Math Musings

I recently gave a talk on Internet Security and Cryptography at the international CMSC conference on Jul 3, 2020. Kindly see the enclosed video for the 5 minute video explaining the essence of Elliptic Curve Cryptography. 

Prof Parimala Raman is the only lady mathematician educated in India who is currently member of all three Indian Academy of Sciences besides winning the prestigious Shanti Swarup Bhatnagar award in 1987. In 2010, she was selected as the plenary speaker at the International Congress of   mathematicians. When the government of India decided to establish eleven chairs to honor women in math and sciences recently, Prof Raman is the only living person in the list. Her early years were spent in Chennai where the love and support of her parents and teachers got her interested in math. In her undergraduate, she seriously considered majoring in Sanskrit poetry but the mathematician won that battle. It is not surprising both the fields require a lot of originality and creativity. She went on to get her doctorate at the prestigious Tata Institute of Fundamental research (TIFR). She always had excellent mentors in her early years of research and that was the sounding board whi...

A ring is a group with additional features in abstract algebra. But first let’s look at the set of integers, set of real numbers, A matrix with 2 rows and 3 columns and the set of complex numbers. If we look at the set of integers(Z), it is closed under addition, subtraction and multiplication meaning that if we perform these operations on two integers, we will get another integer. However, if we divide two integers, we might not get another integer but rather a fraction. In a similar vein, the set of real numbers(R) will be closed on all these four operations provided we do not divide by 0. In the case of  R 2x3   matrix, while it will be closed under addition and subtraction, we cannot multiply two matrices of arbitrary sizes but rather follow the rule and inverse of a matrix does not always exist.   Lastly, if we take two polynomials and divide them, we get a rational function so the set of complex numbers is not closed under division. We alr...

Livingston High School student Rohan Jha won a prize of "200 Pi Dollars." See what that comes out to in U.S. currency here. LIVINGSTON, NJ — For a glimpse into the tongue-in-cheek tone of the Steven H. Strogatz Prize for Math Communication, one needs only look at the prize of "200 Pi Dollars." For those not mathematically inclined, that total – which comes out to about $628 (200 x 3.14159 = $628.32) – was what Livingston high school student Rohan Jha earned via the inaugural award event. The contest was spearheaded by the National Museum of Mathematics (MoMath) in New York City. It seeks to highlight high school students who "celebrate the universality of math" using social media, visual art, writing and dance. The 2020 winners, their projects and the original call for entries can be viewed here. "The purpose of Math Musings, the magazine I started in high school, was to show that math is everywhere, yet many times we are not awar...

In the 1800s, mathematicians knew how to solve equations like these starting from the linear equation. However, what about equations of higher degree? Degrees 5,6,7 and beyond? A young teenager at the time in France, Ēvariste Galois answered this question. . And to do so he used a tool that he called a “group” . Around this time, Carl Friedrich Gauss was making sensational discoveries of his own. He showed a new technique called modular arithmetic which helped him solve many problems in number theory. As it turned out, Modular arithmetic shared many similarities to the groups used by Galois. The 1800s also saw a revolution in geometry. For more than 2000 years, Euclid dominated the scene with his book – “The Elements” but mathematicians began to realize there are other geometries beyond the one devised by the ancient Greeks. It didn’t take long before groups were found to be a useful tool in studying these new geom...

Singular value decomposition (SVD) is one of the jewels of linear algebra. In modern times it has found applications in machine learning and Artificial intelligence. The most important feature of SVD is that of dimension reduction so that we are able to make predictions on very large data sets with a small subspace of variables.   Let’s try to understand how it works through a real-life example. Netflix has many subscribers and they have a collection of movies. When we watch a movie, we often see similar movies being recommended to us next time we go to Netflix and we wonder how did they figure out the kind the movies I like! The answer is SVD . As an example, suppose Netflix has 5 movies and multiple subscribers. Netflix gets the data for every user with a rating going from 0 to 5 with 0 implying the user did not watch a particular movie and 5 implying the user has watched it multiple times. A sample data set will look like this.  Each row in this matrix A ...

First, we need to understand what we mean by a square free number. It essentially means that a whole number is not divisible by the square of a prime number. Let’s take the example of 30. From fundamental theorem of arithmetic, we can write it uniquely in terms of prime numbers as 30= 2 × 3 × 5 Since none of the prime numbers are repeated, 30 is square free. Now suppose we look at number 12. 12= 2 × 2 × 3 In this case the prime number 2 is repeated and hence 12 is not square free . The question we have posed in this blog is -what is the probability that a randomly chosen whole number is square free is not very precise. The reason being that we know there are an infinitely many primes and there is no associated probability distribution. So, for our discussion we will assume that we are talking about a finite set of whole numbers say a million and ask the question what proportion of those are square free. In a more general sense, we can take a big number x ...