Ergodic Theory and Fractals (2024)

Convener(s)

 
Name: Dr Srijanani Anurag Prasad  Dr Shilpak Banerjee
Mailing Address: Assistant Professor
Department of Mathematics and Statistics,
Indian Institute of Technology Tirupati,
Yerpedu – Venkatagiri Road, Yerpedu Post,
Tirupati District, Andhra Pradesh.
Postcode - 517619.		 Assistant Professor
Department of Mathematics and Statistics,
Indian Institute of Technology Tirupati,
Yerpedu – Venkatagiri Road, Yerpedu Post,
Tirupati District, Andhra Pradesh.
Postcode - 517619.
Email: srijanani at iittp.ac.in shilpak at iittp.ac.in

Fractals and Ergodic Theory are two independent but interconnected disciplines of mathematics, each contributing to our knowledge of complex systems and patterns in nature.

Fractals, which exhibit self-similarity at multiple scales, provide a geometric foundation for describing irregular and complicated structures. Fractals were widely known in mathematics thanks to the groundbreaking work of Mandelbrot in the 1970s, which also revealed their prevalence in natural phenomena such as coastlines, clouds, and ferns. The recursive nature of fractals allows for the representation of complexity through simple repeating methods, encouraging discoveries into the intrinsic order of seemingly chaotic systems.

Ergodic Theory, on the other hand, is concerned about the statistical behaviour of dynamic systems over time. It delves into the concept of ergodicity, which occurs when a system's time average equals its ensemble average. This theory offers a basis for comprehending the long-term behaviour of dynamic systems and has applications in information theory, probability, and physics.

In the study of modern theory of dynamical systems, of which Ergodic Theory can be considered as a branch, we often seek out patterns that appear with the evolution of time and sometimes such patterns can even be considered `static’. Certain examples of interest from the world of complex dynamical systems would be Julia sets and Mandelbrot sets. Also, in the study of hyperbolic dynamical systems, structures such as Smale’s horseshoes appear as invariant sets and are often used as crucial tools to further our understanding of such systems. One thing common with these patterns is that they are self-repeating, or fractals! Hence a deeper knowledge of fractals is essential in the understanding of dynamical systems. In this workshop we seek to bring both the communities together and hope it will serve as a launching platform for many future collaborations.

Dates: 

Monday, December 9, 2024 - 19:00 to Saturday, December 21, 2024 - 19:00

Venue: 

Venue Address: 

Department of Mathematics and Statistics, Indian Institute of Technology Tirupati,
Yerpedu – Venkatagiri Road, Yerpedu Post, Tirupati District, Andhra Pradesh. Postcode - 517619.

Venue State: 

Venue City: 

PIN: 

517619

Chrono Order: 

492

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

Selected Applicants: 

