Curves over C and Abel at 26
Graduate Number Theory Seminar, Department of Mathematics, The University of Arizona
In this seminar, I discussed some interesting things I learned while preparing for my comprehensive exam.
Lecture Notes
It is not enough to be in the right place at the right time. You should also have an open mind at the right time. -Paul Erdős
Engaged in Applications
Graduate Number Theory Seminar, Department of Mathematics, The University of Arizona
In this seminar a brief overview of the computer-related applications of number theory was given. One important example I didn’t discuss in this seminar is the work of Lenore Blum leading to cryptographically secure pseudorandom number generator.
Supersingular isogeny Diffie-Hellman
Graduate Number Theory Seminar, Department of Mathematics, The University of Arizona
In this seminar we discussed the mathematics involved in the working of a post-quantum cryptographic protocol based on isogenies between supersingular elliptic curves. Some familiarity with the arithmetic properties of elliptic curves was assumed (for example, see my Fall 2020 RTG presentation slides).
Elliptic Curve Cryptography 2.0
CS Minor pitch, Department of Computer Science, The University of Arizona
In this presentation I discussed pairing-based cryptography, the mathematical concepts involved and its possible applications. I would recommend reading the extended abstract from Crypto 2001 to understand the mathematics involved using a concrete example.
Diophantine Equations
Tucson Math Circle, Department of Mathematics, The University of Arizona
The concepts like gratest common divisor and Euclidean algorithm were introduced. These were used to provide a method for solving linear Diophantine equations in two variables.
Lebesgue Differentiation Theorem
M555 Coursework Presentation, School of Mathematical Sciences, NISER Jatni
Lebesgue differentiation theorem is an analogue, and a generalization, of the fundamental theorem of calculus in higher dimensions.
Tangram
MathToys 5, MathematiX Club, NISER Jatni
The tangram (Chinese word, literally: “seven boards of skill”) is a dissection puzzle consisting of seven flat shapes, called tans, which are put together to form shapes. There are infinite possible arrangements that can be created using the seven pieces of tangram. In 1942, two Chinese mathematicians, Fu Traing Wang and Chuan-Chih Hsiung proved that by means of the tangram exactly thirteen convex polygons can be formed. We verified this statement. We also discussed scissors-congruence, Banach-Tarski paradox, and the snug tangram number problem proposed by Ronald C. Read.
Math-O-Trick
Open Day 2017, Science Activity Club, NISER Jatni
This was a 45-minutes card magic trick performance prepared and performed by 7 students from my college (alphabetic order):
Elliptic Adventures of a Mathematician in Prison
SUMS 74, MathematiX Club, NISER Jatni
This was the seventh lecture in the series of ten lectures (SUMS 68 to SUMS 77) based on the lecture notes of Kyoto University (Japan) weekly mathematics seminars for high school students in December 1994 and January 1996. In this lecture we saw a proof of Poncelet Closure Theorem.
Bachet 3
Lectures on Mathematical Card Tricks, School of Mathematical Sciences, NISER Jatni
This was the third lecture of the three lectures I gave on mathematical card tricks. In this lecture we discussed “Gilbreath Principle”, “The Great Discovery”, “Remembering the Future”, and “A Mathematical Wizard”. These card tricks were later performed by 7 people during the NISER Open Day 2017. I just told the theory, how to perform them was taught by Swaroop and Devashish.
Bachet 2
Lectures on Mathematical Card Tricks, School of Mathematical Sciences, NISER Jatni
This was the second lecture of the three lectures I gave on mathematical card tricks. In this lecture we discussed “Peirce Curiosity”, “Pairs Repaired”, “The Royal Pairs”, and “The Cyclic Number”. These card tricks were later performed by 7 people during the NISER Open Day 2017. I just told the theory, how to perform them was taught by Swaroop and Devashish.
Bachet 1
Lectures on Mathematical Card Tricks, School of Mathematical Sciences, NISER Jatni
This was the first lecture of the three lectures I gave on mathematical card tricks. In this lecture we discussed “Monge’s shuffle”, “Spelling the Spades” and “Gergonne’s Pile Problem”. These card tricks were later performed by 7 people during the NISER Open Day 2017. I just told the theory, how to perform them was taught by Swaroop and Devashish.
First Case of Fermat’s Last Theorem
SRS 4, MathematiX Club, NISER Jatni
Fermat’s Last Theorem (FLT) states that $x^n+y^n=z^n$ has no non-trivial integer solution for $n>2$. It is easy to show that if the theorem is true when $n$ equals some integer $r$, then it is true when $n$ equals any multiple of $r$. Since every integer greater than 2 is divisible by 4 or an odd prime, it is sufficient to prove the theorem for $n=4$ and every odd prime. On $19^{th}$ September 1994, Andrew Wiles announced that he had finally completed the proof of FLT. In this seminar we saw an elementary proof by Sophie Germain (1823) which can be extended to prove FLT for all prime exponents less than 1700.
Geometry Around Us: An Introduction to Non-Euclidean Geometry
Open Day 2016, Science Activity Club, NISER Jatni
Using one of the most remarkable results from Euclidean Geometry “Morley’s Miracle” as motivation I ventured into real-world geometries. Discussed some examples from our surroundings illustrating non-euclidean geometry. The discovery of non-Euclidean geometries posed an extremely complicated problem to physics, that of explaining whether real physical space was Euclidean as had earlier been believed, and, if it is not, to what what type of non-Euclidean spaces it belonged. This problem is still not completely resolved.
Celebrating Uncle Paul’s 103rd Birthday
SRS 3, MathematiX Club, NISER Jatni
We know that the sequence of prime numbers $2, 3, 5, 7, \ldots$ is infinite and the size of its gaps between two prime numbers is not bounded. In fact we can give a sequence of $k$-consecutive composite numbers, $N+2, N+3, N+4, \ldots, N+(k+1)$ where $\displaystyle{N = \prod_{p\leq k+2} p}$. Bertrand conjectured that the gap to the next prime cannot be larger than the number we start our search at. I this talk I paid homage to Paul Erdős (whom I refer to as “Uncle Paul”) by discussing his elegant proof of Bertrand’s conjecture.
Continued fractions in disguise
SUMS 54, MathematiX Club, NISER Jatni
Continued Fractions have fascinated many mathematicians due to their mystical properties. In this talk I discussed one of my personal experiences with continued fractions (without giving their exact mathematical definition). I illustrated two methods (without proof) for solving equations in two variables in integers.
Sperner’s Theorem
M203 Coursework Presentation, School of Mathematical Sciences, NISER Jatni
An application of concept of chain, anti-chain and posets.
Beautiful Repetitions: 5-minute introduction to Iterations & Fractals
Open Day 2015, Science Activity Club, IoP-NISER Bhubaneswar
Fractals constitute a relatively modern discovery; they date to the latter half of the 20th century and may be said to have originated in the work of the French Mathematician Benoit Mandelbrot. The aim was to show how beautiful the simple iterative processes can be, thus providing motivation for studying fractals. Some definitions related to iterations were introduced. I also showed that we encounter iterations and fractals everyday.
Celebrating 110th Birthday of D. R. Kaprekar
SUMS 46, MathematiX Club, NISER Bhubaneswar
Dattarya Ramchandra Kaprekar was an Indian recreational mathematician who described several classes of natural numbers. The motive of this talk was to give a flavor of Elementary Number Theory and Iterations, by discussing the contributions of D. R. Kaprekar. I discussed Kaprekar Numbers, Kaprekar Routines and Kaprekar Sequences. Nothing more than class 10 mathematics was needed to understand this talk.