National Level Webinar

 

On Universal Quadratic forms 

Date: 27^th December, 2021
Time: 11:00 am to 12:30 pm

Mode: Online (Zoom and Live YouTube stream) 

1. Background

December 22^nd is a day of celebration for all Mathematicians in India. This day is celebrated as National Mathematics Day throughout the country. On this day, in 1887, born the legendary mathematician Srinivasa Ramanujan in a small place in Tamil Nadu. Difficulties mattered nothing to this genius, who was committed and passionate towards creating mathematics of his own. G.H. Hardy, one of the finest Mathematicians and also the man who introduced Ramanujan to the formal Mathematics describes his relationship with Ramanujan in his autobiography A Mathematician’s apology as follows: “I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, ‘Well, I have done one thing you ould never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms”.  These words of Hardy are enough to tell the impact Ramanujan had on World Mathematics Community. It is not an exaggeration if we tell that, even after a century Ramanujan has left us physically, most of the current research articles in the areas of q-series, modular equations and partition theory contain the name and ideas of Ramanujan. In recognition of the contribution of Rmanujan to Mathematics, in 2012 Central Government of India declared 22^nd December of every year as the National Mathematics Day in India. Hence this day, 22^nd December is a day of celebration for people working in Mathematics. To commemorate the birth day of Srinivasa Ramanujan, Sarada Vilas College in collaboration with Sri. H. D. Devegowda Government First Grade College oraganized a National level webinar on 27^th December, 2021.  Dr. C. S. Yogananda, Former professor, SJCE, Mysuru was invited as the speaker for the webinar.

2. Formalities

The webinar began with an introduction by Yathirajsharma M. V, Asst. Professor, Sarada Vilas College, who was the anchor. The guests were welcomed by Dr. Vijay Kumar, Asst. Professor, Sri. H. D. Devegowda Government First Grade College. This was followed by the encouraging speeches by the Principals of both the colleges, Dr. Devika M and Dr. G. D. Narayana. A brief introduction of the speaker was given by Ms. Pushpa K, Asst. Professor, Sarada Vilas College. This was followed by the talk by Dr. C. S. Yogananda and an interaction session. Finally the vote of thanks was given by Dr. Saroja Y Talwar, Asst. Professor, Sarada Vilas College.

Dr. C. S. Yogananda

 3. Webinar

The main objective of the webinar was to give glimpses of the way Ramanujan used to work. The author chose universal quadratic forms as the main example. Before beginning the talk on the actual topic, universal quadratic forms, the speaker showed the online repository of works of Ramanujan. It was the website www.ramanujan.sirinudi.org which can be accessed freely by anybody. Speaker mainly used this website to show the work of Ramanujan on universal quadratic forms and simultaneously used a presentation. Title of the presentation was “Universal quadratic forms from Ramanujan to Manjul Bhargava”. Speaker began with the meaning of quadratic forms and then explained how the problem of universal quadratic forms was perused by various mathematicians from Ramanujan to Manjul Bhargava including J. H. Conway.  A universal quadratic form is a positive definite quadratic form with integer matrix which can represent all positive integers. The form which was considered by Ramanujan was the below one:

\(ax^2+by^2+cz^2+dw^2\).

Mathematicians were interested in finding the conditions on a, b, c and d for the above expression to be a universal quadratic form, that is conditions on a, b, c and d such that the above expression generates all natural numbers for various values of x, y, z and w (being natural numbers). Ramanujan showed that there were 55 possibilities on a, b, c and d for the above form to represent all natural numbers. This was true, provided all the coefficients, a, b, c and d were positive. Later, mathematicians became interested in the problem of finding the conditions on generalized quadratic forms to be universal. For instance, J. H. Conway and Schneeberger found out a way that for a positive definite quadratic form with integer matrix to be a universal quadratic form, it is enough to check if it represented all natural numbers from 1 to 15. Similarly Manjul Bhargav perused the same work and proved the 290 conjecture of Conway.  

 

Speaker showing the original paper of Ramanujan

This was a small journey kind of story from Ramanujan to Manjul Bhargav. Speaker finished the talk by showing few pages of Ramanujan’s notebooks (like the taxi cab number’s page and number of representation of 4^th power of some number as sum of 4^th powers of some other numbers). After finishing the talk, speaker asked students’ to interact if any. Speaker himself encouraged students to pursue further by asking a question on sums of squares. The question was this - ' Which is the smallest positive number which can be expressed as sum of two squares in two different ways?' Few students enthusiastically interacted, thought, and responded to the question of the speaker. This was a lesson by the speaker, to question oneself similar kinds of questions which they encountered in any talk.

 

4. Statistics

A total of 352 number of people had registered for the webinar through google forms of which 72 number of people were from out of the state and 244 number were students. Of these registered 352,  304 people gave their feedback about the talk. 

We all hope that the webinar gave some picture, especially to students, of the personality of Ramanujan.