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Mathematics colloquium

Friday, October 29, 2021 - 15:30 to 16:30
Online (GoogleMeet)
Apoorva Khare
IISc Bangalore
Positive semidefiniteness and the critical exponent of a graph

The study of positive semidefinite (psd) matrices, and of their endomorphisms, has its origins in results by Schur, Schoenberg, Rudin and Loewner among others. One refinement involves understanding the powers preserving positivity in a fixed dimension, when applied entrywise. We will see this question and some related variants.


We then focus on the following refinement: given a graph $G$, let $P_G$ denote the cone of psd matrices with zeros according to $G$. Which entrywise powers preserve $P_G$? We show how preserving positivity relates to the geometry of the graph $G$. This leads to a hitherto unexplored graph invariant: the "critical exponent" of $G$. A joint work with D. Guillot and B. Rajaratnam shows how this purely combinatorial invariant resolves the positivity preserver problem for all chordal graphs.the resurrection of some of the old ones.

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