The syllabus of the workshop has been designed in a way so that the major part of the workshop will be devoted to foundational aspects of the proposed courses. A salient feature of the proposed workshop is that besides foundational aspects, the participants will also be exposed to some advanced topics. These topics are mostly missing in the syllabus of colleges/universities though they are foundational to learn many modern aspects of Mathematics. We are planning a basic course on each topic of three to five hours. Whereever possible, this basic course will be followed up by a short advanced course, that in some cases could be a research exposition as well. The basic courses will be given by known experts. Currently, we are communicating with several mathematicians and some confirmed courses are already in cards.

In addition, we are planning to frame one or two plenary talks by eminent mathematicians as per availability of them. These plenary talks will be a overview of a broad theme and would connect several of the basic courses.

CONFIRMED COURSES :

In the following we summarize the confirmed courses. We have also mentioned the number of contact hours for each course. Under each item (a) denote the basic course and (b) the follow up to the basic course.

1. Course Title : Bilinear Forms and Its Applications (5 lectures)

Speaker : Prof. S.A. Katre (The University of Pune)

Course Outline: Lecture 1: Bilinear Forms, Symmetric Forms, Hermitian Forms

Lecture 2: Orthogonality in R^n and C^n w.r.t. the bilinear form

Lecture 3: Euclidean and Hermitian spaces. Spectral theorems for normal, Hermitian and unitary operators on Hermitian spaces and symmetric operators on Euclidean Spaces.

Lecture 4: Classification of Conics and Quadrics using spectral theorem for real symmetric matrices

Lecture 5: Skew-symmetric Forms.

2. Course Title: Hyperbolic Geometry (6 lectures)

Speaker: Dr. Krishnendu Gongopadhyay (IISER-Mohali)

Course Outline : Euclidean similarity geometry. Inversive geometry. Several models of the hyperbolic space. Hyperbolic distance, area and geodesics. The Mobius group: action of PSL(2, ℝ) on the hyperbolic space. Classifying different types of isometries. Hyperbolic triangles, Gauss-Bonnet theorem.

3. Course Title : Lie Groups And Lie Algebra (5 Lectures)

Speaker: Prof. Pralay Chatterjee (IMSc Chennai)

Course Outline: To be announced.

4. Course Title: CANONICAL FORMS (4 Lectures)

Speaker: Dr. Sudip Mazumder (Bangabasi College,Kolkata)

Course Outline : In this course firstly we shall discuss on canonical forms and in the last session I would like to raise and discuss on the following problem: Given a linear map T, when does T preserve an invariant bilinear or Hermitian form.

5. Course Title: Advanced Complex Analysis (4 lectures)

Speaker: Dr. Avijit Pal (IIT Bhilai)

Course Outline: Basic aspects of complex analysis, Branch and cuts, Schwartz’ Lemma, normal families, Hurwitz’s theorem, Riemann Mapping Theorem, Meromorphic function, Analytic continuation, Sheaves.

6. Course Title: Algebraic Topology-I (4 Lectures)

Speaker: Dr. Abhishek Mukherjee (Kalna College)

Course outline: Quotient Spaces, Homotopy of paths, fundamental group, covering spaces, examples of fundamental groups of important spaces, Borsuk-Ulam theorem, The Bisection theorem.

7. Course Title: Algebraic Topology-II (3 Lectures)

Speaker: Dr. Arpan Kabiraj (IIT PALAKKAD)

Course outline: Separation theorems in planes, Simplicial Complexes, Seifart-van Kampen Theorem, Classification of surfaces.

8. Course Title : An invitation to the theory of mapping class groups of surfaces (4 Lectures)

Speaker: Dr. Soumya Dey (Post-Doc Fellow, IMSc Chennai)

Course outline : The plan is to introduce the audience to the basics of mapping class groups of surfaces. Prerequisites for the lecture series would be undergraduate courses on group theory, point-set topology, basic knowledge of manifolds and understanding of the classification theorem for closed connected orientable surfaces.

9. Course Title : Commutative Algebra (4 Lectures)

Speaker : Dr. Ajay Kumar (IIT Jammu)

Course outline: To be announced.

10. Course Title : Category Theory (2 Lectures)

Speaker : Dr. Amartya Goswami (The University of Johannesburg, South Africa)

Course outline : TBA

11. Course Title : An Invitation to Geometric Group Theory (5 Lectures)

Speaker : Dr. Pranab Sardar (IISER Mohali)

Course outline : TBA

*More courses to be announced shortly

PLENARY TALKS :

1. Title of the talk : A sauntering in the number garden..

Speaker : Prof. Kalyan Chakraborty (HRI, Allahabad)

Abstract: This talk is about some well known problems/conjectures in number theory. I will begin by introducing some works of Ramanujan and then go on introducing these conjectures and their current status. This talk is a popular talk and will be accessible to all. A variety of prime numbers will be touched upon and the talk will also deal about certain interesting numbers too.

2. Title of the talk : Cryptography: The science of information security..

Speaker : Prof. Kalyan Chakraborty (HRI, Allahabad)

Abstract: After seeing prime numbers in the first talk this talk will show application in practical life of these prime numbers. Cryptography is the art of sending ``secured'' messages and the prime numbers play a major role in making the message `secured'. This is a very basic introduction to cryptography. We begin by looking into history and then go on introducing the basic concepts of a crypto protocol. I will discuss a couple of protocols. if time permits we talk about `integer factorization' too.

3. Title of the talk : Elliptic curve cryptography and applications-Classical and Post-Quantum paradigms..

Speaker : Prof. Kalyan Chakraborty (HRI, Allahabad)

Abstract: I will begin by recapping a bit of classical crypto which was the main theme in my 2nd talk.

Then introducing elliptic curves, i will move into classical elliptic curve cryptosystems. Then I will

touch upon a bit on quantum crypto and move again to elliptic curve crypto by proposing a new

area of study which seems to be `post quantum'. I will end with few practical applications.

4. Title of the talk: Elliptic Curves :Introduction and an aaplication

Speaker: Prof. Kalyan Chakraborty (HRI, Allahabad)

Abstract : A basic introduction to the arithmetic of elliptic curves will be the main aim. Will talk about the group of rational points in various fields. Then as an application I will present a very brief sketch of how Fermat's Last Theorem is been proved using elliptic curve arithmetic. It would be a very basic lecture.

5. Title of the talk : Mathematics and Periodic Table of Atoms

Speaker : Prof. Ravi Kulkarni (Bhaskaracharya Pratisthana)

Abstract: The Mendeleev’s Periodic Table of Atoms, whose 150th anniversary year was celebrated last year, is at the foundation of modern science. In this lecture we shall point out the role Mathematics has played in its formulation. There are many deeply philosophical issues about the nature of Mathematics involved in this development.

6. Title of the talk : Lie Groups and their representations:The case of SL(2,R)

Speaker : Prof. Dipendra Prasad (IIT Bombay)

Abstract: In this lecture aimed for Masters students, I will discuss the group SL(2,R) as a basic example of a Lie group and an Algebraic group. We will also discuss its representation theory.

7. Title of the talk : Schur Polynomials

Speaker : Prof. Amritanshu Prasad (IMSc Chennai)

Abstract : Schur polynomials are a family of symmetric polynomials that play a prominent role in representation theory and combinatorics. I will begin with Cauchy's definition of Schur polynomials using bi-alternants, then show that they form a basis of symmetric polynomials. I will then explain their importance in representation theory and algebraic combinatorics.

8. Title of the talk : An Invitation to Advanced Complex Analysis

Speaker : Prof. S Ponnusamy (IIT Madras)

Abstract : TBA