Student Probability Seminar

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Landscape Complexity: Counting zeros with the Kac-Rice formula

Speaker: Pax Kivimae, CIMS

Location: Warren Weaver Hall 517

Date: Tuesday, November 29, 2022, 1 p.m.

Synopsis:

Within spin glass theory, some of the most basic questions are: For a symmetric p-tensor with i.i.d entries, how many real eigenvalues does it have? How big is the largest eigenvalue? What does it fluctuate like? For p=2, these are classical questions in random matrix theory. For p>2, these questions get harder as the number of eigenvalues becomes exponentially large, and answers to these questions have only been obtained quite recently. The key tool, however, is a simple identity which reduces these questions back to (now harder) problems in random matrix theory, the so-called Kac-Rice formula. In this talk, we will review this formula, and present some basic examples and new results involving it.