Talk on Ambivalency by Sushrutha H S

Abstract: A finite group is called ambivalent if every element is conjugate to its inverse. This property is closely tied to the structure of conjugacy classes and has natural consequences in representation theory.

In this talk, we examined ambivalency in two of the most fundamental families of groups: the symmetric groups S_n​ and the alternating groups A_n​. For symmetric groups, ambivalency can be described in terms of cycle decompositions and holds for all nnn. In contrast, the alternating groups exhibit a more delicate behavior: ambivalency occurs only for certain values of nnn, and its failure is linked to the splitting of conjugacy classes.

The discussion provided a clear characterization of when A_n​ is ambivalent, supported by concrete examples, and highlighted how this property reflects deeper aspects of the groups’ internal symmetries. The talk was accessible to anyone familiar with the basics of permutation groups. 

Talk on Dirichlet's Theorem by Naman Thakkar

Abstract: The set of integers has infinitely many prime elements. For a positive integer m, integers can be partitioned into disjoint classes, depending on the remainder an integer leaves upon dividing by m. These classes are called 'residue classes modulo m'.

A natural question is, how are the infinitely many primes distributed into these residue classes? Notice, for a given positive integer m, the number of residue classes is m, i.e. finite. Hence, at least one class must contain infinitely many primes. (Why?)

Thus, for a given positive integer m, we want to determine which residue classes contain infinitely many primes. Extracting this information directly from the residue classes is difficult. The idea is to 'transfer' this information from the residue classes into the group of non-zero complex numbers. This is done via 'structure-preserving maps' from the residue classes to complex numbers. Once this is done, the information from complex numbers is extracted using tools from analysis.

Talk on General Topology by Eshwar

Abstract: We will look at the problem of lifting continuity from known settings like Euclidean to more broader cases and how this leads to formulation of axioms of topology. We will look at a few different(all equivalent) formalisms which achieve this. This will reveal a deep relation to convergence itself. We will see how sequences are inadequate to describe continuity or convergence in general settings. And as a consequence discuss their generalisations which do the part, viz. nets and filters. And we will see a final formalism which is based solely in convergence of these new objects. I will reconcile this view with the open set formalism(most popular for good reasons) I will discuss continuity and compactness to demonstrate the value of these objects. Tychonoff's theorem is greatly simplified for an instance.  After this we can look at many ramifications of these constructions which honestly are very varied. For example this can lead to discussion about limits in categories(filtration), filters in logic and so on. Or if the people wish I could further about other matters in general topology. 

Talk on Knots by Sanchit Nayak

Abstract: Knots have been a subject of interest since antiquity, and have led to the development of several mathematical theories. Using the language of topology, one can transform the question “Can this knot be unraveled?” into a rigorous mathematical statement; and using the tools of algebraic topology, one can convert this seemingly difficult problem into a purely algebraic (and more tractable) problem.

In this talk, I will give an overview of the topological methods involved in the study of knots and links. I shall also give an overview of the classification (upto homeomorphism and ambient isotopy) of knots in the plane, the 2-sphere, and the torus. If time permits, I shall also talk about a few special classes of knots in R^3.

The talk is based on a semester project I did in the August 2024 semester, with Prof. Rama Mishra.

Talk by Jigyas Baruah

Abstract: Urn Models are stochastic models that deal with urns and coloured balls, with fixed rules (schema) on how the colour populations change depending on what ball colour is picked from the urn at each turn. The focus of the talk would be to analyse the long term properties of the colour populations depending on the schema and the starting colour populations. We will be analysing various schemas, such as the Pólya-Eggenberger Urn, the Ehrenfest Urn, the Hoppes Urn, and some more.

The prerequisites are fairly low, anyone familiar with introductory probability concepts (random variables, probability distributions etc) should be able to easily follow the talk.

Talk by Devanshu

Abstract: I’ll be sharing my semester project on Analytic Number Theory. I’ll start off with some well-known arithmetical functions and their properties, then introduce the concept of Dirichlet convolution. After that, I’ll prove the Euler Summation formula and show how we can apply these ideas to some problems. No prerequisites are expected as all of these ideas can easily be followed.

Talk by Jetharam Bhambhu

Abstract: This talk explores Alan Turing's groundbreaking 1936 paper that laid the foundation for modern computation theory. Turing introduces the concept of a Turing machine, arguing that any computation performable by a human can be replicated by such a machine. He further demonstrates the existence of a universal machine, capable of simulating any other Turing machine when provided with its description. Finally, using a diagonalization argument, Turing shows that there are questions about the behavior of Turing machines that no machine can decide. This leads to the conclusion that the Entscheidung's problem is undecidable, establishing the limits of mechanical computation.