Commutative Algebra seminar
Speaker: Sayed Sadiqul Islam (IIT Bombay)
Title: Frobenius endomorphism and applications
Time, day and date: 4:00:00 PM, Tuesday, February 4
Venue: Ramanujan Hall

Abstract
We explore the use of the Frobenius homomorphism to study the structure of local cohomology modules over rings of characteristic p. While local cohomology modules are not always finitely generated, we show that they exhibit important finiteness properties under certain conditions. Specifically, we use filter regular sequences to provide a proof that the set of associated prime ideals of the local cohomology modules is finite when the ring R has finite F-representation type or a finite singular locus. We also prove same results for regular rings of characteristic p.

Colloquium

Speaker : Aram Bingham

Day/Date/Time: 4:00 pm, Wednesday, 5th February

Venue : Ramanujan Hall

Title: Kronecker coefficients, polytopes, and complexity

Abstract : The “Kronecker coefficients problem” is one of the last major open  questions in the classical representation theory of symmetric groups. It  asks for a combinatorial rule describing the decomposition of tensor products of irreducible symmetric group representations, and a solution is known only in limited special cases. Kronecker coefficients have also
been the subject of much recent research motivated by the geometric complexity theory (GCT) program of Mulmuley and Sohoni, who hypothesized efficient computation of these numbers as part of a strategy to separate the computational complexity classes P and NP. We will discuss some of the main representation theoretic questions related to GCT and report some progress on computing Kronecker coefficients via discrete volumes of polytopes (joint work with Ernesto Vallejo).

Commutative Algebra seminar
Speaker: Sayed Sadiqul Islam (IIT Bombay)
Title: Frobenius endomorphism and applications ii
Time, day and date: 4:00:00 PM, Tuesday, February 11
Venue: Ramanujan Hall

Abstract
We explore the use of the Frobenius homomorphism to study the structure of local cohomology modules over rings of characteristic p. While local cohomology modules are not always finitely generated, we show that they exhibit important finiteness properties under certain conditions. Specifically, we use filter regular sequences to provide a proof that the set of associated prime ideals of the local cohomology modules is finite when the ring R has finite F-representation type or a finite singular locus. We also prove same results for regular rings of characteristic p.

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.
 

Annual Progress Seminar
Speaker: Aamir Yousuf (IIT Bombay)
Host: Neela Nataraj
Title: Numerical analysis of a thermoelastic diffusion plate model
Time, day and date: 11:00:00 AM, Monday, February 24
Venue: Online (Link in attached file)
Abstract: We investigate the well-posedness of a coupled 
hyperbolic-parabolic system modeling diffusion in thermoelastic plates, 
consisting of a fourth-order hyperbolic PDE for plate deflection and 
second-order parabolic PDEs for the first moments of temperature and 
chemical potential. The unique solvability is established via the 
Galerkin approach, and additional regularity of the solution is obtained 
under appropriately strengthened given data. For numerical 
approximation, we employ the Newmark method for time discretization and 
a C 0 -interior penalty (C0-IP) scheme for the spatial discretization of 
displacement. For the first moments of temperature and chemical 
potential, we use the Crank-Nicolson method for time discretization and 
conforming finite elements for spatial discretization. The convergence 
of the fully discrete scheme is established, achieving quasi-optimal 
rates in space and a quadratic rate in time. Several numerical examples 
will be presented to validate the theoretical findings.
 

Annual Progress Seminar
Speaker: Aratrika Pandey (IIT Bombay)
Host: Ravi Raghunathan
Title: Exact Approximation in the field of formal series
Time, day and date: 4:00:00 PM, Monday, February 24
Venue: Ramanujan Hall
Abstract: We prove a lower bound for the Hausdorff dimension of the set 
of exactly $\psi$-approximable matrices with values in a local field of 
positive characteristic. This is the analogue of the corresponding 
theorem of Bandi and de Saxc\'e over the reals, and is a higher 
dimensional version of a theorem.

