Advanced Studies Institute in Mathematical Physics
Date: July 25 - August 4, 2022
Venue: Institute of Mathematics, Urgench State University
Contact: Zair Ibragimov (California State University, Fullerton)
E-mail:
zibragimov@fullerton.edu
Lecturer 1: Svetlana Jitomirskaya (University of California, Irvine)
Overview
The lectures will be devoted to recent developments in the study of quasiperiodic Schrodinger operators. We will start with a brief overview of the necessary background, beginning with the basics of spectral measures, different spectral types, multiplicative ergodic theorem, and the key elements of the general theory of ergodic operators. After that we will move to elements of Avila's global theory and proofs of arithmetic nonperturbative localization. The suggested background is functional analysis in the volume of e.g. the first six chapters of Reed-Simon and some familiriarity with ergodic theory.
Topics
- Spectral measures and spectral decomposition;
- Simon's Wonderland theorems;
- Lyapunov exponents of linear cocycles, multiplicative ergodic theorem;
- Ergodic Schrödinger operators, integrated density of states;
- Avila's global theory and its quantitative version;
- Spectral transitions for the almost Mathieu operator.
Lecturer 2: Marcelo Disconzi (Vanderbilt University)
Overview
In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein's equations. The topics to be discussed will include: the relativistic free-boundary
Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity. Emphasis will be given on techniques and concepts that can lead a fruitful interaction among mathematicians and physicists.
Topics
- Set-up, review of standard results, physical motivation;
- The relativistic Euler equations: null structures and the problem of shocks;
- The free-boundary relativistic Euler equations with a physical vacuum boundary;
- Relativistic viscous fluids.
Lecturer 3: Ravshan Ashurov
(Institute of Mathematics, Uzbekistan Academy of Sciences)
Inverse problems of determining the order of fractional derivatives of partial differential equations
Abstract
When considering fractional differential equations as a model equation in the analysis of various anomalous processes, the order of fractional derivatives is often unknown and difficult to directly measure, which requires a discussion of the inverse problem of identifying this physical quantity from some indirectly observed information about the solutions. Inverse problems of determining these unknown parameters are not only of theoretical interest but are also necessary for finding a solution to the initial-boundary value problem and studying the properties of solutions. This series of lectures discusses methods for solving such inverse problems for the equations of mathematical physics. It is assumed that students are familiar with the theory of partial differential equations and elements of functional analysis.
Topics
- Fractional derivatives. Mittag-Leffler functions. Sobolev's embedding theorem. Fractional powers of elliptic operators.
- Classical solution of forward problems for subdiffusion equations and the first method for solving inverse problems.
- Generalized solutions of forward problems and the second method for solving inverse problems.
Program Schedule
| Week 1 | July 25 | July 26 | July 27 | July 28 |
|---|---|---|---|---|
| 8:00 - 10:30 | Free time | Free time | Free time | Free time |
| 10:30 - 12:30 |
Discussion/Tutorial Simon Becker |
Discussion/Tutorial Simon Becker |
Discussion/Tutorial Simon Becker |
Discussion/Tutorial Simon Becker |
| 12:30 - 13:30 | Lunch | Lunch | Lunch | Lunch |
| 16:30 - 17:30 |
Lecture 1 Jitomirskaya |
Lecture 3 Jitomirskaya |
Lecture 5 Jitomirskaya |
Lecture 7 Jitomirskaya |
| 17:30 - 18:00 | Coffee Break | Coffee Break | Coffee Break | Coffee Break |
| 18:00 - 19:00 |
Lecture 2 Jitomirskaya |
Lecture 4 Jitomirskaya |
Lecture 6 Jitomirskaya |
Lecture 8 Jitomirskaya |
| 19:00 - 20:30 | Dinner | Dinner | Dinner | Dinner |
Trip to Bukhara (July 29) and Samarkand (July 30). |
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| Week 2 | August 1 | August 2 | August 3 | August 4 |
| 8:00 - 11:00 | Free time | Free time | Ayaz Kala | Free time |
| 11:00 - 12:30 |
Discussion/Tutorial Disconzi |
Discussion/Tutorial Disconzi |
Ayaz Kala |
Discussion/Tutorial Disconzi |
| 12:30 - 13:30 | Lunch | Lunch | Lunch | Lunch |
| 15:00 - 17:00 |
Lectures 1 & 2 Disconzi |
Lectures 3 & 4 Disconzi |
Lectures 5 & 6 Disconzi |
Lectures 7 & 8 Disconzi |
| 17:00 - 17:30 | Coffee Break | Coffee Break | Coffee Break | Coffee Break |
| 17:30 - 18:30 |
Lecture 1 Ashurov |
Lecture 2 Ashurov |
Lecture 3 Ashurov |
Lecture 4 Ashurov |
| 19:00 - 20:30 | Dinner | Dinner | Dinner | Dinner |
