Svetlana Jitomirskaya (UC Irvine)

Stay: November 12-16, 2018

Lectures will be held on November 14, 15 and 16, 2018.

Jitomirskaya’s lectures will be a part of the workshop on Spectral Theory of Quasi-Periodic and Random Operators (November 12–16). The first two lectures will take place on November 12 and the third one on November 13.

BIOGRAPHY: Svetlana Jitomirskaya is working on dynamical systems and mathematical physics. She obtained her Ph.D. from Moscow State University in 1991. She joined the mathematics department at the University of California, Irvine where she became a full professor in 2000. She is best known for solving the “Ten Martini Problem” along with mathematician Artur Ávila. In 2005, she was awarded the Ruth Lyttle Satter Prize in Mathematics, “for her pioneering work on nonperturbative quasiperiodic localization.” She was an invited speaker at the 2002 International Congress of Mathematicians, in Beijing. She received a Sloan Fellowship in 1996. In 2018 she was named to the American Academy of Arts and Sciences.

Series of lectures

CRM Thematic Semester: Mathematical challenges in many-body physics and quantum information

Svetlana Jitomirskaya (UC Irvine)

Wednesday, November 14, 2018 4:00 pm

Centre de recherches mathématiques
Pavillon André-Aisenstadt, Université de Montréal
Room 6254

SLIDESHOW OF THE LECTURE

Anti-concentration bounds for determinants and Anderson localization

We present yet another proof of Anderson localization for the Anderson model.

Thursday, November 15, 2018 4:00 pm

Centre de recherches mathématiques
Pavillon André-Aisenstadt, Université de Montréal
Room 6254

Localization and delocalization for multidimensional quasiperiodic operators

We discuss recent progress on localization and delocalization for quasiperiodic operators, including the case of interacting particles. The talk is based on papers with Liu and Shi, Bourgain and Parnovsky, and Bourgain and Kachkovskiy.

Friday, November 16, 2018 4:00 pm

Centre de recherches mathématiques
Pavillon André-Aisenstadt, Université de Montréal
Room 1140

Sharp arithmetic transitions for 1D quasiperiodic operators

A very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite significant numerical studies and even a discovery of Bethe Ansatz solutions has remained an important open challenge even at the physics level. I will present a complete solution of this problem in the exponential sense throughout the entire localization regime. Namely, I will describe the continued fraction driven hierarchy of local maxima, and a universal (also continued fraction expansion dependent) function that determines local behavior of all eigenfunctions around each maximum, thus giving a complete and precise description of the hierarchical structure. In the regime of Diophantine frequencies and phase resonances there is another universal function that governs the behavior around the local maxima, and a reflective-hierarchical structure of those, phenomena not even described in the physics literature. These results lead also to the proof of sharp arithmetic transitions between pure point and singular continuous spectrum, in both frequency and phase, as conjectured since 1994. This part of the talk is based on the papers joint with W. Liu. Within the singular continuous regime, it is natural to look for further, dimensional transitions. I will present a sharp arithmetic transition result in this regard that holds for the entire class of analytic quasiperiodic potentials, based on the joint work with S. Zhang.