Seminars and Colloquia by Series

Iterated Jackknives, Two-Sided Variance Inequalities, and $\Phi$-Entropy

Series
Stochastics Seminar
Time
Thursday, March 26, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian HoudréGeorgia Institute of Technology

In analogy with the Gaussian setting, we provide, at first, inequalities on the variance of a function of $n$ independent random variables generalizing results obtained for i.i.d. ones. In particular, we obtain various upper and lower bounds on this variance, via the iterated Jackknife statistics, which can be considered as generalizations of the Efron-Stein inequality. Relations with Hoeffding decomposition are then presented. Finally, the case of the $\Phi$-entropy is also considered.

Joint work with O. Bousquet from Google Brain Team.

Heat semigroup approach to isoperimetric inequalities in metric measure spaces

Series
Stochastics Seminar
Time
Thursday, January 30, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Patricia Alonso-RuizTexas A&M University

The classical isoperimetric problem consists in finding among all sets with the same volume (measure) the one that minimizes the surface area (perimeter measure). In the Euclidean case, balls are known to solve this problem. To formulate the isoperimetric problem, or an isoperimetric inequality, in more general settings, requires in particular a good notion of perimeter measure.

The starting point of this talk will be a characterization of sets of finite perimeter original to Ledoux that involves the heat semigroup associated to a given stochastic process in the space. This approach put in connection isoperimetric problems and functions of bounded variation (BV) via heat semigroups, and we will extend these ideas to develop a natural definition of BV functions and sets of finite perimeter on metric measure spaces. In particular, we will obtain corresponding isoperimetric inequalies in this setting.

The main assumption on the underlying space will be a non-negative curvature type condition that we call weak Bakry-Émery and is satisfied in many examples of interest, also in fractals such as (infinite) Sierpinski gaskets and carpets. The results are part of joint work with F. Baudoin, L. Chen, L. Rogers, N. Shanmugalingam and A. Teplyaev.

Probabilistic approach to Bourgain's hyperplane conjecture

Series
Stochastics Seminar
Time
Thursday, January 16, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Arnaud MarsigliettiUniversity of Florida

The hyperplane conjecture, raised by Bourgain in 1986, is a major unsolved problem in high-dimensional geometry. It states that every convex set of volume 1 in the Euclidean space has a section that is lower bounded away from 0 uniformly over the dimension. We will present a probabilistic approach to the conjecture. 

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