# Online Seminar: Svetlana Jitomirskaya (University of California, Irvine) #### calendar\_today Date and Time **June 1, 2021** 10:00AM - 10:00AM EDT #### pin\_drop Location **Zoom** ![Svetlana Jitormirskaya](/sites/g/files/omnuum6611/files/styles/hwp_1_1__720x720_scale/public/mathpicture/files/jitormirskaya_svetlana_01.jpg?itok=gYgWwSIK) **Location:** Zoom **Time:** Tuesday, 10:00 AM (Eastern US), 15:00 (UK/Eire), 16:00 (Central Europe), 22:00 (China) **Title:** “Fractal properties of the Hofstadter's butterfly and singular continuous spectrum of the critical almost Mathieu operator” **Abstract:** Harper's operator—the 2D discrete magnetic Laplacian—is the model behind the Hofstadter's butterfly and Thouless theory of the Quantum Hall Effect. We present a result (with I. Krasovsky) that proves one half of the Thouless' "one half" conjecture from the early 80s: that Hausdorff dimension of the spectrum of Harper's operator is bounded by 1/2 for all irrational fluxes. The model reduces to the critical almost Mathieu family, indexed by phase, and we will also present a complete proof of singular continuous spectrum for this family, for all phases, finishing a program with a long history. The proof is based on a simple Fourier analysis and a new duality-type transform that is also underlying the solution of the Thouless problem. We will also explain how these ideas provide for a very simple proof of zero measure of the spectrum of Harper's operator, a problem previuosly solved by sophisticated dynamical systems techniques. Finally, we discuss recent progress towards the Thouless Catalan conjecture. **Additional Ways to Join** Join by telephone (use any number to dial in) +1 929 436 2866 +1 312 626 6799 +1 669 900 6833 400 669 9381 China Toll-free International numbers available: [https://harvar](https://harvard.zoom.us/u/aclg6kOggb)[d.zoom.us/u/aclg6kOggb](https://harvard.zoom.us/u/aclg6kOggb) One tap mobile: +19294362866,,779283357# US (New York) Join by SIP conference room system Meeting ID: 779 283 357 <779283357@zoomcrc.com> --- Attachments- [ picture\_as\_pdf 2021-06-01\_jitormirskaya\_seminar.pdf ](/sites/g/files/omnuum6611/files/mathpicture/files/2021-06-01_jitormirskaya_seminar.pdf) --- Share on:- [ Facebook ](#) - [ Twitter ](#) - [ Linkedin ](#) Save: [ Add to calendar calendar\_today ](https://mathpicture.fas.harvard.edu/node/1590342/event-feed.ics) Copy link link