Research

Classical Analysis
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Friday, January 31, 2025

Title: Taboos, smash sums, and ideals: the other side of Anderson localization
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

(Very) recently, Hannah Cairns has proven that the Diaconis-Fulton “smash sum” operation on open sets generating a commutative algebra (graded by cardinality) is essentially unique. The smash sum is known to generate the DLA model in the countable case and quadrature domains otherwise. The proof of uniqueness now provides a new way to characterize the support of superharmonic functions associated to the eigenvectors in Anderson localization.

(No prior knowledge of Anderson localization is assumed.)

Friday, January 24, 2025

Title: Around Bourgain's theorem on the Anderson-Bernoulli localization
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

Abstract

In 2004, Jean Bourgain proved that Anderson localization (Phil Anderson, 1958) holds in dimensions \(n\) equal to \(1\) and higher for a Bernoulli distribution of disorder and potentials with compact support. This remains the strongest result yet for discrete distributions of disorder, as opposed to the continuous case (Goldsh'ein-Molchanov-Pastur for \(n=1\) and Frohlich-Spencer, Altshuler, and Efetov for \(n>1\)). We consider a deformation of the Anderson-Bernoulli model which allows to “go around” Bourgain's theorem and prove that dimension \(n=2\) is critical for the discrete distribution of disorder, as well.