Combinatorics Seminar - Fall 2015
Tuesday 1pm-2pm in room 226.
The Combinatorics Seminar is an introductory seminar for combinatorics. The talk should be accesible to first year graduate students.
Speaker: Apoorva Khare (Stanford University)
Title:The critical exponent of a graph
Which functions preserve positive semidefiniteness (psd) when applied entrywise to the entries of psd matrices? This question has a long history, beginning with Schoenberg and Rudin, and continues to be studied to date, for additional modern reasons. In my talk I will focus on an important special case: matrices with zeros according to a graph G, and the powers that preserve positivity when applied entrywise to all such matrices (for fixed G). Our main result classifies this set of powers for an arbitrary chordal graph G. We show how preserving positivity relates to the geometry of the graph, thus providing interesting connections between combinatorics and analysis. This yields a new graph property called the "critical exponent". We then compute the critical exponent for large families of non-chordal graphs. (Joint with D. Guillot and B. Rajaratnam.)