Past Speakers

Date: Oct 31, 2025

Title: The Infinitude of Primes: An Exploration of Proofs.

Abstract: Whenever we look for famous open problems in number theory, there’s a high chance that primes are involved. In this talk, we revisit the foundational fact without which these problems wouldn’t even make sense — that there are infinitely many primes. First proved by Euclid long ago, this simple result encodes surprisingly beautiful and deep ideas. We will explore a few elegant proofs of this timeless theorem. 

Date: Apr 8, 2025

Title: Primes Numbers are supernatural

Abstract: Fermat conjectured that 2^(2^n) + 1 is prime for every natural number n, but Euler disproved this by showing that it fails for n=5. This leads to a broader question: does there exist a function of natural numbers, constructed using a variable n, a finite set of constants, and the operations of addition, multiplication, and exponentiation, that always yields prime values? It is conjectured that no such function exists. In this talk, we will explore the formalization of this conjecture and its implications.

Date: Feb 25, 2025

Title: Theory of Cauchy Principal Value Integrals

Abstract: In this talk, we are going to discuss the definition and develop a few results of the Cauchy principal value integrals. We will mainly be discussing the work of Hardy, who said that, "Definite integrals are of two kinds—finite and infinite. In finite integrals the range of integration is finite, and the subject of integration finite throughout it..." We shall study the integrals of the second kind. This talk will be built on basic concepts in real analysis.

Date: Feb 4, 2025

Title: Introduction to Huber rings

Abstract:  If a ring is endowed with a topology that behaves well with its algebraic structures (i.e., the ring operations are continuous), many interesting properties emerge. Roland Huber introduced a class of topological rings, namely Huber rings, which generalize the notions of Tate rings and adic rings defined by ideals (I-adic topologies on commutative rings). The main focus of the talk is to discuss a few sections of the seminal paper 'Continuous Valuations' by Huber himself, and if time permits, we will construct adic spaces—a category that encompasses many well-known algebraic geometric objects like rigid analytic spaces, formal schemes and schemes.

Date: Jan 21, 2025

Title: Explicit zero free region of the Riemann Zeta function

Abstract:  We will discuss the proof that the Riemann zeta function has no zeros in the region Re(s) ≥ 1 − 1/(55.241 (log |t|)2/3 (log log |t|)1/3) for |t| ≥ 3. We will also discuss the intermediate zero-free region. 

Date: Oct 29, 2024

Title: Differential Nullstellensutz

Abstract:  This talk is an attempt to lead a peek into the world of Differential Algebra. As the name suggests, we will be drawing a lot of inspiration from commutative algebra and will be seeing a lot of similar results but in a differential algebraic setting (will define what that means as well). A course in commutative algebra is, of course, not a pre-requisite to the talk, but you can connect things if you know a couple of results. We will start by defining what a derivation is in general and try to see various algebraic ways to play around with it. The talk won’t contain much of the proves but will try to motivate most of the results with some nice examples and intuitions.

Date: Sep 17, 2024

Title: A Quick Overview of de Rham Cohomology

Abstract: Recall that given a smooth manifold M, there is a unique exterior derivative d on M. A differential form ω on M is called closed if dω = 0, and it is called exact if ω = dη, for some differential form η on M. Since d∘d = 0, every exact form is also a closed form, not conversely. But how can one find the extent to which a closed form is not exact? It turns out that whether every closed form on a manifold is exact depends on the topology of the manifold. This notion is given by the de Rham Cohomology of the manifold. In this talk, I shall define the de Rham Cohomology of a smooth manifold as the extent to which closed forms on a manifold are not exact. I shall discuss some of the basic examples to compute de Rham Cohomology. Also, I shall show that the de Rham Cohomology, possibly, one of the most important diffeomorphism invariants of a manifold.

Date: Aug 27, 2024

Title: Maximality and Prime Ideals

Abstract: Given an abelian category, we know that we can construct the homotopy category of chain complexes. In this talk, I will be showing the existence of a category, namely the derived category, to which there is a functor from the homotopy category which takes quasi isomorphisms (isomorphism in the homology level) to isomorphisms in the derived category, and it is universal with respect to this property. Having said these technical jargons, we will make it again complicated by introducing these techniques to algebraic geometry, specifically in the study of algebraic varieties and its classification. 

Date: Apr 9, 2024

Title: The codification of poetry and the birth of combinatorics.

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.

Date: Mar 19, 2024

Title: Group Action and Orientation

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.

Date: Feb 6, 2024

Title: GROUP ACTIONS AND COUNTING.

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.  

