Skip to Content

Sponsors

No results

Tags

No results

Types

No results

Search Results

Events

No results
Search events using: keywords, sponsors, locations or event type
When / Where
All occurrences of this event have passed.
This listing is displayed for historical purposes.

Presented By: Department of Mathematics

Integrable Systems and Random Matrix Theory Seminar

Spectral curves for matrix models. Take I: a numerical analysis perspective

A spectral curve for a matrix model is, in very loose terms, an equation with unknown being the Cauchy (a.k.a. Stieltjes) transform of the limiting spectral density. Sometimes also known as master loop equation or string equation, it commonly appears as an algebraic equation, hence the name ''curve'' as it determines an algebraic curve. A classical situation is given by the celebrated semicircle law, whose Cauchy transform satisfies a very simple algebraic equation of degree 2.

In this series of two talks we discuss several aspects of spectral curves. In the first of these talks, we explore a problem in numerical analysis, namely constructing quadrature rules for highly oscillatory integrals, to revisit several old results and learn how we can use techniques from random matrix theory to get insight into the distribution of the corresponding complex quadrature points. This first talk is based on joint work with Andrew Celsus (University of Cambridge).

In the second talk, we take a slight more abstract perspective. Starting from an ''ideal'' spectral curve, which is known to exist in several examples, we show how to reconstruct a variational problem for the eigenvalue distribution, ultimately giving a characterization of the limiting eigenvalue distribution for a random matrix model with external source. A key aspect here is the determination of a ''free boundary'' for the variational problem, which is done with the help of quadratic differentials on the algebraic curve. This second part is based on joint work with Andrei Martinez-Finkelshtein (Baylor University). Speaker(s): Guilherme Silva (University of Michigan)

Explore Similar Events

  •  Loading Similar Events...

Back to Main Content