# ATMW Probabability and Representation Theory (2016)

Venue: |
The Institute of Mathematical Sciences, Chennai |

Dates: |
7 |

Convener(s) |
Speakers, Syllabus and Time table |
Applicants/Participants |

Name |
Prof. Arvind Ayyer | Prof. K N Raghavan |

Mailing Address |
Assistant Professor Department of Mathematics Indian Institute of Science Bangalore 560 012 |
Professor H Mathematics Group The Institute of Mathematical Sciences, CIT Campus, Taramani Chennai 600 113 |

** **

There were five lecture courses in all. Listed below in separate subsections are the synopses of the courses in the words of the respective speakers.

2.1. **Vershik-Okounkov theory for representations of the symmetric group: 4 lectures ****by Murali Srinivasan of IIT Mumbai**. The first two lectures covered basic (ordinary) representation theory of finite groups. The third and fourth lectures covered the Vershik- Okounkov approach to the representation theory of the symmetric group. These results

were used to determine eigenvalues of the transitions matrices involved in the transpose- top-to-random and the random-transposition models of card shuffling.

2.2. **Markov chains: 2 lectures by Antar Bandyopadhyay of ISI Delhi**. Topics covered:Discrete Markov chains with countable state space, Examples including two state chain,Simple Symmetric Random Walk, random walk on d-dimensional integer lattice, Ehren-fest chain, and card shufflings. Classification of states, recurrence and transience; irre-ducibility, decomposition of state space into irreducible classes, examples. Stationary distributions, limit theorems, positive and null recurrence. Definition of periodicity, ape-riodic chains. Limit theorems for aperiodic irreducible chains, proof for positive recurrent case using coupling. Total variation distance and definition of mixing time.

2.3.** Computer algebra: 2 lectures by Amritanshu Prasad of IMSc**. Topics covered: In-troduction to Sage, basics of python programming, construction of some transition ma-trices using Sage. Irreducible representations of symmetric groups in Sage. Use of Sage’s

symmetrica interface to compute character values of symmetric groups.

2.4. **Semigroup theory: 2 lectures by Pooja Singla of IISc**. Topics covered: Interplay between representation theory of semigroup algebras and Markov chains on semigroups.Semilattice and Left regular band semigroups. Eigenvalues of Tsetlin library.

2.5. **Card shuffling models: 4 lectures by Arvind Ayyer of IISc**. The following random walk models on the symmetric group were discussed, with emphasis on the stationary distribution, eigenvalues of transition matrices, and mixing times:

(1) Random transpositions

(2) Transpose-top-to-random model

(3) Tsetlin library

(4) Top-to-random insertions

(5) Random-to-random insertions

There were four half-hour research talks as follows:

• Amritanshu Prasad (IMSc): Representations of the symmetric group with non-trivial determinant

• Sushil Bhunia (IISER, Pune): z-classes in unitary groups

• Ashish Mishra (CMI): Vershik-Okounkov approach to representations of the wreath product

• Hassain M (IISc): Representations of SL_{2} (Z/16Z) and SL_{2} (F_{2} [t]/t^{4} )

## Speakers and Schedule

Timings | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
---|---|---|---|---|---|---|

09:00 – 09:30 | Breakfast | |||||

09.30 – 11:00 | Representation Theory 1 |
Representation Theory 2 |
Vershik-Okounkov Theory 1 |
Vershik-Okounkov Theory 2 |
Model 3: Tsetlin Library |
Model 4: Random-to-random shuffle |

11:00 – 11:30 | Coffee/Tea break | |||||

11.30 – 13:00 | Markov Chain Theory 1 |
Markov Chain Theory 2 |
Semigroup Theory 1 |
Semigroup Theory 2 |
Research talks 1 and 2 |
Discussion |

13:00 – 14:30 | Lunch break | |||||

14:30 – 16:00 | Computer Algebra 1 |
Model 1: Random Transpositions |
Computer Algebra 2 |
Model 2: Top-to-random shuffle |
Research talks 3 and 4 |
Free |

16:00 – 16:30 | Coffee/Tea break | |||||

16.30 – 17:30 | Discussion | Discussion | Discussion | Discussion | Discussion |

## Local organisers

Arvind Ayyer (IISc Bangalore)

Amritanshu Prasad (IMSc Chennai)

K N Raghavan (IMSc Chennai)

** **

Actual Participants |

Sr. |
SID |
Full Name |
Gender |
Affiliation |
City |
Position in College/ University |
University/ Institute M.Sc./ M.A. |
Year of Passing M.Sc./ M.A |
Ph.D. Deg |

1 | 5593 | Mr. Tulasi Ram Reddy Annapareddy | Male | Indian Insitute of science | Bengaluru | Junior Research Associate | Indian Statistical Institute | 2010 | |

2 | 5664 | Mr. Prashant Vasantrao Uswadkar | Male | Shiv Nadar University | Gautam Buddha Nagar | PhD student | University of Pune | 2010 | |

