Zhenghe Zhang Assistant professor Office: Skye 256 Phone: 951-827-3210 Email: zhenghe.zhang@ucr.edu SeminarResearch 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 Schrodinger operators and dynamics of the associated Schrodinger cocycles. Currently, I am focusing on positivity and large deviation estimates for the Lyapunov exponent for different types of ergodic base dynamics. In mathematical physics, these two properties are strong indications of Anderson Localization. I am particular interested in quasiperiodic potentials and potentials generated by hyperbolic dynamics. Papers Publications:- 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), to appear in*Transactions of the AMS.* - Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger Operators
2013 preprint, extended in 2018 and 2019 (32 pages), to appear in*Journal of Spectral Theory.* - 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:- Positive Lyapunov Exponents and a Large Deviation Theorem for Continuum Anderson Models, Briefly
(with V. Bucaj, D. Damanik, J. Fillman, V. Gerbuz, T. VandenBoom, and F. Wang),*2019 preprint*(6 pages). - 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 260 Seminar-Ergodic Theory and Spectral Theory, Spring 2018-Present
- Math 148-Introduction to chaotic and complex dynamical systems, Spring 2019
- Math 150A-Intermediate Analysis, Winter 2019
- NASC 093-Freshman Advising Seminar, Fall 2018
- Math 211A-Ordinary Differential Equations, Fall 2018
- Math 046-Introduction to Ordinary Differential Equations, Summer 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 - AMS Fall Western Sectional Meeting, special session Dynamical Systems and Ergodic Theory (organizer, with N. Haydn and H. Hu)
UC Riverside, Nov 9-10, 2019 - 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 |