Commutative Algebra seminar
Speaker: Sai Krishna P M S (IIT Bombay)
Host: Tony Puthenpurakal
Title: Number of generators of a Cohen-Macaulay ideal
Time, day and date: 4:00:00 PM, Tuesday, February 18
Venue: Ramanujan Hall

Abstract

Commutative Algebra seminar
We look at some results related to the bound on the number of generators of an ideal of a Noetherian local ring. Under the assumption that the ideal is Cohen-Macaulay, we get a bound on the number of generators of an ideal in a Noetherian local ring (R) in terms of the embedding dimension of the ring, the dimension, and the multiplicity of the quotient ring. We extend this result to any Cohen-Macaulay ideal of a Noetherian ring.

Title : Harmonic functions on groups

Date and Time: 18 February, 2025.  4 PM - 5 PM

Venue: Classroom 216, department of Mathematics

Abstract : In this talk, we will discuss about finitely generated groups
with a probability measure and the associated theory of harmonic
functions. We shall study the notions of (weak) Liouville property and
strong Liouville property, and see a characterisation of strong Liouville
property. Further, we shall discuss about Green speed, Green metric and
observe their relations with strong Liouville property. Finally, we study
slowly growing harmonic functions and how they descend onto finite index
subgroups. This is based on an ongoing joint work with Mayukh Mukherjee
and Soumyadip Thandar.
 

Speaker: Pradyumna Datta Poduri
Title: Eigenvalues of Kneser graphs

Abstract: A computation of the eigenvalues of Kneser graphs using
equitable partitions will be presented and then this will be used to prove
the
Erdős–Ko–Rado theorem.

Those interested are cordially welcome.

Speaker: Omkar Ramadas
Date: Feb 19
Venue: Room 105
Time: 11:45 AM to 12:30 PM.
Title: Degrees and list chromatic numbers

Abstract: The list chromatic number is a notion introduced by Erdos, Rubin
and Taylor in their seminal paper. I will introduce this new variant of
the chromatic number in my seminar and will prove the lower bound on it
given by Noga Alon using probabilistic methods.
 

Speaker: Bhawani Singh

Title: GNS Representation and its Applications

Abstract: In this seminar, we will start with some basic notions of
C*-algebras and present the Gelfand–Naimark–Segal (GNS) construction,
providing a brief proof of the GNS representation theorem. We will
explicitly construct the GNS triple for C(X), where X is a compact
Hausdorff space, and examine its crucial application: proving that every
abstract C*-algebra is isometrically isomorphic to a closed subalgebra of
the bounded linear operators on a Hilbert space.

Title: Stinespring dilation theorem and its applications.

Date and time: 19 February, 2025. 4:20 pm - 5 pm

Venue: Class room 113 (Mathematics Department)

Abstract:  Stinespring's dilation theorem is a fundamental result in
operator algebra, stating that every completely positive (CP) map on a
C*-algebra can be realized as the compression of a *-homomorphism on an
extended Hilbert space.

In this talk, I will begin with a brief overview of completely positive
maps, illustrating their significance with examples. We will then outline
key steps in the proof of Stinespring’s theorem and explicitly construct
Stinespring representations for certain CP maps. Additionally, we will
discuss the unitary equivalence of minimal Stinespring representations and
see how the theorem generalizes the GNS construction.