IIT Jammu Mathematics Colloquium
IIT Jammu Mathematics Colloquium
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The IIT Jammu Mathematics Colloquium is directed at students, postdocs, and faculty working in the fields of pure and applied Mathematics & Statistics. It aims to present expository lectures or mini-courses that appeal to our wide range of audience at IIT Jammu. One of the main objectives of these colloquia is to inform non-specialists and graduate students about recent research trends, ideas, and results in various areas of Mathematics, rather than focusing on technical details for specialists.
This semester (i.e. 2023-24, Sem-1), the IIT Jammu Mathematics Colloquium will take place every Wednesday/Friday afternoon from 3 to 4 pm. Invited external speakers will deliver exciting talks in online mode, while the faculty, graduate students, and visitors of IIT Jammu will present talks in offline mode in alternate weeks. Please note that the timing for a few talks may be adjusted depending on the convenience of renowned speakers from institutes around the globe.
If you are interested in giving a talk at the Colloquium or wish to receive emails with news about the online talks and reminders, please drop an email to arvind.kumar@iitjammu.ac.in
Current/Upcoming talks:
Current/Upcoming talks:
11-08-23 (Online)
Speaker: Prof. Anuradha Sharma, IIIT Delhi
Title: Enumeration Formulae for Self-dual, Self-orthogonal and LCD Codes over Finite Commutative Chain Rings
Abstract: In this talk, we will present enumeration formulae for all self-orthogonal, self-dual and linear with complementary dual (LCD) codes of an arbitrary length over finite commutative chain rings. These enumeration formulae are useful in classifying these three classes of codes up to equivalence, which we will illustrate in certain special cases.
We will also show that the class of LCD codes over finite commutative chain rings is asymptotically good and that every free linear code over a finite commutative chain ring with at least five elements is equivalent to an LCD code. Besides this, we will explicitly determine all inequivalent LCD [n,1,d]-codes and [n,n-1,d]-codes over finite commutative chain rings for 1 \leq d \leq n. This is a joint work with Monika Yadav.
16-08-23
Speaker: Dr. Nupur Patanker, IIT Jammu
Title: Generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes
Abstract: Linear codes form a large family of error-correcting codes. Due to their algebraic properties, these are the most studied codes from the mathematical point of view. Recently, Jaramillo, Pinto and Villarreal introduced various linear codes called toric codes over hypersimplices and squarefree affine evaluation codes and calculated their basic parameters.
In this talk, we will give an overview of our work on the generalized Hamming weights of these codes. At the beginning of this talk, we will define linear code and its basic parameters. Then we will see the construction of toric codes over hypersimplices and square-free affine evaluation codes and the results on their basic parameters. Then we will answer the questions on the number of zeroes of squarefree polynomials of a certain degree in the affine torus. This, in turn, answers our question on the generalized Hamming weights of toric over hypersimplices and squarefree affine evaluation codes under certain cases.
This is a joint work with Dr. Sanjay Kumar Singh.
23-08-23 (Online)
Speaker: Prof. Anish Ghosh, TIFR Mumbai
Title: An Introduction to the Geometry of Numbers
Abstract: I will provide an introduction to Minkowski's geometry of numbers and discuss various applications of this beautiful theory. Recently, the subject has experienced a renaissance, and I will explain some recent developments.
30-08-23
Speaker: Dr. Rajiv Kumar, IIT Jammu
Title: Graphs whose independence complex is shellable.
Abstract: In 2005, Herzog-Hibi proved that if a bipartite graph arises from posets, then its independence complex is shellable. We generalize this work for $r$-partite graphs. In this talk, we discuss the proof of the result of Herzog-Hibi, and we will try to understand its generalization. This is a joint work with Dr. Ajay Kumar. The talk will be elementary in nature.
15-09-23 (Online)
Speaker: Prof. Arindama Singh, IIT Madras
Title: Maths in CS
Abstract: I am planning to give an overview of some important topics from maths which are used and applied in computer science, and exactly where they are used.
22-09-23 (Online)
Speaker: Prof. Arindama Singh, IIT Madras
Title: Maths in Image Denoising.
Abstract: Starting from the problem of denoising, how and which types of maths are used to solve the problem. This will be more technical than the first one.
