Student Colloquia

Coordinator: Prof. Stephan Baier

Objectives

We initiated our student colloquia at the Department of Mathematics at RKMVERI with the following objectives.

• Giving MSc and PhD students the opportunity to study a mathematical problem or topic of their choice outside the usual curriculum. There are no marks for this exercise, and it is fine if the chosen topic is entertaining.
• In some cases, this can lead to a little research project. However, it is not a requirement to come up with something new.
• Giving the students the opportunity to present their findings to their peers in the form of a colloquium talk.
• Training the student’s capabilities to pick up mathematics independently and convey the material in a lively way to a general audience.
• Stimulating discussions about interesting mathematical questions among the students and between students and faculty.

The Context

The curriculum offered to the MSc students is rather standardized. However, in mathematics, there are a countless amount of topics and problems which students usually do not meet during their MSc studies. Interesting questions can be found at all levels. It is useful for the students to get in touch with such topics outside their courses. They can point in different directions and have different flavors: being entertaining, stimulating research, establishing known results using surprising methods, providing insight into advanced topics, connecting to fields outside pure mathematics (such as physics, data science or even art). Most important is that this exercise is enjoyable for the students. In the end, the students present their study projects to their peers (and also the faculty). This trains their ability to convey mathematics to a general audience and react to questions in real time. They will need to do this plenty of times during their future careers. So giving talks will be for their own benefit and the benefit of the audience which, in this way, learns about new topics as well. Not only students, but also we, the faculty, can benefit from these lectures because there are also many things which have never crossed our paths.

The Practice

All of our students do a project in their fourth semester which is completed by a talk of about 30 minutes. These presentations are mandatory. However, this report focuses on voluntary student colloquium talks. From the time when we initiated this format, it has developed as follows.

• In February 2019 we held an initial session in which we, the faculty, distributed topics to students which we found suitable for an interesting talk of about one hour. We also offered the possibility to the students to pick a problem of their own choice. This was followed by a series of exciting talks in the spring semester 2019. These talks were taken up again in autumn 2019.
• During the corona pandemic, communication was limited, and our regular colloquium was interrupted for some time. We initiated a series of online colloquia in 2021. 6 of them were presented by MSc and PhD students.
• When regular offline lectures restarted at the beginning of 2022, several MSc students showed a keen interest in presenting talks. It was particularly delightful that we didn’t need to push them, but rather they approached our faculty with their wish to present lectures. These presentations were of high quality. Altogether, we had 4 student talks in 2022, one of them involving two students.
• In 2023, we had a lecture series by our former MSc student Sourajyoti Maiti on the group law of elliptic curves. This complemented material on elliptic curve cryptography covered in this year’s cryptography course. It connected naturally with a recent project which he did at the HRI Prayagraj, in which he studied the theory of elliptic curves in depth. Later in 2023 our PhD student Aishik Chattopadhyay gave a minicourse about Hensel’s lemma in number theory.
• In addition, as of the beginning of 2022, we hold a weekly number theory seminar. However, this is about advanced topics and restricted to PhD students, postdocs and faculty working in number theory. All members of our group have presented lectures in this seminar. Currently, we have a speaker from outside which presents his lectures online.

Evidence of Success

Our students have been very attracted to the student colloquia and interacted intensely with the speakers. Since the speakers are of the same level as the audience, in terms of their career stage, the atmosphere in these talks is particularly relaxed. One can feel the enjoyment of both speakers and audience. So far, all student talks were well-prepared and well-presented. Our faculty has been happy to assist the students with their lecture preparations and discuss their questions on the material. However, the presenters showed a high level of independence, which was a pleasant experience. We learned a lot from these talks and were sometimes astonished about particular results.

Problems Encountered and Resources Required

As already mentioned above, the corona pandemic limited our communication and interrupted the usual university life. After the offline lectures restarted, we arranged the student colloquia in a rather informal manner. These talks arose mainly from personal communication between students and faculty and didn’t follow a standardized schedule. It would be good to take up our regular programme again, which we initiated before the corona pandemic. These talks should remain voluntary, but it would be better to give this format a structure again, offering a variety of topics at the beginning of the semester and having these talks on a regular basis. Moreover, their frequency should be increased. We have all resources we need: A class room equipped with all necessary technical devices and enthusiastic students and faculty.

