on Chi-squared approximation.

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