25-09-23
Speaker: Prof. B. Rajeev, IISER Trivandrum
Title: An Introduction to Brownian Motion and Stochastic Calculus
Abstract: In this talk we introduce the Brownian motion process via the ‘Invariance Principle’ and derive the distribution of the maximum of the process. In the second half of the talk we will discuss the elements of stochastic calculus viz. the Itˆo integral and Itˆo’s formula. Finally we derive a stochastic partial differential equation for Brownian motion.
13-10-23
Speaker: Mr. Bijender, IIT Jammu
Title: Stanley-Reisner complex of the facet ideal of a simplicial tree
Abstract: In this talk, we will discuss the simplicial complexes, with a particular focus on simplicial trees. Additionally, we'll explore the concept of vertex decomposability within simplicial complexes.
20-10-23 and 23-10-23
Speaker: Prof. M. Manickam, IISER Bhopal
Title: On Ramanujan congruence
Abstract: In the first talk, we will cover the basic theory of modular forms of integral and half-integral weights. Subsequently, we will derive Ramanujan congruences for modular forms of weight 12.
In the second talk, we will generalize the Ramanujan congruence to modular forms of integral weight and half-integral weight forms.
03-11-23
Speaker: Prof Trygve Johnsen, UiT: The Arctic University of Norway
Title: Higher weight spectra of generalized Reed-Muller codes RM_q(2,2)
Abstract: We analyze the matroid of a parity check matrix of the Reed-Muller code RM_q(2,2) obtained by evaluating all polynomials of degree at most 2 over F_q at the points in (F_q)^2. We choose a geometric viewpoint and study how zero sets of conics in P^2 intersect (the complement of) the line at infinity, and combine combinatorics and homological algebra to find algebraic invariants that in the end will give us all the higher weight spectra.
14-12-23, 15-12-23 and 18-12-23
Speaker: Prof Fernando Pinero, University of Puerto Rico
Title: Introduction to Coding Theory and Evaluation Codes
Abstract: In this series of lectures we introduce basic notions of coding theory. In particular, we introduce some fundamental families of codes, such as Reed-Solomon codes, BCH codes, binary Goppa codes. We also present Petersen's decoding algorithm as well as the list decoders of Sudan and Sudan-Guruswami. If time permits, we also talk about LDPC and LRC codes from evaluation codes.
21-12-23
Speaker: Prof R. K. Sharma, IIT Delhi
Title: Group Based Cryptography
Abstract: In this talk, we discuss some group based cryptosystems. These groups may be finite or infinite, cyclic, abelian or non abelian.
Past Talks:
Past Talks:
24-07-2023
Speaker: Prof. Pratima Panigrahi, IIT-Kharagpur
Title: Some spectral results of distance-regular graphs and minimal cages
Abstract: From a given graph G, several matrices are constructed by researchers to study different properties within graph theory as well as in other areas. Computation of eigenvalues and eigenvectors of these matrices comes under the area of spectral graph theory. Here we shall mainly focus on eigenvalues of distance matrix associated with distance regular graphs and in particular minimal cages. Distance regular graphs were introduced by Biggs, in the year 1974, while studying so called distance transitive graphs. Random walks on certain distance regular graphs correspond to important models for the diffusion of particles. The Ehrenfests’ urn model that was proposed to explain the second law of thermodynamics corresponds to random walks on the hypercube, which is an example of distance regular graphs. For positive integers k and g, a minimal (k, g)-cage graph is a k-regular graph with girth g and minimum possible number of vertices. We obtain a quotient matrix Q(G)(which is of smaller order) of distance matrix D(G) of a distance regular graph G so that all the eigenvalues of Q(G) are eigenvalues of D(G). We determine a polynomial p(x) for minimal cages so that p(λ) is an eigenvalue of distance matrix of a minimal cage, where λ is an adjacency eigenvalue of the same minimal cage. Finally, we discuss some open problems associated with the above results.
26-05-23
Speaker: Dr. Manmohan Vasistha, IIT Jammu
Title: Lagrange multiplier method and its applications
Abstract: In this talk, we shall discuss the Lagrange multiplier method and some of its applications in linear algebra.
19-05-23
Speaker: Dr. Manish Kumar Pandey, SRM University, Andhra Pradesh (Online mode)
Title: GL_2 automorphic forms and associated L-function.
