Spring 2024
Talk:7 (16.4.2024)
Title:- Colon Structure of Associated Primes of Monomial Ideals
Speaker:- Siddhi Balu Ambhore
Abstract:- In commutative algebra, the associated primes of the quotient ring serve as an essential tool to understand various algebraic properties of the ring. The associated primes can also be viewed as colon ideal (also known as ideal quotients). This talk will focus on the associated primes of monomial ideals. We will explicitly express the associated primes as a colon ideal for monomial ideals. We will see how this expression aligns with the combinatorial properties of graphs in the case of edge ideals.
Talk: 6 (9.4.2024)
Title:- The codification of poetry and the birth of combinatorics
Speaker:- Dr. Sumukha S
Abstract:- Historically, mathematics has developed through different branches. One of the most important and interesting among them is combinatorics. Interestingly, the development of combinatorics in ancient India is closely related to the codification of poetry. In this talk, we will give a brief introduction to Indian classical poetry. Then, we will explore the interesting mathematics arising from it, providing examples from Pingala's Chandashastra.
Talk:5 (19.3.2024)
Title:- Group Action and Orientation.
Speaker:- Abhijit Manna
Abstract:- In the last talk, we defined all the prerequisites ( such as the orientation of a smooth manifold and its equivalent conditions, the smooth action of a group on a smooth manifold, etc.). In this talk, we will continue from the smooth action of a group on a smooth manifold. Firstly, we will see that the resultant orbit space M/G of a smooth action of a group G on a smooth manifold M admits a smooth structure so that the quotient map π: M → M/G is a local diffeomorphism. We will then discuss that if M is oriented, so is the orbit space M/G, and vice versa. Finally, as examples, we will discuss the orientability of the real-projective space, the Möbius strip, and the Klein bottle.
Talk:4 (5.3.2024)
Title:- Group Action and Orientation.
Speaker:- Abhijit Manna
Abstract:- We will begin the talk with recalling the orientation of a smooth manifold, and some of its equivalent characterizations. Then we will discuss the smooth action of a group G on a smooth manifold M, and that the resultant orbit space M/G admits a smooth structure so that the quotient map π : M → M/G is a local diffeomorphism. We will then see that if M is oriented the so is the orbit space M/G, and the vice versa. Finally we will discuss the orientability of the real-projective space, the Möbius strip, and the Klein bottle.
Talk:3 (13.2.2024)
Title:- Sieve of Eratosthenes and its application
Speaker:- Jagannath Sahoo
Abstract:- Sieve methods are powerful tools to solve many number theoretic problems. In this talk, we will first discuss the classical sieve of Eratosthenes, which gives an algorithmic method to find primes until a certain number $x$. We will also see Legendre's version of this sieve and apply it to obtain an upper bound for the number of primes up to $x$. Twin primes are pairs of primes that differ by $2$. We state a general version of the sieve of Eratosthenes and use it to prove a celebrated result about twin primes by Viggo Brun, which says that the sum of the reciprocals of the twin primes converges.
Talk:2 (6.2.2024)
Title:- GROUP ACTIONS AND COUNTING
Speaker:- Akhil Surendran
Abstract:- Group action is a powerful technique which has been used in almost every branch of mathematics, it pops up when there are some symmetries playing around. In this talk we will have a glance how Group action is used in combinatorics to solve some counting problems. Painful, but have to do some bookkeeping, we begin by discussing the definition of a group and group actions, and simple examples of both, such as the group of symmetries of a square and this group’s action upon a vertex. We proceed to then define both an orbit and a stabilizer, and prove the Orbit-Stabilizer Theorem, which is central to proving Burnside’s Lemma. Subsequently, we exemplify how Burnside’s Lemma can help us solve combinatorial problems. Namely, we compute the number of distinct colourings of a geometric pattern and the number of distinct necklaces that can be made with coloured beads by utilizing Burnside’s Lemma.
Talk:1 (23.1.2024)
Title:- Introduction to Spherical Varieties
Speaker:- Pavan Adroja
Abstract:- A spherical variety X is a variety with an action of a connected reductive algebraic group G, which contains an open B-orbit, where B is a Borel subgroup of G. In this talk, I will start with some preliminaries of variety, and algebraic groups. After that, I will introduce spherical variety and some examples.