Talks given by students during the year 2025 are listed below.

Spring 2025

EP2505: Wigner Matrices - The Semicircular Law and its Variants

Speaker: Saraswata Sensarma (M. Math, 2026)
Abstract: Random Matrix Theory was introduced as a model for studying “typical” matrices of a given nature. The initial investigations were made by Wishart (1928) for random covariance matrices and Wigner (1955) for random hermitian matrices. Today, RMT boasts connections with topics ranging from integrable particle systems to number theory, with applications in data science, telecommunication, and theoretical physics to name a few.
The semicircular law (SCL) was among the first universality results for random matrices. Proven by Wigner in his seminal paper, this law governs the behavior of the bulk of the eigenvalues of Wigner matrices. In the coming years, several other techniques were introduced for the proof of the SCL, some of which have since been used to prove stronger universality phenomena for the spectrum.
In this talk, we wish to give a sketch of Wigner’s original proof - which involved the moment method and an inverse moment computation. We briefly discuss some other methods for the proof of SCL, and how they are applicable in a broader setting. We then talk about more intricate universality results - a local variant of the SCL, eigenvalue rigidity and eigenvector delocalisaion (quantum unique ergodicity). We end with some remarks on non-Hermitian Matrices.
Pre-requisites: Some results from Probability III will be stated and used. To follow along, a knowlegde of Probability I and some linear algebra would be sufficient.
Video: [TBA]
Date and Time: [TBA]
Venue: [TBA]

EP2504: Ramanujan’s Circle Method, and application to Partition Number

Speaker: Srijeet Bhattacharjee (B. Stat, 2026)
Abstract: Srinivasa Ramanujan was one of the greatest Indian mathematicians, who has left us with a great number of interesting and beautiful results. One of his most famous contribution is the asymptotic expression of the partition number. Few years later, Rademacher gave an exact expression of it, in terms of a series. However, the ingenious “Circle Method”, that Ramanujan used in the proof, has ever since been refined and applied to a vast plethora of number theoretic problems, and is still heavily being used in modern day research.
We would state Ramanujan’s version of the Circle Method, and prove Rademacher’s Series of Partition Number.
Pre-requisites: Some very basic understanding of complex analysis (2nd year onwards have seen this much in the context of Characteristic Functions), however this can be safely ignored.
Notes: Available here.
Video: (Not Recorded)
Date and Time: Tuesday, 1st April 2025, 4:30 PM - 6:00 PM
Venue: Room 401, 4th floor, SN Bose Bhavan

EP2503: Convergence of random series

Speaker: Ayan Ghosh (B. Stat, 2026)
Abstract: Discussion on upper and lower bounds on Kolmogorov’s maximal inequality, Hence proving Kolmogorov’s three series theorem. Discussion on rates of convergence. Finally a theorem (probably by Levy): Convergence of the series $\sum_{n = 1}^{\infty} X_n$ in distribution implies almost sure convergence of the series, provided $X_i$’s are independent.
Video: (Not Recorded)
Date and Time: Tuesday, 18th March 2025, 4:30 - 6:00 PM
Venue: Room 508, 5th floor, SN Bose Bhavan

EP2502: A Short Proof of Kolmogorov’s SLLN

Speaker: Atmadeep Sengupta (B. Stat, 2026)
Abstract: Strong law of large number gives a justification of thumbs rule of average. It states that sample mean converges to original mean almost surely. The original and traditional proof is due to A. N. Kolmogorov. The alternative elementary proof by N. Etemadi bypasses stuffs like maximal inequalities and convergence of random series. In this talk, we will discuss this proof and have some insight on the elegance of the proof. After that, we will have a proof of another statistic, namely median. We will have a short proof of almost sure convergence of sample median to true median, in the case of unique median.
Notes: Available here.
Video: (Not Recorded)
Date and Time: Wednesday, 12th March 2025, 4:30 PM - 5:30 PM (IST)
Venue: Room 401, 4th floor, SN Bose Bhavan

EP2501: Dirichlet’s Theorem of Primes in an Arithmetic Progression

Speaker: Srijeet Bhattacharjee (B. Stat, 2026)
Abstract: We know that there are infinitely many primes in natural numbers. But are there infinitely many primes in any arithmetic progression? More generally which arithmetic progressions contain infinitely many primes? We would prove Dirihclet’s theorem on primes in an arithmetic progression. We will introduce the concepts of Dirichlet characters and Dirichlet $L$-functions, which gave rise to other general $L$-functions and studying their properties is an area of the bulk of active research in Analytic number theory and even connected to the Riemann Hypothesis
Notes: Available here.
Video: (Not Recorded)
Date and Time: Friday, 7th March 2025, 4:30 PM - 5:30 PM (IST)
Venue: Room 401, 4th floor, SN Bose Bhavan