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Department Colloquium: Geometric central limit theorems on non-smooth spaces

Event Type
Lecture
Sponsor
Department of Mathematics
Location
245 Altgeld Hall
Date
Apr 25, 2024   2:30 pm   Please note the unusual time.
Speaker
Ezra Miller (Duke University)
Contact
Alexander Yong
E-Mail
ayong@illinois.edu
Views
27
Originating Calendar
General Events - Department of Mathematics

The central limit theorem (CLT) is commonly thought of as
  occurring on the real line, or in multivariate form on a
  real vector space.  Motivated by statistical applications
  involving nonlinear data, such as angles or phylogenetic
  trees, the past twenty years have seen CLTs proved for
  Fr├ęchet means on manifolds and on certain examples of
  singular spaces built from flat pieces glued together in
  combinatorial ways.  These CLTs reduce to the linear case
  by tangent space approximation or by gluing.  What should a
  CLT look like on general non-smooth spaces, where tangent
  spaces are not linear and no combinatorial gluing or flat
  pieces are available?  Answering this question involves
  figuring out appropriate classes of spaces and measures,
  correct analogues of Gaussian random variables, how the
  geometry of the space (think "curvature") is reflected in
  the limiting distribution, and generally how the geometry
  of sampling from measures on singular spaces behaves.  This
  talk provides an overview of these answers, starting with a
  review of the usual linear CLT and its generalization to
  smooth manifolds, viewed through a lens that casts the
  singular CLT as a natural outgrowth, and concluding with
  how this investigation opens gateways to further advances
  in geometric probability, topology, and statistics.  Joint
  work with Jonathan Mattingly and Do Tran.

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