 Final list of confirmed participants

Sr.n SID First Name Gender Affiliation Position in College/University University/Institute M.Sc./M.A. Year of Passing M.Sc./M.A Ph.D. Degree Date
1 58527 Mr. Kifayat Ullah Lone Male SRM Institute of Science and Technology, Kattankulathur, Chennai Research Scholar SRM Institute of Science and Technology Kattankulathur 2020  
2 58571 Mr. Viswanathan S Male TIFR PhD KSoM 2022  
3 58616 Mr. Thirupathi P Male Indian Institute of Science Education Research PhD Thiruvalluvar University 2019  
4 58637 Ms. Pooja Rani Female Indian Institute of technology (BHU) PhD M.Sc 2018  
5 58650 Mr. Bharath Krishna S Male Indian Institute of Science Education and Research Thiruvananthapuram Integrated PhD student Indian Institute of Science Education and Research Thiruvananthapuram 2024  
6 58653 Ms. Rameshwari Gupta Female University of Hyderabad, Telangana PhD Student Guru Ghasidas Vishwavidyalaya, Bilaspur 2019  
7 58667 Mr. Manoj Kumar Male Indian Institute of Technology, Mandi PhD Panjab University, Chandigarh 2021  
8 58740 Mr. Himanshu Lekharu Male Jawaharlal Nehru University PhD student Jawaharlal Nehru University 2021  
9 58852 Mr. Utsav Dewan Male Indian Statistical Institute, Kolkata PhD Student Indian Statistical Institute, Kolkata 2020  
10 58874 Mr. Himanshu Baranwal Male IIT Madras Research Scholar IIT Madras    
11 58898 Mr Kamalakannan Vetriselvan Male The Gandhigram Rural Institute (Deemed to be University) Phd The Gandhigram Rural Institute (Deemed to be University) 2020  
12 58908 Ms. Sushmita Yadav Female Indian Institute of Technology Jodhpur PhD Student Indian Institute of Technology Jodhpur 2020  
13 58952 Mr Subith G Male IISER TRIVANDRUM Phd Calicut University 2020  
14 59013 Mr Swapnil Deepak Malegaonkar Male SRM University AP PhD Student Hyderabad Central University 2015  
15 59113 Mr Anubrato Bhattacharyya Male Presidency University PhD Tifr-CAM 2020  
16 59202 Ms Sharvari Vijay Pisal Female Mount Carmel College, Bengaluru MSc Student Mount Carmel College, Bengaluru    
17 59337 Ms. Moumita Manna Female IIT Bhubaneswar PhD Banaras Hindu University 2020  
18 59353 Mr Aaqib Ayoub Bhat Male Aligarh Muslim University, Aligarh, U.P. India PhD Aligarh Muslim University 2020  
19 59404 Ms. Anju Sundaran Female National Institute of Technology Calicut PhD University of Calicut 2020  
20 59940 Mr Aaryan Dharmesh Shah Male National Institute of Technology, Rourkela, Odisha, India Junior Research Fellow Indian Institute of Science Education and Research Bhopal 2024  
21 59569 Mr Ramesh Kumar Male Department of Mathematics, University of Delhi Ph.D. Hindu College, University of Delhi 2018  
22 59674 Mr Mohd Sharique Male Aligarh Muslim University Research scholar Aligarh muslim university 2022  
23 59713 Mr. Anurag Kumar Male National Institute of Technology Raipur PhD Ravenshaw University, Cuttack, Odisha 2017  
24 59975 Ms. Richa Jain Female Visvesvaraya National Institute of Technology, Nagpur PhD Dr. Harisingh Gour Viswavidyalaya, Sagar, M.P. 2020  
25 60002 Miss Khushi Mishra Female Indian Institute of Science Education and Research (IISER) Mohali Integrated PhD Student Indian Institute of Science Education and Research Mohali 2024  
26 60063 Mr. Arkajit Pal Choudhury Male Indian Statistical Institute, Kolkata SRF Mathematics Indian Statistical Institute, Kolkata 2020  
27 60256 Ms Shweta Wadhwani Female University of Delhi PhD K.B.Patil college, Vashi 2018  
28 60461 Mr. Krishna Mani Nath Male Gauhati University PhD Student Gauhati University 2023  
29 60471 Mr. Gourav Verma Male Dr B R Ambedkar National institute of technology , Jalandhar Phd University of Delhi 2022  
30 59850 Mrs Sivaranjani Ramasamy Female KPR Institute of Engineering and Technology Research Scholar M.Sc 2006  
31 59798 Mr. Subhamoy Mondal Male Presidency University PhD Student M.Sc. 2017  
32 58390 Mr. Alamgir Hossain Male Presidency University PhD Presidency University 2015  
33 58682 Ms. Akshita Aggarwal Female Central University of Punjab, Bathinda. MSc Student (completed) M.Sc Physics 2024  
34 59599 Mr. Rohit Nageshwar Male University of Delhi PhD Student M.Sc. 2020  
35 60267 Mr. Pranab Jyoti Hazarika Male Gauhati University PhD Tezpur Central University 2020  
36 60035 Mr. Gurubachan Gurubachan Male Indian Institute of Technology Jodhpur PhD Indian Institute of Technology Delhi 2020  
37 60278 Mr. Jomin K J Male Indian Institute of Information Technology Lucknow PhD M. Sc. 2017  
38 58530 Mr Priyanshu Chourasiya Male IISER Pune Bs Ms student IISER Pune    
39 58676 Mr. Naveenkumar Sureshkumar Yadav Male B. K. M. Science College, Valsad Asst. Professor The Maharaja Sayajirao University of Baroda 2015  
40 60059 Ms Monica S Female Vellore Institute of Technology,Vellore PhD Manonmaniam Sundaranar University 2019 08/01/2023
41 60260 Ms. Sakshi Sharma Female Dr B R Ambedkar National Institute of Technology, Jalandhar PhD Dr B R Ambedkar National Institute of Technology Jalandhar 2020  
42 60384 Ms. Charukeshini Saravanakumar Female Sri G.V.G. Visalakshi College For Women MSc Student Bharathiar University -  Sri G.V.G. Visalakshi College For Women Awaiting Result  
Local Participants
1 58723 Ms. Aiswarya T Female IIT Tirupati PhD scholar University of Calicut 2021  
2 59438 Ms Dipanwita Mandal Female IIT Tirupati PhD Student IIT Tirupati 2022  
3 60254 Ms. Pritika Raj Female IISER Tirupati BSMS Student   Appeared  
4 60258 Mr. Rajat Kumar Male IIT Tirupati PhD IIT TIRUPATI 2021  
5 60415 Mr. Joydip Roy Male IIT Tirupati MSc IIT Tirupati Appeared  

 

 

How to Reach: 

IIT Tirupati is located on the Yerpedu–Venkatagiri Highway. It is flanked by picturesque mountains and is 22 km from Tirupati town, 16 km from Renigunta Junction, 13 km from Srikalahasti Railway Station, and 14 km from the Tirupati International Airport.  Tirupati is a well-connected city with robust rail, road, and air networks. Here’s how to reach the IIT Tirupati campus:

1. From the Bus Station:
You can catch a "Tirupati-Venkatagiri" bus, which stops right at the campus entrance. Alternatively, take a "Tirupati-Srikalahasti" bus, get off at "Yerpedu junction", and then hire an auto-rickshaw. Yerpedu is just 1 km from the campus.

2. From the Railway Station:
There are two railway stations in Tirupati - Renigunta Junction and Tirupati Station. Renigunta is the closest one, about 12 km from the campus. You can take one of the aforementioned buses from there or opt for an auto-rickshaw (though be cautious as they can charge higher fares). Tirupati Station is only half a kilometre from the bus station, so you can easily walk there and catch a bus. 

3. From the Airport:
There are generally no RTC buses from the airport, so your best option is to take a taxi to reach the campus.

Safe travels!  

 

School Short Name: 

etf

Last Date Application: 

Monday, August 15, 2022

School Type: 

NCMW

Separate faculty form: 

0