3 | 5679 | Mr. Debashish Bose | Male | Shiv Nadar University | Gautam Buddha Nagar | Asst. Prof. | IIT Kanpur | 2001 | 12/1/2010 |

4 | 5681 | Mr. Subhajit Ghosh | Male | Indian Institute of Science | Bangalore | Int. PhD | Indian Institute of Science, Bangalore | 2015 | |

5 | 5752 | Mr. Dipankar Roy | Male | Indian Institute of Science | Bangalore | PhD | IIT Kharagpur | 2015 | |

6 | 5753 | Mr. Pradeep Kumar | Male | Central University Of South Bihar | Patna | MPhil-PhD Student | Central University of South Bihar | 2014 | |

7 | 5870 | Ms. Rashmi Jain | Female | IISER , Mohali | Mohali | MSc Student | IISER , Mohali | Appeared / Awaiting Result | |

8 | 5934 | Ms. Priyanga Ganesan | Female | National Institute of Science Education and Research | Khurda | Integrated M.Sc - fourth year student | National Institute of Science Education and Research | Appeared / Awaiting Result | |

9 | 5935 | Mr. Sanjoy Kumar Jhawar | Male | Indian Institute of Science Bangalore | Bangalore | PhD | IIT Delhi | 2014 | |

10 | 5994 | Ms. Yasmeen Shameem Akhtar | Female | Indian Institute of Science Education and Research | Pune | PhD Student | S.P. Pune University | 2010 | |

11 | 6002 | Mr. Soumitra Das | Male | North-Eastern Hill University | Shillong | Phd Student | North-Eastern Hill University | 2012 | |

12 | 6036 | Mr. Pritam Majumder | Male | Tata Institute of Fundamental Research | Mumbai | PhD | IIT Kanpur | 2014 | |

13 | 6199 | Mr. Jyotirmoy Ganguly | Male | IISER, Pune | Pune | Ph.D | University of Hyderabad | 2013 | |

14 | 6200 | Mr. Harshit Yadav | Male | IIT, Kanpur | Kanpur | B.S. student | N/A | Appeared / Awaiting Result | |

15 | 6214 | Mr. Nihar Prakash Gargava | Male | IIT, Kanpur | Kanpur | Student | None | Appeared / Awaiting Result | |

16 | 5630 | Mr Uday Bhaskar Sharma | Male | The Institute of Mathematical Sciences | Chennai | Senior Research Fellow | University of Hyderabad M.Sc. | 2011 | |

17 | 5637 | Mr Sridhar Poojyam Narayanan | Male | Institute of Mathematical Science | Chennai | PhD | BITS Pilani | 2012 | |

18 | 5887 | Dr. Kumarapandiyan G | Male | Indian Statistical Institute, Chennai Center | Chennai | Post Doctoral Fellow | Pondicherry University | 2011 | 1/21/2015 |

19 | 6230 | Mr. Krishanu Roy | Male | Institute of Mathematical Sciences | Chennai | JRF | Institute of Mathematical Sciences | 2015 | |

20 | 6311 | Ms. Ananya Lahiri | Female | Chennai Mathematical Institute | Chennai | DST INSPIRE fellow | 2005 | ||

21 |
Prof. Anupam Singh |
IISER Pune |
|||||||

22 |
Mr. Pritam Majumdar |
TIFR, Mumbai. |
|||||||

23 |
Prof. Suresh Govindarajan |
IIT Madras |
|||||||

24 |
Prof. S. Krishnan |
IIT Bombay |
|||||||

25 |
Mr. Muthuvel Murugan |
CMI, Chennai |
|||||||

26 |
Mr. Anish Mukherji |
CMI, Chennai |
|||||||

27 |
Ms. N. Karimilla Bi |
IMSc, Chennai |
|||||||

28 | Mr. Debajyoti | Chennai Mathematical Institute |

**How to reach**

The Chennai neé Madras, and to IMSc, the Institute of Mathematical Sciences. Here is some information on getting to the Institute. Address and phone numbers are

THE INSTITUTE OF MATHEMATICAL SCIENCES

C I T CAMPUS, TARAMANI, CHENNAI 600 113

Telephone Numbers

Here are some maps to give you a rough idea of the layout of Madras and the area around the Institute.

- A simplified schematic map of Madras, showing the major roads and Hotels and marking the area in which the CIT Campus is located.
- A postscript version of this figure if you want to download it to print at your end.
- A detailed Madras map.
- A gzipped postscript version of this figure
- A schenatic map of the location of IMSc within CIT Campus and its environs
- Detailed map of the location of IMSc within CIT Campus and other landmarks in the Greater Adyar area.

The Institute of Mathematical Sciences is located in a campus-like area variously known as the ``CIT Campus'' or the ``CPT Campus''. Many people in Madras may not have heard of the Institute, but almost everyone would have heard of the M.G.R. Film City and Tidel Park which are further down in the same campus area. IMSc has two complexes. One houses the main Institute building, the library building and the new building. The other has the students' hostel and Guest House. These two complexes are across the road from each other.