Convener(s)
| Name: | Prof. Jaydeb Sarkar | Dr. Arup Chattopadhyay | Dr. Amit Maji |
| Mailing Address: | Professor, Statistics and Mathematics Unit, ISI Bangalore, Bangalore– 560059, Karnataka, India |
Associate Professor, |
Associate Professor, Department of Mathematics Indian Institute of Technology Roorkee Roorkee-247 667, Uttarakhand, India |
| Email: | jaydeb at gmail.com | arupchatt at iitg.ac.in | amit.maji at ma.iitr.ac.in |
Operator algebras and operator theory, central areas of functional analysis, interact significantly with fields such as topology, complex analysis, harmonic analysis, and group theory. Originally motivated by problems in quantum mechanics and spectral theory, these subjects provide a powerful framework for studying linear operators on Hilbert and Banach spaces. Over time, they have also played a crucial role in fields such as non-commutative geometry, representation theory, and mathematical physics.
The aim of this programme is two-fold. We start with the basics of functional analysis and operator theory, which includes topics in C^*-algebras and basic of Hardy space theory. Then we move our discussion to a part of multivariable operator theory, namely dilation theory and Nevanlinna-Pick interpolations, a classical topic in modern operator theory. In the second part, we will discuss some basics of von Neumann algebras, including the GNS construction. At the end, we will give a brief flavour of trace formulae in operator theory as a part of non-commutative analysis.
This AIS program is aimed at postgraduate and Ph.D. students. It will introduce key classical and modern developments through mini-courses and problem sessions. The courses will be largely self-contained and accessible, requiring only a basic background in complex analysis and functional analysis.
Dates:
Venue:
Venue Address:
Department of Mathematics
Indian Institute of Technology Roorkee
Roorkee-247 667, Uttarakhand, India
Venue State:
Venue City:
PIN:
Syllabus:
|
Name of the Speaker with affiliation |
No. of Lectures |
Detailed Syllabus |
|
Prof. B V Rajarama Bhat (ISI Bangalore Centre) |
6 hrs |
Basics of (C^*)-algebras, the Gelfand–Naimark theorem, continuous functional calculus, the spectral theorem for normal operators, positive linear functionals, representations and the GNS construction, completely positive maps, and the Stinespring dilation theorem. |
|
Dr. Bata Krishna Das (IIT Bombay) |
9 hrs |
RKHS, Sums and products of kernels, Multiplier algebras on RKHS, Wold decomposition of isometries, transfer function realization of Schur functions and BCL factorization of the Hardy shift, Isometric Dilation of single and n-tuple of commuting contractions, Ando dilation using BCL model and refined von Neumann inequality on distinguished varieties. |
|
Prof. Jaydeb Sarkar |
9 hrs |
H^p spaces, Beurling's theorem, Model spaces, Characteristic functions, Commutant lifting theorem, Toeplitz operators, Hankel operators, Schur functions and representations of Schur functions, Nevanlinna-Pick interpolations, Schur functions and interpolation on polydisc and ball. |
|
Dr. Ved Prakash Gupta (Jawaharlal Nehru University) |
9 hrs |
Von-Neumann Algebra, von Neumann’s double commutant theorem and Kaplansky’s density theorem, normal linear functionals, geometry of projections in a von Neumann algebra and type decomposition of von Neumann algebras, abelian von Neumann algebras, the GNS construction. |
|
Dr. Arup Chattopadhyay |
6 hrs |
Trace class operators, Schatten Class, Second-order trace formulas, Carey-Helton-Howe-Pincus trace formula, Singular traces and perturbation formulae for tuples of commuting unitaries, Trace formula for contractions. |
Tutorial Assistants:
|
S. No. |
Name |
Affiliation |
|
1 |
Dr.Sibaprasad Barik |
ISI Bangalore |
|
2 |
Dr. Samir Panja |
ISI Bangalore |
|
3 |
Mr. Arghya Chongdar |
ISI Bangalore |
Time Table:
|
Day |
Date |
Lecture 1 (9.30 |
Tea (11.05 to |
Lecture 2 (11.30 |
Lunch (1.05 |
Tutorial (2.30 |
Tea (3.35 to |
Tutorial (4.00 |
Snacks 5.05 |
|
|
|
(name of the speaker |
|
(name of the speaker |
|
(name of the speaker + tutors) |
|
(name of the speaker + tutors) |
|
|
Tues |
15/12/26 |
BVRB |
|
BKD |
|
BVRB+ ARC+SP |
|
BKD+SB+SP |
|
|
Wed |
16/12/26 |
BVRB |
|
BKD |
|
BVRB+ ARC+SP |
|
BKD+SB+SP |
|
|
Thu |
17/12/26 |
BVRB |
|
BKD |
|
BVRB+ ARC+SP |
|
BKD+SB+SP |
|
|
Fri |
18/12/26 |
BVRB |
|
BKD |
|
BVRB+ ARC+SP |
|
BKD+SP+SB |
|
|
Sat |
19/12/26 |
BKD |
|
BKD |
|
BKD+ ARC+SP |
|
BKD+SB+SP |
|
|
SUNDAY: OFF |
|||||||||
|
Mon |
21/12/26 |
JS |
|
VPG |
|
JS+SB+SP |
|
VPG+ARC+SP |
|
|
Tues |
22/12/26 |
JS |
|
VPG |
|
JS+SB+SP |
|
VPG+ARC+SP |
|
|
Wed |
23/12/26 |
JS |
|
VPG |
|
JS+SB+SP |
|
VPG+ARC+SP |
|
|
Thu |
24/12/26 |
JS |
|
VPG |
|
Lecture by AC |
|
VPG+ARC+SP |
|
|
Fri |
25/12/26 |
JS |
|
VPG |
|
Lecture by AC |
|
AC+SB+SP |
|
|
Sat |
26/12/26 |
JS |
|
VPG |
|
Lecture by AC |
|
VPG+ARC+SP |
|
|
Sun |
27/12/26 |
AC |
|
AC |
|
AC+SB+SP |
|
AC+SB+SP |
|
**The timetable has been adjusted due to the unavailability of accommodation.
Full forms for the abbreviations of speakers and tutors:
- BVRB: B V Rajarama Bhat
- JS: Jaydeb Sarkar
- BKD: Bata Krishna Das
- VPG: Ved Prakash Gupta
- AC: Arup Chattopadhyay
- SB: Sibaprasad Barik
- SP: Samir Panja
- ARC: Arghya Chongdar
Selected Applicants:
How to Reach:
TBA