Mathematics Colloquium
Date:Lincoln NE 68508
Affiliation: UNL
Local Hosts: Carina Curto and Vladimir Itskov
Title: A topological invariant of symmetric matrices
Refreshments will be served in 348 Avery 3:30-4:00 pm.
The talk is free and open to the public.
Additional Public Info:
<b>Abstract:</b> One common difficulty encountered when building a bridge between scientific experiment and theory is that experimentally observable variables are often related to what we theoretically believe is “really happening” by some unknown nonlinear transformation. Many traditional, linear algebra-based data analysis techniques are designed to be invariant under changes of coordinate system but sensitive to the relative scale of the data points, and thus such nonlinearities dramatically complicate their use. Here we offer one potential answer to the question, “What tools are more natural in such a setting?” Given a symmetric real matrix, we construct a filtered sequence of graphs which is invariant under the action of the group of monotonic increasing transformations on the matrix. Using a recent technique from algebraic topology called “persistent homology”, we extract quantitative measures of this combinatorial object which we use to show that certain correlation structures between the matrix entries — such as those arising from distances in Euclidean space — can be readily distinguished from random controls. As an application, we show that data recorded from the rat hippocampus is consistent with the existence of a Euclidean “coding space” for the observed network across a variety of behaviors.
Refreshments will be served in 348 Avery 3:30-4pm.
The talk is free and open to the public.
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