Anupam Kumar Singh

Summer Workshop in Mathematics (27 July - 1 August 2009) IISER, Pune

Aim :
The aim of this workshop is to motivate students to take up Mathematics for research. We have invited eminent mathematicians of our country to speak in this workshop. We also want to create awareness about Mathematics education, in general. Thus, keeping this in mind we have requested each mathematician to be available in the afternoon to interact with audience.

Organisation :

Executive Organising Committee :
1. Prof. R. Sujatha (TIFR, Mumbai)
2. Prof. L. Shashidhara (IISER, Pune)
3. Prof. R. Ramachandran (IISER, Pune)
Local Organising Committee :
1. Dr. Anupam Kumar Singh (IISER, Pune)
2. Dr. Rama Mishra (IISER, Pune)

Invited Speakers :

Prof. Ramanan (CMI, Chennai) : Topology and Geometry
Prof. R. Sujatha (TIFR, Mumbai) : Number Theory
Prof. C. S. Dalawat (HRI, Allahabad) : Number Theory
Prof. Raja Sridharan (TIFR, Mumbai) : Algebra
Prof. R. Sridharan (CMI, Chennai) : Algebra
Prof. Siddhartha Bhattacharya (TIFR, Mumbai) : Algebra
Prof. Dinesh Thakur (University of Arizona) : Number Theory
Prof. S. A. Katre (University of Pune) : Number Theory
Prof. Siddhartha Gadgil (IISc, Bangalore) : Topology
Prof. P.N.Srikanth (TIFR center Bangalore) : Partial Differential Equations
Prof. H. Bhate (University of Pune) : Algebra
Prof. N. Raja (TIFR, Mumbai) : Computer Science
Prof. D. Dhar (TIFR, Mumbai) : Statistical Physics
Dr. Anupam Kumar Singh (IISER, Pune) : Algebra
Dr. Aniruddha Naolekar (IISER, Pune) : Algebraic Topology
Dr. Anita Naolekar (IISER, Pune) : Algebraic Topology
Dr. Ayan Mahalanobis (IISER, Pune) : Cryptography/Group theory
Dr. Rama Mishra (IISER, Pune) : Low dimensional Topology/knot theory
Dr. Soumen Maity (IISER, Pune) : Discrete Mathematics/Combinatorics
Suhas Pandit (IISER, Pune) : Geometric Group theory
Dr. Pranay Goel (IISER, Pune) : Maths-Biology

Schedule :
There will be usually two lectures (one hour each) in the morning and discussion sessions (compulsory to participate) in the afternoon at 2:30 PM . The tentative schedule is below. Note that some of the lecture could be reshuffled according to availability of speakers.

Date and Time Speaker Title Abstract

27 July 2009, 10:00 AM Inauguration

27 July 2009, 10:30 AM Prof. Sridharan, CMI Chennai Enumeration of some combinatorial patterns in Ancient India Not received yet

27 July 2009, 11:45 AM Prof. Raja Sridharan, TIFR Mumbai Pell's Equation We will consider the Equation of Pell from a historical context and outline the historical contributions of the Indian Mathematicians to this topic and relate it to more recent Developments like the Units of Quadratic Fields, Pfister forms etc.

28 July 2009, 10:00 Noon Prof. P. N. Srikanth, TIFR Bangalore Partial Differential Equations: why study them Not recieved yet

28 July 2009, 11:30 AM Prof. D. Thakur, University of Arizona Good Approximations Not received yet

29 July 2009, 10:00 AM Prof. Siddhartha Gadgil, IISc Bangalore One can not hear the shape of a drum The notes produced by a drum depend on its shape and size. The isospectral problem asks whether two drums of different shapes can produce the same set of notes. In this talk, we present a simple example of such a pairs of drums (due to Buser, Conway, Doyle and Semmler). We also discuss some related mathematics, in particular the so called reflection principle.

29 July 2009, 11:30 AM Prof. C. S. Dalawat, HRI Allahabad p-adic numbers Around a hundred years ago, a new number system was discovered. it is called the system of p-adic numbers, and there is one such system for every prime number p. We will give an elementary introduction to these numbers and show what they good for.

30 July 2009, 10:00 AM Prof. Siddhartha Bhattacharya, TIFR Mumbai Perron-Frobenius theorem In this talk we will consider large matrices with strictly positive entries. We will prove a result about eigenvalues and eigenvectors of such matrices, and discuss its application in internet search engines and tournaments.

30 July 2009, 11:30 AM Prof. D. Dhar, TIFR Mumbai Some Enumeration Problems from Statistical Physics In this talk, I will discuss how the main problem in equilibrium statistical physics can be reduced to that of finding the number of different states available to a given system having energy in a given range E_1 < E < E_2. We start by enumerating the number of random walks of N steps on a line, or on a square lattice. The number of walks having a specified endpoint R has many applications in the theory of diffusion. And it introduces the technique of Generating Functions for enumeration problems. More complicated problems are like self-avoiding walks. We discuss upper and lower bounds for the number of self-avoiding walks, and introduce the computer algorithms for enumeraion. Finally, I discuss the number of partitions of an integer N to illustrate the power of the method of generating functions. Some harder problems will be listed, but left as home-wok problems.

