Advanced Studies Institute in Mathematical Physics & Pluripotential Theory
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, USA)
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,
USA)
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)
overview
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.
Lecturer 4: Norman Levenberg (Indiana University, Bloomington)
OVERVIEW
Pluripotential theory, the study of plurisubharmonic functions in several complex variables, has been utilized in recent years in other areas of mathematics. In this minicourse, we will first develop some of the needed basic tools, e.g., extremal plurisubharmonic functions, complex Monge-Amperé measures, and generalized Vandermonde matrices. We will proceed by analogy to the univariate case, logarithmic potential theory in the complex plane C, instead of providing detailed proofs.
Topics
(1) Basics of pluripotential theory; (2) Random polynomials and random polynomial
mappings
Lecturer 5: Turgay Bayraktar (Sabanci University, Turkey)
OVERVIEW
We will use tools and techniques developed in Professor Levenberg's lectures to discuss recent results on asymptotic behavior of zero sets of sequences of random polynomials and random polynomial mappings in C^d, d > 1; on random point processes on compact sets K subset of C^d and associated large deviation principles; and on applications to complex dynamics such as approximating the equilibrium measure K of certain compact sets K by a sequence of dynamically generated Brolin measures associated to a sequence of polynomials exhibiting a certain regular behavior on K.
Topics
(1) Point processes and large deviation principles; (2) Applications to complex dynamics.
Program Schedule
| Week 1 | July 25 | July 26 | July 27 | July 28 |
|---|---|---|---|---|
| 10:00 - 12:00 |
Discussion/Tutorial Simon Becker |
Discussion/Tutorial Simon Becker |
Discussion/Tutorial Simon Becker |
Discussion/Tutorial Simon Becker |
| 12:30 - 13:30 |
Lunch |
Lunch |
Lunch |
Lunch |
| 13:30 - 15:30 |
Lecture 1 (Zoom) Norman Levenberg |
Lecture 2 (Zoom) Norman Levenberg |
Lecture 1 (Zoom) Turgay Bayraktar |
Lecture 2 (Zoom) Turgay Bayraktar |
| 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 |
| 19:30 - 21:30 | Dinner | Dinner | Dinner | Banquet for Svetlana |
Trip to Bukhara and Samarkand (July 29-31). |
||||
| Week 2 | August 1 | August 2 | August 3 | August 4 |
| 8:00 - 10:30 | Free time | Free time | Ayaz Kala | Free time |
| 10:30 - 12:00 |
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 - 21:00 | Dinner | Dinner | Dinner | Banquet for Marcelo |
List of Student Participants
- Nurali Akramov (National University of Uzbekistan)
- Dilshod Atajanov (Urgench State University)
- Shoira Atanazarova (Romanovski Institute of Mathematics)
- Azizbek Azamatov (Urgench State University)
- Aygul Babajanova (Romanovski Institute of Mathematics)
- Iroda Boltaeva (Urgench State University)
- Anabel Camarena (Sonoma State University)
- Camila Carmona (Sonoma State University)
- Devraj Duggal (University of Minnesota - Twin Cities)
- Matthew Enlow (University of Nebraska - Lincoln)
- David Evans (Sonoma State University)
- Swapnil Garg (University of California, Berkeley)
- Jesse Hulse (Syracuse University)
- Omar Hurtado (University of California, Irvine)
- Khursandbek Kamolov (Urgench State University)
- Umid Khoytmetov (Romanovski Institute of Mathematics)
- Brian Luczak (Vanderbilt University)
- Andrei Mandelshtam (Stanford University)
- Izak Oltman (University of California, Berkeley)
- Joshua Paik (Penn State University)
- Sherzod Sadullaev (Urgench State University)
- Alberto Takase (University of California, Irvine)