Abstract: In this talk, we will briefly introduce the theory of GL_2 automorphic forms in the classical language. Then we will look at the Fourier-Whittaker expansion and spectral expansion. We will attach the L-function to a Hecke-Maass cusp form. If time permits we will describe some of the unsolved problems. The basic reference for the talk will be the book of Goldfeld named Automorphic Forms and L-functions for the group GL_n(R).
10-05-23
Speaker: Dr. Arvind Kumar, IIT Jammu
Title: Generalizations of Ramanujan congruence.
Abstract: Ramanujan made a series of influential conjectures in his 1916 paper "On some arithmetical functions" on what is now called the Ramanujan tau function. In the same paper, he also proved a notable congruence mod 691 for the Ramanujan tau function. The existence of such congruences opened the door for many modern developments in the theory of modular forms. For example, it led to Serre and Swinnerton-Dyer developing a geometric theory of mod p modular forms.
For newforms of prime level, some partial results about the existence of such congruences are known. In this talk, we discuss and refine some of those results.
26-04-23
Speaker: Dr. Rohit Kumar Mishra, IIT Gandhinagar (Online mode)
Title: 2D V-line Tensor Tomography
Abstract: The reconstruction problem of tensor fields has been studied in many classical works of integral geometry. The underlying problem involves the inversion of integral transforms of unknown tensor fields, including the longitudinal (Doppler) ray transform, mixed ray transform, transverse ray transform, and momentum ray transform of different orders. We consider a generalization of these transforms, in which the linear integration path is replaced either by V-lines (broken rays) or stars (a finite union of rays emanating from a common vertex). We first focus on the case of vector fields and present several exact closed-form inversion formulas along with some numerical simulations. Then we move to the case of symmetric 2-tensor fields and discuss recent developments in the area.
19-04-23 and 21-04-23
Speaker: Dr. Prasant Singh, IIT Jammu
Title: An Application of Point-Line Incidence of Grassmannian in Decoding Grassmann Codes.
Abstract: Grassmann Codes, the algebraic-geometric codes associated with Grassmann varieties over finite fields, have been an important object amongst coding theorists since their discovery in the late nineties. Several interesting properties of these codes are known but no nontrivial decoding algorithm for these codes was explored until recently. In this talk we will recall some basic notions of coding theory and introduce Grassmann Codes. We will see certain interesting properties of the Point-Line incidence of Grassmannian. At the end, we will use this incidence geometry to obtain a decoding algorithm for Grassmann Codes. We only assume basic linear algebra and finite fields. This talk is based on a joint work with Peter Beelen.
12-04-23
Speaker: Prof. Eknath Ghate, TIFR, Mumbai (Online mode)
Title: Modular representation theory of GL_2 over a finite field using calculus.
Abstract: The mod p representation theory of GL_2(F_q) where F_q is a finite field with q = p^f elements is a classical subject going back to Richard Brauer. We use some differential operators (and even some sort of integration) to establish some new isomorphisms in the theory. One corollary of our work is that in the case of principal series we are able to deduce some new periodicity results. This is joint work with Arindam Jana.
31-03-23
Speaker: Dr. Abhay A. Soman, University of Hyderabad (Online mode)
Title: Totally positive field extensions
Abstract: The talk will focus on 'Totally positive field extensions'. These field extensions are defined in terms of extensions of semi-orderings. I will give examples of totally positive field extensions, and their connection to quadratic form theory. I will also indicate a study of the behavior of central division algebras over totally positive field extensions. The talk will be accessible to M.Sc. and Ph.D. students. The talk is based on joint work with P. Mandal (IITB) and R. Preeti (IITB).
22-03-23 and 23-03-23
Speaker: Prof. Dilip Patil, IISc Bangalore
Title: A criterion for rational points using trace forms
Abstract: In these two lectures, I will discuss the number of K-rational points and the signature of the trace form of a finite K-algebra over a real closed field K. The main tools are symmetric bilinear forms, hermitian forms, trace forms, generalized trace forms and their types and signatures. I will prove a criterion for the existence of K-rational points by using generalized trace forms. Some preliminaries and motivation will also be provided. As an application, I will try to sketch a proof of the Pederson-Roy-Szpirglas theorem about counting common real zeros of real polynomial equations.
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