Seminar
Speaker: Dr. Rajeev Das
Host: Radhendushka Srivastava
Title: Design and Applications of Robust Metaheuristics for Constrained 
Optimization Problems
Time, day and date: 4:00:00 PM, Monday, February 24
Venue: Online (https://meet.google.com/dtw-ztpe-jnc)
Abstract: The presentation includes a detailed discussion of my Doctoral 
research based on the development and applications of efficient 
metaheuristic algorithms for solving constrained optimization problems 
(COPs). The primary focus is on the Quadratic Approximation guided Jaya 
Algorithm (JaQA) and its enhanced variant, the Self-Adaptive 
Multi-Population Quadratic Approximation guided Jaya (SMP-JaQA). These 
two algorithms address challenges in solution diversity and balancing 
exploration-exploitation, making them highly effective in solving 
complex real-world optimization problems. The presentation begins with 
an introduction to COPs and the significance of metaheuristics in 
overcoming traditional optimization limitations. A comprehensive 
discussion on the research motivation, problem statement, and the 
identified gaps in existing methodologies follows. The design and 
implementation of these two proposed methodologies is elaborated upon, 
along with their computational complexity analysis. Experimental 
validation is carried out using constrained benchmarks from CEC 2006, 
CEC 2010, and CEC 2017. The outcomes demonstrate the superior 
performance of the proposed algorithms when compared to state of-the-art 
techniques, validated through statistical analysis (say, Friedman Rank 
Test and Wilcoxon Signed Rank Test), exploration-exploitation analysis, 
convergence analysis, etc. Real-life applications of the research are 
also highlighted in areas such as single-stage and two-echelon uncertain 
Fixed Charge Transportation Problems (FCTP), dairy feed ration 
optimization, mechanical design and electronics design optimization. The 
discussion concludes with an overview of research contributions, key 
findings, and potential future research directions. Moreover, insights 
into the publication status of my research so far is also provided 
showcasing peer-reviewed journal and conference publications along with 
communicated works.

Analysis seminar
Speaker: Kapil Jaglan (IIT Ropar)
Host: Prachi Mahajan
Title: A study on Planar Harmonic Mappings and Minimal surfaces
Time, day and date: 5:00:00 PM, Monday, February 24
Venue: Online (https://meet.google.com/woo-psdw-wdj)
Abstract: This talk explores the properties of univalent harmonic 
functions from a geometric function theoretic perspective and 
establishes connections between these functions and minimal surfaces. 
The talk consists of four sections followed by future direction, with 
the initial section serving as an introduction and containing basic 
definitions and results from the existing literature.

In the second section, our main aim is to determine a geometric 
condition under which a locally univalent harmonic mapping f defined on 
the unit disk D is univalent, and maps D onto a linearly accessible 
domain of order β for some β ∈ (0, 1). A linearly accessible domain (a 
non-convex domain) is important because, under certain sufficient 
conditions stated by Dorff et al., minimal graphs over these domains are 
area-minimizing. As a consequence, we derive sufficient conditions for f 
to map D onto a linearly accessible domain of order β, in the form of a 
convolution result, and a coefficient inequality. By extending the ideas 
of Dorff et al., we construct one-parameter families of globally 
area-minimizing minimal surfaces over a linearly accessible domain of 
order β.

In the third section, we explore the properties of odd univalent 
harmonic functions. Our starting point of investigation is to obtain the 
sharp coefficient estimates for odd univalent functions exhibiting 
convexity in one direction. We then advance our investigation to more 
generalized classes, including major geometric subclasses of 
sense-preserving univalent harmonic mappings. We examine the growth 
pattern of odd univalent harmonic functions and extend the range of 'p' 
for which these functions belong to the Hardy space h^p. Our results, in 
particular, add to the understanding of the growth pattern between odd 
univalent harmonic functions and the harmonic Bieberbach conjecture.

In the final section, we deal with the fundamental problem of 
determining the location of the zeros of complex-valued harmonic 
polynomials. The best-known results available in this direction are up 
to harmonic trinomials only. The exploration of the zeros of a general 
harmonic polynomial has been limited due to various challenges. Our 
research takes a leap further in identifying the regions encompassing 
the zeros of a general harmonic polynomial of arbitrary degree using 
various techniques, such as the matrix method and certain other matrix 
inequalities. Additionally, we employ the harmonic analog of the 
argument principle to examine the distribution of zeros, which we 
demonstrate through illustrative examples.

Number theory seminar
Speaker: D. Surya Ramana (Harish-Chandra Research Institute)
Host: Ravi Raghunathan
Title: The Large Sieve with sieving by prime powers.
Time, day and date: 11:30:00 AM, Friday, February 28
Venue: Ramanujan Hall
Abstract  The Large Sieve inequality for prime powers provides an upper 
bound for the cardinality of a subset of the integers in a real 
interval, given the cardinalities of the subsets of 
$\mathbb{Z}/p^{\nu}\mathbb{Z}$ in which the reduction of the set modulo 
$p^{\nu}$ are known to lie, for various primes powers $p^{\nu}$. We give 
a simple proof of this inequality via the principle of the Selberg 
sieve.