Advanced Studies Institute in Analysis on Fractal Spaces & Dynamical Systems
Date: August 4-9, 2023
Venue: Urgench State University (Uzbekistan)
Contact: Zair Ibragimov (California State University, Fullerton)
E-mail:
zibragimov@fullerton.edu
Full funding to participate in the Advanced Studies Institute is available for graduate students from U.S. institutions who are U.S. citizens or permanent residents. Please click the Apply botton below to access the online application form. We accept participants on a rolling basis (first come - first served). Application deadline is June 14, 2023.
Key Lecturer: Mario Bonk
University of California
Los Angeles
Overview
A Thurston map is a branched covering map on a topological 2-sphere for which the forward orbit of each critical point under iteration is finite. Each such map gives rise to a fractal geometry on its underlying 2-sphere. The study of these maps links diverse area of mathematics such as dynamical systems, classical conformal analysis, Teichmüller theory, and analysis on metric spaces. In my lectures I will give an introduction to this circle of ideas.
Topics
- Branched coverings and Thurston maps
- Thurston’s characterization of postcritically-finite rational maps
- Visual metrics of expanding Thurston maps
- Basic notions of quasiconformal geometry
- The visual sphere of an expanding Thurston map
Key Lecturer: Mikhail Lyubich
State University of New York
Stony Brook
Overview
Holomorphic Dynamics is a theory of iteration of rational endomorphisms of
the Riemann sphere. This theory was founded in the early 20th century in
the classical work of Fatou and Julia and has become a central theme in
Dynamics in the past fourty years. Methods and ideas of the classical
geometric function theory, hyperbolic geometry, and ergodic theory have
found a beautiful interplay in this area.
Topics
- Classical Fatou-Julia Dichotomy
- Sullivan's No Wandering Domains Theorem
- Structural Stability and lambda-lemma
- Basic structure of the Mandelbrot set
- Distribution of preimages and periodic points
Invited Lecturers
Sergiy Merenkov
City College of New York and CUNY Graduate Center
Title:
Geometry and dynamics versus regularity for manifold automorphisms.
Abstract: I plan to discuss how regularity of automorphisms of manifolds influences the geometry and dynamics of their minimal sets and vice versa. A particular attention will be given to smooth and quasiconformal homeomorphisms of n-tori, n>=2, and hyperbolic surfaces. The results to be presented are higher dimensional versions of Denjoy's theorem on topological conjugacy for circle diffeomorphisms.
Pekka Pankka
University of Helsinki, Finland
Title:
From quasiregular mappings to quasiregular curves.
Abstract: Quasiregular mappings are a non-injective quasiconformal mappings and they may be viewed as a higher dimensional generalization of holomorphic mappings. I will first discuss some of the recent results in their global theory. Then I will discuss a new class of mappings generalizing both quasiregular mappings and holomorphic curves, called quasiregular curves.
Dzmitry Dudko
State University of New York at Stony Brook
Title:
From Denjoy's Theorem to the Renormalization Theory.
Abstract:
Renormalization is a powerful tool for understanding dynamical systems on various scales. We will introduce some of the renormalization ideas, often illustrating them on pictures. In the first talk, we will review Denjoy's Theorem on rotation number and then proceed to finer renormalization questions involving circle maps. In the second talk, we will discuss the period-doubling renormalization of real quadratic polynomials and make an analogy between the associated adding machine attractor and circle maps.
Sabya Mukherjee
Tata Institute of Fundamental Research, India
Title: Fractals arising from antiholomorphic maps and reflection groups.
Abstract: We will explicate topological and geometric structures of fractal sets arising from iterations of antiholomorphic rational maps (complex conjugates of rational maps of the Riemann sphere) and Kleinian reflection groups. We will also explain, mostly at the level of examples, how these conformal dynamical systems can be combined to manufacture richer systems with unusual dynamical and complex-analytic properties.
Program Schedule
August 4 | August 5 | August 6 | August 7 | August 8 | August 9 | |
---|---|---|---|---|---|---|
10:00 - 10:50 |
Lecture 1
Bonk |
Lecture 3
Bonk |
Lecture 5
Bonk |
Lecture 1
Lyubich |
Lecture 3
Lyubich |
Lecture 5
Lyubich |
11:00-11:30 | Coffee Break | Coffee Break |
Coffee Break |
Coffee Break |
Coffee Break | Coffee Break |
11:30-12:20 |
Lecture 2
Bonk |
Lecture 4
Bonk |
Lecture 6
Bonk |
Lecture 2
Lyubich |
Lecture 4
Lyubich |
Lecture 6
Lyubich |
12:30-13:30 | Lunch | Lunch |
Lunch |
Lunch |
Lunch | Lunch |
14:00-14:50 |
Lecture 1
Merenkov |
Lecture 2
Merenkov |
Khiva |
Lecture 1
Dudko |
Lecture 2
Dudko |
Ayaz Kala |
15:00-15:50 |
Lecture 1
Pankka |
Lecture 2
Pankka |
Khiva |
Lecture 1
Mukherjee |
Lecture 2
Mukherjee |
Ayaz Kala |
16:00-19:00 | Free Time & Discussions |
Free Time & Discussions |
Khiva | Free Time & Discussions |
Free Time & Discussions |
Ayaz Kala |
19:00-21:00 | Dinner | Dinner | Banquet | Dinner | Dinner | Banquet |
Travel Dates and Cultural Program
Option 1
July 31 - August 1 : Arrival to Tashkent, local sightseeing
August 2 : Tashkent-Samarkand (Afrosiyob Train 770: Departure at 08:59, Arrival at 11:07)
August 3: Samarkand-Tashkent (Afrosiyob Train 763: Departure at 16:08, Arrival at 18:30)
August 3: Tashkent-Urgench (Uzbekistan Airways Flight HY57: Departure at 20:25, Arrival at 21:55)
August 4-9: Workshop in Urgench
August 9 : Urgench-Tashkent (Uzbekistan Airways Flight HY53: Departure at 22:55, Arrival at 12:15am next day)
August 10 - August 11 : Departure from Tashkent
August 4 : Arrival to Urgench at 07:00 (via Istanbul onTurkish Airlines Flight TK262)
August 4-9: Workshop in Urgench
August 10: Urgench-Bukhara (Uzbekistan Airways Flight HY52: Departure at 08:20, Arrival at 09:10)
August 11: Bukhara-Samarkand (Afrosiyob Train 769: Departure at 15:24, Arrival at 16:59)
August 12: Samarkand-Tashkent (Afrosiyob Train 761: Departure at 17:33, Arrival at 19:43)
August 13 : Local sightseeing, Tashkent-Urgench (Uzbekistan Airways Flight HY57: Departure at 20:25, Arrival at 21:55)
August 14: Departure from Urgench at 08:20 (via Istanbul onTurkish Airlines Flight TK263)
Travel Logistics
Uzbekistan Airways
has non-stop flights to Tashkent from New York (JFK), Frankfurt (FRA), London (LHR), Milan (MXP), Paris (CDG), Rome (FCO), Tel Aviv (TLV), Istanbul (IST) and Dubai (DXB) as well as non-stop flight to/from Urgench (2-3 flights/day).
Turkish Airlines
has non-stop flights to Istanbul from major U.S. and European cities and well as non-stop flights to/from Tashkent (daily) and to/from Urgench (M,F) .
Visa to Uzbekistan
U.S. passport holders needs a visa to travel to Uzbekistan and electronic visas are available (costs $20, validity 30 days, issued in 2-3 business days). Visa applications are processed through the official website of the Ministry of Foreign Affairs of Uzbekistan at https://e-visa.gov.uz/main
.