Monday, August 17, 2020

Andrews' approach to conjecture the Rogers-Ramanujan identities

 If something is worth doing, then I suppose it is worth doing again. 

I had previously written an article on how to discover the Rogers-Ramanujan identities. That was based on ideas of Dick Askey. In this talk I presented an introduction to partitions  presenting many results of Euler and ending with George Andrews' approach to discover the Rogers-Ramanujan identities. This approach was given in his Number Theory book, and it seems that it is not as well known as it should be. A notation that my collaborator Hartosh Singh Bal and I use to gain intuition is also explained here. 

 



Abstract


The two Rogers-Ramanujan identities were sent by Ramanujan to Hardy in a letter in 1913. As an example, here is the first Rogers-Ramanujan identity:

$$1+\frac{q}{(1-q)}+\frac{q^4}{(1-q)(1-q^2)}+\frac{q^9}{(1-q)(1-q^2)(1-q^3)}+\cdots $$

$$=\frac{1}{(1-q)(1-q^6)(1-q^{11})\dots}\times \frac{1}{(1-q^4)(1-q^{9})(1-q^{14})\dots}$$


They look less forbidding when interpreted in terms of partitions, which is how MacMahon considered them. A partition of a number  is a way of writing it as an unordered sum of other numbers. Unordered means that $2+3$ and $3+2$ are considered the same. For example, 

$$5 = 4+1 + 3+2 = 3+1+1 =  2+1+1+1$$ 

are partitions  of $5$. (Two partitions of $5$ are missing in this list; can you find them?) The theory of partitions is an attractive area of mathematics, where many complicated formulas are rendered completely obvious by making the `right picture'. However, while each side of the Rogers--Ramanujan identities are represented naturally in terms of partitions, they are still far from obvious.


In this talk, we will introduce partitions, explain how to enumerate them systematically, represent them graphically, and write their generating functions. We present an experimental approach  to discover the Rogers-Ramanujan identities. This approach is due to Professor George Andrews of Penn State University.


Thursday, April 30, 2020

Lest we forget them (Edited) - By P. K. Ghosh

An email Asoke and I sent with a link to P K Ghosh's article "Lest we forget them". I have previously written a book review of PKG's novel: On returning to desh.



IITs of India are known worldwide as excellent academic institutions. For several decades, IIT Kanpur was the leader in this group. Those who were fortunate to have studied there, would fondly remember the ambience, the Halls of Residence, the Central Library, the canteens, the lecture hall complex...and certainly the teachers. They were great men, and sought to ignite the spark in kindred spirits among younger souls. Among them, Prof. P. K. Ghosh stands out as one who always maintained a high standard in teaching and research, and demanded the same from his students. The attached booklet gives a glimpse of his time in the Chemistry Department at IIT Kanpur, where he and his students created one of the highest resolution optical spectrometers in the country, an unusual programmable microprocessor for training, data acquisition and display, wrote several monographs which have been praised internationally, among other things. Prof. Ghosh has recounted those days in a delightful booklet "Lest We Forget Them."

Who are the people whom he doesn't want us to forget? What else does he not wish to forget? The mood. The vision. The bravado. The students who will go on to achieve great success and renown.

For details, you must read the booklet.

Here is a link: 

Best wishes,

Asoke and Gaurav 

PS. We would appreciate it if viewers and readers can identify themselves and comment below. Any messages to PKG will be forwarded to him along with your contact details. I will have to `approve' the comment. This is to avoid spam.

PPS. This is an edited version. Some pictures and letters have been edited from here.If you know PKG personally, he may share with you the complete article. Please send a request for the same to Asoke. 

--
Asoke Chattopadhyay
Professor
Department of Chemistry
University of Kalyani
Kalyani 741235
(MSc integrated, IIT Kanpur) 
asoke dot chattopadhyay at gmail


Gaurav Bhatnagar
bhatnagarg at gmail com