Zhenghe Zhang Assistant professor Office: Surge 256 Phone: 951-827-3210 Email: zhenghe.zhang@ucr.edu SeminarStarting Spring 2018, I am running a seminar entitled " Ergodic Theory and Spectral Theory".Click the title in the line above for schedule of talks. Research InterestsMy research interests center around the dynamical object, the Lyapunov exponent, and lie across dynamical systems, spectral theory, and mathematical physics. Concretely, I study spectral analysis of ergodic type of Schrodinger operators. Currently, I am focusing on uniform positivity and uniform large deviation estimates for the associated Lyapunov exponent for various types of ergodic base dynamics, where uniformities are in the energy parameter. The two properties immediately imply certain regularities of the Lyapunov exponent and are strong indications of the Anderson Localization phenomenon. Examples of ergodic base dynamics I am interested in are irrational rotations on the unit circle (also called quasiperiodic motion, typical elliptic dynamics) or the doubling map (typical hyperbolic dynamics). By comparing the mechanisms that lead to uniform positivity and LDT of the Lyapunov exponent over these two types of base dynamics, one may detect the huge gap between almost periodic dynamics and strongly mixing dynamics as well as the big difference between elliptic dynamics and hyperbolic dynamics. Papers Publications:- Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator (with J. Fillman and D. C. Ong),
*Comm. Math. Phys.***351**(2017), (37 pages). - Cantor spectrum for a class of $C^2$ quasiperiodic Schrödinger operators (with Y. Wang),
. 2017, no. 8, (37 pages).*Int. Math. Res. Notices* - Uniform positivity and continuity of Lyapunov exponents for a class of $C^2$ quasiperiodic Schrödinger cocycles (with Y. Wang),
*J. Funct. Anal*.**268**(2015), (61 pages). - Singular density of states measure for subshift and quasi-periodic Schrödinger operators (with A. Avila and D. Damanik),
*Comm. Math. Phys.***330**(2014), (30 pages). - Positive Lyapunov exponents for quasiperiodic Szegő cocycles
*,**Nonlinearity***25**(2012), (27 pages).
Preprints:- Localization for the one-dimensional Anderson model via positivity and large deviations of the Lyapunov exponent
(with V. Bucaj, D. Damanik, J. Fillman, V. Gerbuz, T. VandenBoom, and F. Wang),*2017 preprint*(45 pages, submitted). - Uniform hyperbolicity and its applications to spectral analysis of 1D discrete Schrödinger Operators
*2013 preprint*(23 pages, rewrote and extended in 2018, submitted). - Uniform positivity of the Lyapunov exponent for monotone potentials generated by the doubling map,
*2016 preprint*(13 pages).
Papers in Preparation:- Work related to positivity, continuity, and LDT of the Lyapunov exponent and Anderson Localization for
potentials generated by hyperbolic base dynamics (with A. Avila and D. Damanik).
Teaching:- Math 046-Introduction to Ordinary Differential Equations, Summer 2018(upcoming)
- Math 260 Seminar-Ergodic Theory and Spectral Theory, Spring 2018
- Math 046-Introduction to Ordinary Differential Equations, Spring 2018
- Math 046-Introduction to Ordinary Differential Equations, Winter 2018
- Math 151A-Advanced Calculus I, Fall 2017
Past Teaching at Rice University (2014-2017):- Math 211-Ordinary differential equations and linear algebra, Spring 2017
- Math 382-Complex analysis, Spring 2017
- Math 521-Advanced topics in real analysis, Fall 2016
- Math 435-Dynamical systems, Spring 2016
- Math 523-Functional analysis, Fall 2015
- Math 211-Ordinary differential equations and linear algebra, Fall 2015
- Math 112-Calculus and its applications, Spring 2015
- Math 322-Introduction to analysis II, Spring 2015
- Math 211-Ordinary differential equations and linear algebra, Fall 2014
Meetings - Workshop on Between Dynamics and Spectral Theory (organizer, with W. Yessen),
Simons Center for Geometry and Physics, June 6-10, 2016
Linkssince 10/27/2014 |