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.
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.
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.
In the first talk we gave a brief outline of the contents of the course. In the rest of the semester we will get deeper into some topics. In the coming lecture ( and the next one) we will discuss Kazhdan property T and its connections with expanders and with first cohomology groups. No prior knowledge will be assumed.
The expander Chernoff bound states that random walks over expanders are good samplers, at least for a certain range of parameters. In this talk we will be interested in “Parity Samplers” that have the property that for any test set, about half of the sample sets see the test set an *even* number of times, and we will check whether random walks over expanders are good parity samplers. We will see that:
1. Random walks over expanders fare quite well with the challenge, but, 2. A sparse Random complex does much better.
The "PCP theorem" says that problems in NP are hard in a robust or stable way.
I will give a brief intro to PCPs (and explain the acronym) and then try to outline a proof of the PCP theorem based on "agreement expansion" which is a form of high dimensional expansion.
My aim is to show how high dimensional expansion is inherently present in PCP type questions.
Abstract:
I will introduce the notion of (PCP)-agreement expansion which is an important building block in PCPs constructions.
I will then show that a high dimensional expanders imply PCP-agreement expanders.
based on Joint work with Irit Dinur
Speaker : Tatiana Nagnibeda (University of Geneva)
Abstract: The definition of a Ramanujan graph extends naturally to infinite graphs: an infinite graph is Ramanujan if its spectral radius is not larger than (and hence equal to) the spectral radius of its universal covering tree. As with infinite families of finite graphs, it is interesting and non-trivial to understand, how much Ramanujan graphs resemble trees. I will discuss some results in this direction obtained in a joint work with Vadim Kaimanovich, by investigating ergodic properties of boundary actions of free groups.
Speaker: Steven Damelin (The American Mathematical Society)
Abstract:
A classical problem in geometry goes as follows. Suppose we are given two sets of $D$ dimensional data, that is, sets of points in $R^D$. The data sets are indexed
by the same set, and we know that pairwise distances between corresponding points are equal in the two data sets. In other words, the sets are isometric. Can this correspondence be extended
to an isometry of the ambient Euclidean space?
In this form the question is not terribly interesting; the answer has long known