List of Colloquia hosted in the recent past

Sl. No.	Speaker	Date	Details
1.	Sayan Ganguly (B2030108), MSc in Mathematics	16.06.2022	Title: ‘Brief Introduction to Fouier Series’
2.	Tamoghnna Kar (B2030117), MSc in Mathematics	26.05.2022	Title: ‘Mobius transformations’
3.	Subhadeep Rana (B2030109), MSc in Mathematics	23.05.2022	Title: ‘An Approach to proving the Fundamental Theorem of Algebra via Linear Algebra’
4.	Aniket Bhattacharyea (B2030091), MSc in Mathematics	19.05.2022	Title: ‘Easy or hard? Finding solutions to an equation using elliptic curves’
5.	Saayan Mukherjee (B2030102), MSc in Mathematics	10.05.2022	Title: ‘The converse of the intermediate value theorem’
6.	Sugata Mandal (B1950009), PhD in Mathematics	18.11.2021	Title: ‘On the n-subspace of alternating forms and calculating s_n(F) for various fields’
7.	Anup Haldar (B1950005), PhD in Mathematics	22.10.2021	Title: ‘Adelic approach to number theory’
8.	Arkaprava Bhandari (B1950006), PhD in Mathematics	17.09.2021	Title: ‘Binary quadratic forms and Bhargava’s cubes’ Abstract: In this talk we will look into the classical theory of binary quadratic forms with integer coefficients by Gauss and Dirichlet and its modern interpretation by Manjul Bhargava. We will see how Dirichlet’s composition gives a group structure to the finite set of equivalence classes of primitive binary quadratic forms when the discriminant is negative or non-square positive. To this end, we establish a group isomorphism to the narrow class group of the oriented orders of the same discriminant. For the general case of non-zero discriminants, we will follow the work of Manjul Bhargava, where he considers cubes of integers to represent forms, generalises the idea of oriented orders and introduces the notion of oriented quadratic rings. Bhargava proved that for any non-zero discriminant, the finite set of the equivalence classes of primitive binary quadratic forms is group isomorphic to the narrow class group of the oriented quadratic ring of the same discriminant.
9.	Snehashis Mukherjee (B18919), PhD in Mathematics	10.09.2021	Title: ‘Quantum Spatial Ageing algebra and its representations’ Abstract: In 2016, V.V.Bavula and T. Lau introduced the quantum version of 1-Spatial Ageing algebra namely the Quantum Spatial ageing algebra. They studied the prime ideals, primitive ideals and weight modules over this algebra for the generic case. In this lecture we will show that this algebra becomes a polynomial identity algebra for the non generic case. We will also compute its PI-degree and finally classify all the simple modules over the algebra to show the relation between PI-degree and dimensions of simple modules over an algebra.
10.	Sanu Bera (B18918), PhD in Mathematics	03.09.2021	Title: ‘Polynomial Identity Algebras and Representations’
11.	Dwaipayan Mazumder (B17907), PhD in Mathematics	06.08.2021	Title: ‘Diophantine Approximation with Prime Restriction in Real Quadratic Number Fields’ Abstract: The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude of primes $p$ satisfying $||\alpha p||\le p^{-\nu}$. The latest record in this regard is Kaisa Matom\”aki’s landmark result $\nu=1/3-\varepsilon$ which presents the limit of currently known technology. Prof. Marc Technau and Prof. Stephan Baier produced an analogue of Bob Vaughan’s result $\nu=1/4-\varepsilon$ for all imaginary quadratic number fields of class number 1. In the present article, we see an analog of the last-mentioned result for real quadratic fields of class number 1 under a certain Diophantine restriction.
12.	Shyamal Datta (B2050015), PhD in Mathematics	30.07.2021	Title: ‘Vector measure, a brief discussion’ Abstract: The theory of vector measure is a generalisation of the complex valued signed measures. We shall see how the theory deals with Banach space valued measures, their variation,semivariation, absolute continuity w.r.t. finite non negative measures and most importantly, their range.
13.	Aniket Bhattacharyea (B2030091), MSc in Mathematics	16.06.2021	Title: ‘Hopf fibrations and Hurwitz-Radon Numbers’
14.	Susmita Seal (B1950010), PhD in Mathematics	11.06.2021	Title: ‘Stability results of small diameter properties in Banach spaces’
15.	Aayushmaan Chakrabarti (B1930127), MSc in Mathematics	23.04.2021	Title: ‘The zeta function and Ramanujan polynomials’
16.	Aniket Bhattacharyea (B2030091), MSc in Mathematics	02.04.2021	Title: ‘A proof of Morley’s theorem’
17.	Sanu Bera (B18918), PhD in Mathematics	14.08.2019	Title: ‘Finite dimensional representations of the Lie algebra sl(2,C)’
18.	Kanoy Kumar Das (B18914), PhD in Mathematics	07.08.2019	Title: ‘Unique factorization in regular local rings’