NCMW - Ergodic Theory and Fractals (2024)

Speakers and Syllabus


 

Name of the Speaker with their Affiliation

No. of Lectures

Detailed Syllabus

Dr. Shrihari Sridharan
IISER Thiruvananthapuram

6 x 1.5

hrs

Title: Ergodic Theory
  1. Measure preserving maps, Examples.
  2. Recurrence and unique ergodicity
  3. Ergodicity and its characterisations
  4. Examples of ergodic maps
  5. Ergodic theorems I
  6. Ergodic theorems II

Dr. Sabyasachi Mukherjee
TIFR Mumbai

6 x 1.5

hrs

Title: Conformal measures on Julia and limit sets

  1. Introduction to rational dynamics: I
  2. Introduction to rational dynamics: II
  3. Conformal dynamics of Kleinian groups
  4. Hausdorff dimension of rational Julia and Kleinian limit sets
  5. Patterson-Sullivan measures on Kleinian limit sets
  6.  Sullivan conformal measures on rational Julia sets

 

Dr. Nishant Chandgotia
TIFR-CAM Bangalore

6 x 1.5

hrs

Title: Thermodynamic Formalism - a quick introduction

  1. A crash course in information theory
  2. What is a shift of finite type? Examples and why we care.
  3. Probability measures on shifts of finite type: Some examples and their properties
  4. Entropy - topological and measure theoretic.
  5. The variational principle
  6. Dobrushin Lanford Ruelle theorem

 

Prof. A.K.B.Chand
IIT Madras

6 x 1.5

hrs

Title: Fractal Interpolations

  1. Fractal Interpolation Functions and Fractal Splines
  2. Different types of Fractal Surfaces
  3. Fractal Approximation
  4. Applications of Fractal Functions
  5. Fractal Radial Basis Functions
  6. Applications of Fractal Radial Basis Functions

 

 

Dr. Amit  Priyadarshi
IIT Delhi

6 x 1.5

hrs

Title: Various Fractal Dimensions
  1. Hausdorff measure and Hausdorff dimension
  2. Other Fractal Dimensions: Box Counting Dimension, Packing Dimension, Assoud Dimension
  3. Properties and Relationship between various dimensions
  4. Iterated Function Systems (Finite, Countable, etc.) and their attractors
  5. Decomposition of functions in terms of fractal dimensions of their graphs
  6. Connection of Hausdorff dimension of attractors with Perron- Frobenius Theory

 

Dr. Senthil Raani
IISER Behrampur

6 x 1.5

hrs

Title: Fourier Transform and fractals
  1. Ahlfors David regular sets and Hausdorff dimension
  2. Energies and Frostman's Lemma
  3. Fourier transform
  4. Energy integrals and Fourier Dimension
  5. Application to Distance set problems
  6. Ball averages and Strichartz theorems

 

Prerequisite: Real Analysis, Complex Analysis, Measure Theory, Point-Set Topology
References:

  1. Kenneth Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley–Blackwell, 2014.
  2. Pertti Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, 2012.
  3. Pertti Mattila, Fourier Analysis and Hausdorff Dimension, Cambridge University Press, 2016.
  4. M.F. Barnsley, Fractals Everywhere, Academic Press, Orlando, Florida, 1988.
  5. Peter Massopust, Fractal Functions, Surfaces and Wavelets, Academic Press, 1995.
  6. C.S. Aravinda and V.S. Bhat, Elements of Dynamical Systems: Lecture Notes from NCM School, Edited by A Nagar, R Shah, S Sridharan, Chapter 3, Springer Nature Singapore.
  7. P Walters, An Introduction to Ergodic Theory, Springer-Verlag New York, Inc. 1982.
  8. David Ruelle, Thermodynamic Formalism, Cambridge University Press, 2004.
  9. Robert Burton and Jeffrey E. Steif, Non-uniqueness of measures of maximal entropy for sub shifts of finite type, Ergodic Theory and Dynamical Systems, 14(2):213-235, 1994.
  10. Burton, R., Steif, J.E. New results on measures of maximal entropy. Israel J. Math. 89, 275–300, 1995.
  11. Keller G. Equilibrium States in Ergodic Theory. Cambridge: Cambridge University Press; 1998.
  12. Milnor J., Dynamics in One Complex Variable: Princeton University Press, 2006.
  13. Przytycki F. and Urbanski M., Conformal Fractals: Ergodic Theory Methods: 371: Cambridge University Press, 2009.
  14. Nicholls P.J., The ergodic theory of discrete groups, Cambridge University Press, 2008.

Names of the tutors with their affiliations:
    1. Dr Shilpak Banerjee, IIT Tirupati
    2. Dr Sharvari Tikekar, TIFR Mumbai
    3. Ms M Megala, IIT Tirupati
    4. Ms Harsha Gopalakrishnan, IIT Tirupati


Time Table

 Tentative timetable, mentioning names of the speakers and tutors with their affiliations:

Day

Date

Lecture 1

(9:30–11:00)

Tea
(11:00
to
11:30)

Lecture 2

(11:30–13:00)

Lunch (13:00 to
14:00)

Lecture 3

(14:00–15:30)

Tea
(15:30
to
16:00)

Discussion (16:00-17:00)

Snacks
(17:00 to
17:30)

Mon

09/12/24

AP

AC

SR

AP+ MM+ HG

Tues

10/12/24

AP

AC

SR

AC+ MM +HG

Wed

11/12/24

SR

AP

AC

SR+ MM+ HG

Thu

12/12/24

SR

AP

AC

AP+ MM+ HG

Fri

13/12/24

AC

SR

AP

AC+ MM +HG

Sat

14/12/24

AC

SR

AP

SR+ MM+ HG

Mon

16/12/24

SS

NC

SM

SS+ SB+ ST

Tues

17/12/24

SS

NC

SM

NC+ SB+ ST

Wed

18/12/24

SM

SS

NC

SM+ SB+ ST

Thu

19/12/24

SM

SS

NC

SS+ SB+ ST

Fri

20/12/24

NC

SM

SS

NC+ SB+ ST

Sat

21/12/24

NC

SM

SS

SM + SB+ ST

 Full forms for the abbreviations of speakers and tutors:

Speakers:
SS: Dr Shrihari Sridharan
SM: Dr Sabyasachi Mukherjee
NC: Dr Nishant Chandgotia
AC: Prof. A.K.B. Chand
AP: Dr Amit Priyadarshi SR: Dr Senthil Raani
Tutors:
SB: Dr Shilpak Banerjee
ST: Dr Sharvari Tikekar MM: M Megala
HG: Harsha Gopalakrishnan

File Attachments: