Statistical Mirror Symmetry

This is a talk for the fourth edition of the Information Geometry and Affine Differential Geometry Conference (IGADG IV) at Chongqing University of Technology. There were a number of experts in complex differential geometry in the audience, and as such there is a noticeable difficulty spike about 13 minutes in when I start discussing complex geometry. But hopefully the first third of the talk is reasonably accessible even without that background.

Abstract: In complex and symplectic geometry, mirror symmetry is a duality between Calabi-Yau manifolds, in which two distinct spaces have closely related geometry. This idea has played an important role in string theory as well as enumerative algebraic geometry. In this talk, we will discuss a related duality known as statistical mirror symmetry. For any exponential family, this theory provides a relationship between two canonical Kähler geometries defined on the tangent bundle of the associated statistical manifold. Our focus will be the family of normal distributions, where statistical mirror symmetry gives a correspondence between the unit ball in ℂ2 and the Siegel-Jacobi space. Finally, we will discuss how this theory relates to Kähler-Ricci flow and give a conjectural application in number theory.

This was my first time recording a lecture in a single take and it was definitely a challenge to multi-task between reading my notes, clicking through slides and trying to speak clearly. I'm reasonably happy with how it turned out and recording was really good practice for the actual talk, but there are a few hiccups here and there.

Errata:
At 7:40 I meant to say that the variance of the first pair is much smaller than the second pair, but stated it backwards.
At 9:21, I referred to the eta parameters as sufficient statistics, which is incorrect. I meant to say "natural parameters."
At 12:37, the slide shows the Hessian metric using the variable theta. It should be written in terms of u, but it is a common habit to conflate the sufficient statistics with their expected values in this context.
At 35:03, it is more standard to refer to this function as the "modular discriminant" instead of "Ramanujan Delta function."


Chapters:
0:00 Introduction
1:15 The information geometry of the normal family
12:50 Construction of statistical mirror symmetry
16:20 The relationship to mirror symmetry for Calabi-Yau manifolds
23:02 The dual Kähler metric: complex hyperbolic space
27:30 The primal Kähler metric: The Siegel-Jacobi space
33:38 A recap of the story thus far
34:14 Automorphic forms, lifts, and an open question
39:17 Kähler-Ricci flow and anti-bisectional curvature
45:07 Artwork credits


This video was partially funded by Simons Collaboration Grant 849022 "Kähler-Ricci Flow and Optimal Transport." As a final note, you can find sheet music for the outro here: https://differentialgeometri.files.wo...


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