Smooth quasiperiodic SL(2,\R)-cocycles (II)-Sharp transition space for the continuity of the Lyapunov exponent.

Speaker: 

Lingrui Ge

Institution: 

UCI

Time: 

Thursday, January 28, 2021 - 10:00am to 11:00am

Location: 

https://uci.zoom.us/j/95453091338?pwd=L1hXRTRremw4YjFQY3I3NHZGNUdKZz09

We construct discontinuous points of the Lyapunov exponent of quasiperiodic Shr\"odinger cocycles in Gevrey space $G^{s}$ with $s>2$. In contrast, the Lyapunov exponent has been proved to be continuous in $G^{s}$ with $s<2$ by Klein and Cheng-Ge-You-Zhou. This shows that $G^2$ is the transition space for the continuity of the Lyapunov exponent.

Smooth quasiperiodic SL(2,\R)-cocycles (I)-Global rigidity results for rotations reducibility and Last's intersection spectrum conjecture.

Speaker: 

Lingrui Ge

Institution: 

UCI

Time: 

Thursday, January 21, 2021 - 10:00am to 11:00am

Location: 

https://uci.zoom.us/j/97333959480?pwd=VExLT1d2Q0F6SkovT3hmbGhBZ05HUT09

For quasiperiodic Schr\"odinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schr\"odinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy by Avila-Fayad-Krikorian. From spectral theory side, the ``Schr\"odinger conjecture"  has been verified by Avila-Fayad-Krikorian and the ``Last's intersection spectrum conjecture" has been proved by Jitomirskaya-Marx. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, some open problems in this aspect were raised by  Fayad-Krikorian and Jitomirskaya-Marx. In this paper, we prove the above mentioned results for ultra-differentiable potentials.

On the abominable properties of the Almost Mathieu operator with Liouville frequencies

Speaker: 

Mira Shamis

Institution: 

Queen Mary, london

Time: 

Thursday, January 14, 2021 - 12:00pm to 1:00pm

Location: 

zoom https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09

 

We show that, for sufficiently well approximable frequencies, several spectral characteristics of the Almost Mathieu operator can be as poor as at all possible in the class of all discrete Schroedinger operators. For example, the modulus of continuity of the integrated density of states may be no better than logarithmic. Other characteristics to be discussed are homogeneity, the Parreau-Widom property, and (for the critical AMO) the Hausdorff content of the spectrum. Based on joint work with A. Avila, Y. Last, and Q. Zhou

THE UMPTEEN OPERATOR

Speaker: 

Sasha Sodin

Institution: 

Queen Mary, london

Time: 

Thursday, November 12, 2020 - 12:30pm to 1:30pm

Location: 

zoom https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09

Abstract: 

It was found in the 1990s that special linear maps playing a role in the representation theory of the symmetric group share common features with random matrices. We construct a representation-theoretic operator which shares some properties with the Anderson model (or, perhaps, with magnetic random Schroedinger operators), and show that indeed it boasts Lifshitz tails. The proof relies on a close connection between the operator and the infinite board version of the fifteen puzzle.
No background in the representation theory of the symmetric group will be assumed. Based on joint work with Ohad Feldheim.

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