AIS Quadratic Forms (2014)

Brief Description of School :
A quadratic form has an innocuous definition, a homogeneous polynomial in degree 2, but has far-reaching connections to many parts of mathematics; linear algebraic groups, division algebras, modular forms and class field theory, to name a few. Quadratic form may also be used to assess how complicated a field is. The main focus of the workshop is the study of various cohomological invariants associated to quadratic forms so as to classify quadratic forms in some cases. This will provide a background to study numerical invariants such as level, Pythagorean number and the recent progress on u-invariant. In this AIS, we intend to learn basics of quadratic forms so as to be able to learn more about these invariants in a follow-up workshop a year later.

We will follow the well-known book Introduction to quadratic forms over fields by T. Y. Lam. We are interested in the algebraic theory of quadratic forms. So we expect a prospective participant to be familiar with linear algebra over arbitrary fields and abstract algebra including Galois theory.

Syllabus and Lecturer
Prerequisites: Linear algebra over arbitrary fields and Galois theory. This material can be fin many books, for instance, in:

1. Artin: Algebra
2. Dummit, Foote: Abstract Algebra.
3. Jacobson: Basic Algebra.
4. Lang: Algebra.

Syllabus: T. Y. Lam: Introduction to quadratic forms over fields

References:

1.  Larry J. Gerstein: Basic quadratic forms.
2.  O'Meara: Introduction to quadratic forms.
3.  Serre: A course in arithmetic.

 Speakers/Tutors :

1. A. Chandrakant Sharma, IISER Mohali (Ch. 1. Foundations)
2. Anuradha Garge, University of Mumbai (Ch. 2. Introduction to Witt rings)
3. Shripad Garge, IIT Mumbai (Ch. 3, 4. Quaternion algebras and the Brauer-Wall group)
4. Maneesh Thakur, ISI Delhi (Ch. 5. Clifford algebras)
5. Ch. V. G. Narasimha Kumar, IIT Hyderabad (Ch. 6. Local fields and global fields)
6. Amit Kulshrestha, IISER Mohali (Ch. 11. Field invariants)

There  will  also be guest lectures by senior  mathematicians

1. Anupam Singh, IISER Pune (T)
2. Anuradha Nebhani, IIT Mumbai (T)
3. Anand Sawant, TIFR Mumbai (T)
4. Anirban Bose, ISI Bangalore (T)
5. Preena Samuel, University of Hyderabad (T)
6. Pratyusha Chattopadhyay, ISI Bangalore (T)

Typical daily schedule:

Young researchers.
1. Dilpreet Kaur, IISER Mohali (Quadratic forms over fields of characteristic 2)
 

How to reach

• By Air: Arrive at Chandigarh airport. There are many of connecting flights from Delhi/Mumbai to Chandigarh.
• By Train: Nearest railway stations are
  
  1. Chandigarh (Station Code: CDG, 14 Km from IISER)
  2. Ambala (Station Code: UMB, 50 Km from IISER)
  3. Mohali (Station Code: SASN, 2 Km from IISER): There are very few connections to Mohali. One useful connection from Delhi is Una Jan-Shatabdi (Train No. 12057)
• From Airport/Chandigarh Railway station to IISER: Take a Radio taxi/auto (phone numbers: 0172-4141414, 0172-4242424, 0172-4646464) to IISER Mohali. Almost all the drivers know the location and way to IISER. Neighbouring locations are Indian School of Business (ISB) Mohali and National Institute of Pharmaceutical Education and Research (NIPER) Mohali.  The distance between airport and IISER is about 10 kms and between Chandigarh Railway station and IISER is about 14 Kms. Mohali Railway station (called Sahibzada Ajit Singh Nagar station or SAS Nagar station) is just 2 km from IISER.