Speaker: Omkar Javadekar
Abstract: Lefschetz properties arise in several branches of mathematics including algebraic topology, combinatorics, and commutative algebra. In this talk, we will introduce the Weak Lefschetz Property (WLP) and the Strong Lefschetz Property (SLP) for Artinian algebras. We will see examples (and non-examples) of classes of algebras satisfying these properties. In the remaining time, we will see some of the applications of these properties.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Umesh Shankar
Abstract: We have seen isomorphisms. We have seen homomorphisms. Now we will look at bijections. The RSK correspondence gives a purely bijective proof of a result from representation theory.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Brahadeesh Sankarnarayanan
Abstract: Paul Erdős liked to speak of "The Book", in which God kept the most elegant proofs of each theorem. As Erdős once said, "You don't have to believe in God, but you should believe in The Book." In this talk, I will present one such proof from The Book.
The problem: take a planar graph, and let each vertex be given a list of five colors. Can you color all the vertices such that each vertex receives a color from its assigned list and adjacent vertices are given different colors?
This problem was independently considered by Vizing (1976) and by Erdős–Rubin–Taylor (1979). In 1994, Carsten Thomassen published an elegant half-page proof showing that the answer is positive, and we will see this "Book" proof in the talk.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Suraj Panigrahy
Abstract: In this seminar, we delve into the fascinating realm of geometric puzzles and challenges, specifically focusing on three intriguing problems involving points on a plane. These problems present captivating scenarios where we examine the possibility of satisfying certain conditions. I shall pose some basic questions to stimulate discussion and engagement. Join us as we embark on this captivating journey into the realm of geometric puzzles.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Brahadeesh Sankarnarayanan
Abstract: Last week, we saw a construction of the hyperreal field *R using ultrafilters. In this talk, we will take a first look at the transfer principle, and see some simple ways in which it is used to do non-standard analysis. Some highlights include a non-standard proof of the infinitude of primes, and a non-standard viewpoint on uniform continuity.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Brahadeesh Sankarnarayanan
Abstract: In 1960, Abraham Robinson showed that infinitesimal calculus could be put on a rigorous footing, on par with the epsilon–delta formulation. This talk will be a gentle introduction to the non-standard analysis introduced by Robinson. We will look at a construction of the hyperreal field *R that uses ultrafilters. We will also discuss the transfer principle, which allows one to move back-and-forth between the "standard" reals R and the hyperreals *R.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Aditya Dwivedi
Abstract: In this talk we will look at two Galois extensions, (Fp)alg over Fp and Qalg over Q. We completely determine the Galois group of the first extension, but for Qalg over Q we move down to the "largest" abelian extension of Q over Q. We will also see the construction of inverse limits and use that to compute the Galois groups.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Soumadeep Bhowmick
Abstract: We will discuss the problem of assessing agreement between two raters while the ratings are given separately in 2-point nominal scale and critically examine some features of Cohen's kappa statistic, widely and extensively used in this context. We point out some undesirable features of Cohen's kappa statistic and, in the process, we will talk about two modified kappa statistics. Properties and features of these statistics are explained with illustrative examples.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Priyanka Magar
Abstract: The notion of a fiber bundle, its generalisation called fibration with a dual notion cofibration, plays a vital role in computational topology. We begin the talk by introducing the fiber bundle. While discussing examples, we will discuss the famous once-open "Hopf-invariant one problem". Further, we see a few special classes of the fiber bundle, called vector and principal bundles. We conclude the talk by discussing fibrations, cofibrations, and their applications.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Unnati Nigam
[This is a continuation of the previous talk.]
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Unnati Nigam
Abstract: In this talk, we will discuss the extension to the Topp–Leone distribution, using the Generalized DUS transformation. Various desirable statistical properties of the proposed distribution like hazard rate function, moments, entropy, stress-strength reliability etc. will be discussed. Some ways of parameter estimation for the distribution will be presented. A step-by-step algorithm will be discussed to produce a random sample from the distribution followed by some simulation experiments to study the long-term behavior of the estimators. Lastly, we will discuss the application of the proposed distribution by fitting a real-life dataset over some existing distributions in the same range. We will go through all the relevant properties conceptually, followed by their effect on the proposed distribution.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Sai Krishna P M S
Abstract: We will continue the previous talk by briefly describing the epimorphism/embedding problem and seeing a proof of the Abhyankar–Moh–Suzuki (AMS) Theorem using the Abhyankar–Moh (AM) Theorem.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Sai Krishna P M S
Abstract: Suppose k is a field of characteristic zero. The AM Theorem states that if k[f,g] = k[X], where f, g are two non-constant polynomials in the polynomial ring k[X], then either the degree of f divides the degree of g or vice versa. In this talk, we will see a proof of the AM Theorem due to Nowicki and some applications and generalizations of it.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Suraj Panigrahy
Abstract: We will see how some 'basic' results in number theory can be proved using tools of analysis. Surprising, isn't it? We will 'not' prove many results but rather we will see how number theory and analysis are related. We will see a proof-idea of the infinitude of primes using tools of analysis. We will also see 'some' of the following, if time permits:
Addition Problems
Four Square Theorem
Crazy Dice
A splitting problem
An identity of Euler
Time: 2:30–3:30 PM
Venue: Room 216
Speaker: Makadiya Deepkumar
Abstract: In this talk, we will discuss the representations of 𝔰𝔩(2,F). First, I will define all the background that is needed, and then we will proceed to prove the main theorem of the talk.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Ankita Dargad
[This is a continuation of the previous talk.]
Time: 3:00–4:00 PM
Venue: Room 105
Speaker: Ankita Dargad
Abstract: Combinatorial Game Theory (CGT) studies strategies and mathematics of sequential games with perfect information. It largely covers two-player games that have proper rules and players tries to achieve winning conditions such as chess or tic-tac-toe. This talk will start with playing a game called Nim, then we will introduce the set of short combinatorial games along with partial order and group structure on that set. We will also go through the Fundamental Theorem and some characteristics of short games.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Priyabrata Mandal
Abstract: This talk will start with the quadratic forms and ordering, semiordering over formally real fields. Later we define totally positive field extensions and will discuss such field extensions. If time permits, we will discuss the pythagorean index of a central simple algebra.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Rati Ludhani
Abstract: In this talk, we will discuss what are error-correcting codes and how does this theory of codes impact our lives.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Umesh Shankar
[This is a continuation of the previous talk.]
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Umesh Shankar
Abstract: You can run from pigeons but you can never hide. In this talk, we'll speak about housing pigeons in an insufficient number of boxes and the profound mathematical effects that result from it.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall
Speaker: Suraj Panigrahy
Abstract: We will define what is an 'integer' inside a number field. We will see some properties and then we will compute the 'integers' of a Quadratic Number Field. If time permits, we will see discuss more about Quadratic Number Fields.
Time: 2:30–3:30 PM
Venue: Ramanujan Hall