31 July 2009, 10 AM Prof. N. Raja, TIFR Mumbai The Axiomatic Technique in Logic & Mathematics This talk will provide a very short introduction to the elements of the axiomatic technique, and some of its associated notions such as consistency, completeness, categoricity, and independence.

31 July 2009, 11:30 AM Prof. R. Sujatha, TIFR Mumbai The Mysteries of the prime numbers Not received yet

1 August 2009, 9:30 AM Prof. Ramanan, CMI, Chennai Symmetries in every day life and in the Sciences Not received yet

1 August 2009, 11:00 AM Prof. H. Bhate Fourier Analysis on Groups Not received yet

1 August 2009, 12:00 Noon Prof. S. A. Katre, Pune University The Mobius Function In this talk we shall discuss the Mobius function, its properties, its importance in the theory of arithmetic functions and its application to factoring polynomials of the type x^n-y^n over integers.

Prerequisite :
We have requested all our speakers to keep the lectures elementary and target interdisciplinary audience. Hence, as such, there is no prerequisite for this workshop except enthusiasm and readiness.

Date and Time	Speaker	Title	Abstract
27 July 2009, 10:00 AM		Inauguration
27 July 2009, 10:30 AM	Prof. Sridharan, CMI Chennai	Enumeration of some combinatorial patterns in Ancient India	Not received yet
27 July 2009, 11:45 AM	Prof. Raja Sridharan, TIFR Mumbai	Pell's Equation	We will consider the Equation of Pell from a historical context and outline the historical contributions of the Indian Mathematicians to this topic and relate it to more recent Developments like the Units of Quadratic Fields, Pfister forms etc.
28 July 2009, 10:00 Noon	Prof. P. N. Srikanth, TIFR Bangalore	Partial Differential Equations: why study them	Not recieved yet
28 July 2009, 11:30 AM	Prof. D. Thakur, University of Arizona	Good Approximations	Not received yet
29 July 2009, 10:00 AM	Prof. Siddhartha Gadgil, IISc Bangalore	One can not hear the shape of a drum	The notes produced by a drum depend on its shape and size. The isospectral problem asks whether two drums of different shapes can produce the same set of notes. In this talk, we present a simple example of such a pairs of drums (due to Buser, Conway, Doyle and Semmler). We also discuss some related mathematics, in particular the so called reflection principle.
29 July 2009, 11:30 AM	Prof. C. S. Dalawat, HRI Allahabad	p-adic numbers	Around a hundred years ago, a new number system was discovered. it is called the system of p-adic numbers, and there is one such system for every prime number p. We will give an elementary introduction to these numbers and show what they good for.
30 July 2009, 10:00 AM	Prof. Siddhartha Bhattacharya, TIFR Mumbai	Perron-Frobenius theorem	In this talk we will consider large matrices with strictly positive entries. We will prove a result about eigenvalues and eigenvectors of such matrices, and discuss its application in internet search engines and tournaments.
30 July 2009, 11:30 AM	Prof. D. Dhar, TIFR Mumbai	Some Enumeration Problems from Statistical Physics	In this talk, I will discuss how the main problem in equilibrium statistical physics can be reduced to that of finding the number of different states available to a given system having energy in a given range E_1 < E < E_2. We start by enumerating the number of random walks of N steps on a line, or on a square lattice. The number of walks having a specified endpoint R has many applications in the theory of diffusion. And it introduces the technique of Generating Functions for enumeration problems. More complicated problems are like self-avoiding walks. We discuss upper and lower bounds for the number of self-avoiding walks, and introduce the computer algorithms for enumeraion. Finally, I discuss the number of partitions of an integer N to illustrate the power of the method of generating functions. Some harder problems will be listed, but left as home-wok problems.
31 July 2009, 10 AM	Prof. N. Raja, TIFR Mumbai	The Axiomatic Technique in Logic & Mathematics	This talk will provide a very short introduction to the elements of the axiomatic technique, and some of its associated notions such as consistency, completeness, categoricity, and independence.
31 July 2009, 11:30 AM	Prof. R. Sujatha, TIFR Mumbai	The Mysteries of the prime numbers	Not received yet
1 August 2009, 9:30 AM	Prof. Ramanan, CMI, Chennai	Symmetries in every day life and in the Sciences	Not received yet
1 August 2009, 11:00 AM	Prof. H. Bhate	Fourier Analysis on Groups	Not received yet
1 August 2009, 12:00 Noon	Prof. S. A. Katre, Pune University	The Mobius Function	In this talk we shall discuss the Mobius function, its properties, its importance in the theory of arithmetic functions and its application to factoring polynomials of the type x^n-y^n over integers.

Participation :
Confirmed Participants
This workshop is mainly for students at IISER. We can accommodate about 10-15 students/teachers from Pune as well. Interested people (including IISER, students) who want to attend this workshop please send an email to Dr. Anupam Kumar Singh (email: anupamATiiserpuneDOTacDOTin) expressing their interest. No help, such as, TA/DA, transport, accommodation etc. is provided for the participation.