Date: Nov 8, 2023

Title:  An overview of V-line transform and some generalizations.

Abstract:  In inverse problems related to integral geometry, one of the fundamental goals is to recover the scalar field (or, more generally, tensor field) from its integral along some suitable trajectories. In this talk, we will focus on the integrals along V-shaped trajectories, known as V-line transforms. We will start with a motivation for V-line transforms. Then, we will discuss some generalized V-line transforms for vector fields in the plane and provide some geometrical proofs. If time permits, we will discuss some other generalized V-line transforms.

Date: Sep 27, 2023

Title:  An Overview of Inverse Boundary Value Problems

Abstract:  In this talk, we will discuss the inverse boundary value problem that determines the internal coefficients of the differential equation from measurements made at the region's boundary and its applications. We will further introduce Calderón's Problem (also called the inverse conductivity problem) and discuss some main tools for proving the uniqueness of the inverse boundary value problems

Date: Sep 13, 2023

Title:  Counting Ideals in Numerical Semigroups

Abstract:   If $S$ is a numerical semigroup, let $m(S,k)$ denote the number of ideals of $S$ with codimension $k$ and let $n(S,k)$ denote the number of ideals of $S$ with conductor $k$. We compute the generating function of the sequence $m(S,k)$ for all numerical semigroups of embedding dimension $2$ and for $S = \langle 3,n+2,2n+1\rangle$. We also prove that the sequence $n(S,k)$ becomes stationary after a certain term and compute these stationary terms for numerical semigroups of the form $\langle n,n+1 \rangle$.

Date: Aug 16, 2023

Title: Fundamental Solution to the Laplace Operator

Abstract: The concept of weak solution is an important aspect of the theory of partial differential equations; in this regard, we will study the basics of Distributions and convolution of functions and generalize it to convolution of distributions. Using these concepts, we will construct a fundamental solution to the Laplace operator.

Date: May 3, 2023

Title: Primes in Short Intervals

Abstract: It is well-known that the number of primes up to $x$ is approximately $x/ log x$. If primes are "well-distributed" over positive integers, one would expect that in an interval $(x,x+h(x))$, there are $h(x)/ log x$ many primes. In other words, we may ask that for which functions $h$ we should expect this to hold. Well, it is not easy to answer this question, but what we can do is understand for which functions this does not hold. We will answer this using Maier's matrix method and discuss some consequences of this elegant method.

Date: Apr 5, 2023

Title: Orthogonal Decompositions and Twisted Isometries

Abstract:  We introduce the notions of Twisted Isometries and Doubly Twisted Isometries on Hilbert Spaces. We show the existence of Orthogonal Decomposition for the above classes of isometries. We also identify tuples of isometries that admit a Wold-Von Neumann Decomposition.

Date: Mar 22, 2023

Title: Distribution Theory and its applications to PDE

Abstract: Distributions are an important tool in modern analysis, especially in the field of partial differential equations. Distributions are generalized functions that allow for operations, such as differentiation and convolution, on objects that fail to be functions. The talk aims to introduce the basic theory of distributions.

Date: Feb 1, 2023

Title: The Frobenius Problem: An Algebraic Approach

Abstract: This problem was discussed by Georg Ferdinand Frobenius (1849-1917) in the late 1800s. The problem is finding the largest integer, which can not be written as a non-negative integral combination of the given natural numbers with one as their greatest common divisor. This integer is called the Frobenius number. This problem seems fundamental in arithmetic but is known as NP-hard. In this lecture, we will discuss the proof of the fact that the Frobenius number of a numerical semigroup is the degree of the Hilbert series (as a rational function) of the associated semigroup algebra.

Date: Jan 11, 2023

Title: An Introduction to Game Theory

Abstract: Game theory is the study of mathematical models of strategic interactions among rational agents. It has applications in all fields of social science, as well as in logic, systems science, and computer science. In this talk, we will formally define what are two-player and n-player games and demonstrate some examples, such as Prisoner’s dilemma. We will cover the concepts of payoff matrices, zero-sum games, pure strategy, and mixed strategy. We will define the concept of Pure Strategy Nash Equilibrium (PSNE) and Mixed Strategy Nash Equilibrium (MSNE), and we will cover more topics if time permits. This talk will not assume any prior knowledge of algorithms or complexity theory from the audience.

Date: Nov 15, 2022

Title: Euler Gamma Function and its various generalizations

Abstract: We will define Euler’s gamma function and its various generalized and related special functions. Also, introduce its important properties like a functional equation, difference equation, reflection formula, etc. We will also look for a result on the generalized Mittag-Leffler functions and technique used in its proof.