# Simons Collaboration on Algorithms and Geometry Monthly Meeting, November 2019

Date & Time

#### Agenda

 9:30 - 10:15 AM Breakfast 10:15 - 11:15 AM Jeff Cheeger, Combinatorial formulas via analysis on geometric realizations – Part I 11:15 - 11:30 AM Break 11:30 - 12:30 PM Jeff Cheeger, Combinatorial formulas via analysis on geometric realizations – Part II 12:30 - 2:00 PM Lunch 2:00 - 3:00 PM Ronen Eldan, A new approach to concentration inequalities for Boolean functions, and an isoperimetric inequality conjectured by Talagrand – Part I 3:00 - 3:15 PM Break 3:15 - 4:15 PM Ronen Eldan, A new approach to concentration inequalities for Boolean functions, and an isoperimetric inequality conjectured by Talagrand – Part II
• Abstracts

Jeff Cheeger
Combinatorial formulas via analysis on geometric realizations

A compact manifold $$X$$ equipped with a triangulation, $$T$$, has a canonical geometric realization $$X,T,h(T))$$ which is built out of regular flat simplices, all of whose edges have the same length, say $$1$$. Since simplicial automorphisms of $$(X,T)$$ correspond to isometries of $$(X,T,h(T))$$ any analytic or geometric invariant of $$(X,T, h(T))$$ provides a combinatorial invariant of $$(X,T)$$. We will discuss this general principle in the context of the classical problem (circa 1955) of finding a local combinatorial formula for the signature $${\rm Sig}(X^{4k})$$ of a compact oriented $$4k$$-manifold $$X^{4k}$$ with triangulation $$T$$. Here, $${\rm Sig}(X^{4k})$$ is certain integer valued topological invariant of $$X^{4k}$$ which changes sign under change of orientation. A local formula is one which computes some measurement of $$T$$ at every vertex $$v$$ and then adds up the results. Corresponding local formulas for $${\rm Sig}(X^{4k})$$ involving a certain polynomial in the curvature tensor are well known if in place of $$T$$, $$X^{4k}$$ is equipped with a smooth riemannian metric $$g$$. The challenge is to extend a methodology which works for $$(X^{4k},g)$$ to the case of $$(X^{4k},T, h(T))$$ in which the canonical metric $$h(T)$$ is only piecewise smooth (indeed, piecewise flat). As such, $$h(T)$$ does not have a curvature tensor. The methodology in question is the heat equation proof of the Atiyah-Singer index theorem.

Ronen Eldan
A new approach to concentration inequalities for Boolean functions, and an isoperimetric inequality conjectured by Talagrand.

We revisit several classical inequalities which relate the influences of a Boolean function to its variance – the Kahn-Kalai-Linial (KKL) inequality and its generalizations by Friedgut and Talagrand, and the relation between influences and noise stability by Benjamini-Kalai-Schramm. We will introduce a new method towards the proofs of these inequalities (based on stochastic calculus and the analysis of jump processes). Our method resolves a ’96 conjecture of Talagrand, deriving a bound which strengthens both Talagrand’s isoperimetric inequality and the KKL inequality. Our method also produces robust versions of some of the aforementioned bounds. Joint work with Renan Gross.

• Contacts

Travel and Hotel Assistance
Christophe Vergnol
Protravel
chrisvgroup@protravelinc.com
(646)-747-9767

General Meeting Assistance
Emily Klein
MPS Event Coordinator, Simons Foundation
eklein@simonsfoundation.org
(646) 751-1262

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