AIS Ergodic Theory and Dynamical Systems (2017) - Speakers and Syllabus

Dynamical systems is a very active area of research in Mathematics today. This school provides an opportunity to young researchers to learn various foundational courses that make up the theory of Dynamical systems and Ergodic theory.
In this workshop, we intend to cover topics from real dynamics, topological dynamics, symbolic dynamics and complex dynamics. One of the interesting aspects is the re-ordering of positive integers by Sharkovskii, that explains the chaotic behaviour of certain real dynamical systems. Recurrence is one of the underlying aspects that describes any kind of dynamics which forms the core of Topological Dynamics. Having studied various dynamical systems, it is essential to understand the distribution of various kinds of orbits of points; thus naturally paving way for Ergodic theory. And a useful tool to study most dynamics – Symbolic Dynamics. We intend to lecture not only on discrete- time dynamical systems but also on continuous time dynamical systems.
In the last week of the workshop, we shall look at Complex Dynamics wherein we discuss the Fatou-Julia dichotomy of rational functions and transcendental functions based on Montel’s normality criterion. The workshop culminates with a course on arithmetic dynamics; where we discuss Manin’s conjecture for group varieties and optimal distribution of points following Lubotzky-Phillips-Sarnak and Clozel’s results.

Name of the Speaker with affiliation No. of Lectures Detailed Syllabus
Prof V. Kannan (VK), University of Hyderabad 6 Real Dynamics: Interval dynamics, Stability of periodic points, Sharkovskii’s theorem, Li-Yorke chaos, Logistic maps, Bifurcation, Transitivity, Sensitivity, Devaney’s chaos, Markov maps, Generalisations to higher dimensions.
Prof C.R.E. Raja (CRE), ISI Bangalore 6 Topological Dynamics: Topological transformations, Recurrence, Applications – Van der Waerden’s theorem, Minimality, Equicontinuity, Mixing, Distal and proximal systems, Enveloping semigroups.
Prof SiddharthBhattacharya (SB), TIFR Mumbai 6 Symbolic Dynamics: Shifts and subshifts, Subshifts of finite type and sofic shifts, Factors, irreducibility, entropy, shift equivalence, dimension groups, Multi-dimensional shifts.
Prof C.S. Aravinda (CSA), TIFR Bangalore 6 Basic Ergodic Theory: Measurable dynamics, Measure preserving dynamics, Poincare recurrence, Ergodicity, Unique ergodicity, von Neumann ergodic theorem, Birkhoff ergodic theorem, Horocycle and geodesic flows on closed surfaces of constant negative curvature.
Prof Kaushal Verma (KV), IISc Bangalore 6 Complex Dynamics: Quadratic polynomials, Mandelbrot set, Rational maps, Transcendental maps, Fatou-Julia dichotomy, Various characterisations of the dichotomy, Quadratic-like maps and renormalisation.
Prof Anish Ghosh (AG), TIFR Mumbai 6 Arithmetic Dynamics: Manin conjecture for group varieties, Quantitative density of rational points, optimal distribution of points on spheres following Lubotzky-Phillips-Sarnak and Clozel. Useful tools from Geometry and Harmonic analysis on semisimple groups in Dynamics.

Time-Table

Day Date 
 Lecture 1 (9.30–11.00) Tea
(11.05–11.25)
Lecturer 2 (11.30–1.00) Lunch
(1.05–2.25)
Tutorial (2.30–3.30) Tea
(3.35-3.55)
Tutorial (4.00-5.00) Snacks (5.00-5.30)
    (name of the speaker)   (name of the speaker)   (name of the speaker & tutor)   (Name of the speaker & Tutor)  
Mon Dec 4 VK   VK   VK & PC   VK & PC  
Tues Dec 5 VK   VK   VK & PC   VK & PC  
Wed Dec 6 VK   VK   VK & PC   VK & PC  
Thu Dec 7 CRE   CRE   CRE & MC   CRE & MC  
Fri Dec 8 CRE   CRE   CRE & MC   CRE & MC  
Sat Dec 9 CRE   CRE   CRE & MC   CRE & MC  
Sunday : Off
Mon Dec 11 SB   CSA   SB & PS   CSA & NA  
Tues Dec 12 SB   CSA   SB & PS   CSA & NA  
Wed Dec 13 SB   CSA   SB & PS   CSA & NA  
Thu Dec 14 SB   CSA   SB & PS   CSA & NA  
Fri Dec 15 SB   CSA   SB & PS   CSA & NA  
Sat Dec 16 SB   CSA   SB & PS   CSA & NA  
Sunday : Off
Mon Dec 18 KV   AG   KV & TN   AG & RK  
Tues Dec 19 KV   AG   KV & TN   AG & RK  
Wed Dec 20 KV   AG   KV & TN   AG & RK  
Thu Dec 21 KV   AG   KV & TN   AG & RK  
Fri Dec 22 KV   AG   KV & TN   AG & RK  
Sat Dec 23 KV   AG   KV & TN   AG & RK  

 Tutorial Assistants:

S. No. Name Affiliation
1 Nikita Agarwal (NA) IISER-Bhopal
2 P. Chiranjeevi (PC) IISER-Trivandrum
3 Manoj Choudhuri (MC) HRI, Allahabad
4 Rajkumar Krishnan (RK) SNU, Delhi
5 Tarakanta Nayak (TN) IIT-Bhubaneswar
6 Puneet Sharma (PS) IIT Jodhpur