Results 201 - 210 of 24651

This talk will be a gentle introduction to the topic of property (T) and algebraic spectral gaps.
We will begin by giving a brief overview of property (T) and move on to Ozawa’s characterization via an algebraic spectral gap for the group Laplacian in the augmentation ideal.
Then I will describe a numerical approach to proving property (T) via such sums of squares and discuss certification, which turns an approximate numerical solution into the existence of a true, exact solution. This approach has led to new proofs of property (T) for classical examples such as SL_n(Z), n>2, and proving property (T) for Aut(F_n), n>3.

Two widely used Ricci curvature notions for general graphs are Ollivier and Bakry-Emery curvature. They are defined locally on the edges and vertices of a graph and they can be computed with very reasonable effort. In this talk, I will introduce these discrete curvature notions and discuss implications of curvature restrictions to global invariants like diameter and Laplace spectrum of the underlying graphs. These relations are motivated by similar results in the setting of Riemannian manifolds (Bonnet-Myers and Lichnerowicz), and it is natural to ask whether all spaces, for which these curvature/diameter and curvature/eigenvalue inequalities hold with equality, can be classified. If time permits, I will also mention such kind of results.

I will first introduce and review the localization schemes framework of Chen and Eldan for proving functional inequalities for Markov chains. Then I will explain how "trickling-down" from high-dimensional expansion fits into the framework and give some examples. The latter part is based on joint work with Nima Anari and June Vuong.

High-dimensional expanders (HDX) generalize expander graphs and have found various applications in math and TCS. One approach to proving that a simplicial complex is an HDX uses trickle-down theorems, where expansion of certain low-dimensional links implies expansion properties throughout the complex. In this talk, we will discuss some recent trickle-down theorems for partite HDX, with applications to approximate sampling. We will also show how these theorems arise from the theory of Lorentzian and log-concave polynomials. Joint work with Kasper Lindberg and Shayan Oveis Gharan.