HD-Combinatorics

2017 Dec 25

HD-Combinatorics: Shai Evra, "Bounded degree high dimensional expanders"

2:00pm to 4:00pm

In the recent theory of high dimensional expanders, the following open problem was raised by Gromov: Are there bounded degree high dimensional expanders? For the definition of high dimensional expanders, we shall follow the pioneers of this field, and consider the notions of coboundary expanders (Linial-Meshulam) and topological expanders (Gromov). In a recent work, building on an earlier work of Kaufman-Kazhdan-Lubotzky in dimension 2, we were able to prove the existence of bounded degree expanders according to Gromov, in every dimension.
2017 Nov 20

Leonard Schulman, "Analysis of a Classical Matrix Preconditioning Algorithm"

2:00pm to 3:00pm

Location: 

Room 130, Feldman Building, Givat Ram
There are several prominent computational problems for which simple iterative methods are widely preferred in practice despite an absence of runtime or performance analysis (or "worse", actual evidence that more sophisticated methods have superior performance according to the usual criteria). These situations raise interesting challenges for the analysis of algorithms. We are concerned in this work with one such simple method: a classical iterative algorithm for balancing matrices via scaling transformations. This algorithm, which goes back to Osborne and
2017 Oct 23

HD-Combinatorics: Nati Linial, "High-dimensional permutations"

2:00pm to 4:00pm

Location: 

Israel Institute for Advanced Studies (Feldman building, Givat Ram), Eilat Hall
This is a survey talk about one of the main parts of what we call high-dimensional combinatorics. We start by equating a permutation with a permutation matrix. Namely, an nxn array of zeros and ones where every line (=row or column) contains exactly one 1. In general, a d-dimensional permutation is an array [n]x[n]x....x[n] (d+1 factors) of zeros and ones in which every line (now there are d+1 types of lines) contains exactly one 1. Many questions suggest themselves, some of which we have already solved, but many others are still wide opne. Here are a few examples:
2017 Nov 13

HD-Combinatorics: Shmuel Weinberger, "L^2 cohomology"

2:00pm to 4:00pm

Location: 

Room 130, Feldman Building, Givat Ram
Abstract: I will give an introduction to the cohomology of universal covers of finite complexes. These groups are (for infinite covers) either trivial or infinite dimensional, but they have renormalized real valued Betti numbers. Their study is philosophically related to the topic of our year, and they have wonderful applications in geometry, group theory, topology etc and I hope to explain some of this.
2017 Nov 20

HD-Combinatorics: Ran Levi, "Neuro-Topology: An interaction between topology and neuroscience"

3:00pm to 4:00pm

Location: 

Room 130, Feldman Building, Givat Ram
Abstract: While algebraic topology is now well established as an applicable branch of mathematics, its emergence in neuroscience is surprisingly recent. In this talk I will present a summary of an ongoing joint project with mathematician and neuroscientists. I will start with some basic facts on neuroscience and the digital reconstruction of a rat’s neocortex by the Blue Brain Project in EPFL.
2018 Jan 01

HD-Combinatorics: Alan Lew, "Spectral gaps of generalized flag complexes"

2:00pm to 4:00pm

Location: 

Eilat Hall, Feldman Building (IIAS), Givat Ram
Abstract: Let X be a simplicial complex on n vertices without missing faces of dimension larger than d. Let L_k denote the k-Laplacian acting on real k-cochains of X and let μ_k(X) denote its minimal eigenvalue. We study the connection between the spectral gaps μ_k(X) for k ≥ d and μ_{d-1}(X). Applications include: 1) A cohomology vanishing theorem for complexes without large missing faces. 2) A fractional Hall type theorem for general position sets in matroids.
2017 Sep 11

IIAS Seminar: Nikolay Nikolov, "Gradients in group theory"

11:00am to 12:00pm

Location: 

Feldman building, Room 128
Abstract: Let G be a finitely generated group and let G>G_1>G_2 ... be a sequence of finite index normal subgroups of G with trivial intersection. We expect that the asymptotic behaviour of various group theoretic invariants of the groups G_i should relate to algebraic, topological or measure theoretic properties of G. A classic example of this is the Luck approximation theorem which says that the growth of the ordinary Betti numbers of sequence G_i is given by the L^2-Betti number of (the classifying space) of G.

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