Simons Collaboration on Algorithms and Geometry Monthly Meeting, November 2019

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.