Gaurish Korpal

Gaurish Korpal

PhD Candidate

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A list of all the posts and pages found on the site. For you robots out there is an XML version available for digesting as well.

Pages

Posts

Reading and annotating papers

3 minute read

Published:

Following is the comparison of tech specs of my new tablet (USD 315 + taxes) with the older tablet (USD 280 + taxes):

Higher Arithmetic Computations

15 minute read

Published:

In this post I discuss the options available for doing computational experiments in advanced number theory.

Space filling curve

4 minute read

Published:

In this post I would like to share my old writeup about there existence of a continuous surjective map from $\mathbb{R}$ to $\mathbb{R}^2$. In other words, we will prove that that Hilbert’s Curve is a continuous surjective map. The proof presented here was explained by Dr. Manas Ranjan Sahoo, 3 years ago on 15th April 2016.

Richert Theorem

2 minute read

Published:

In 1852, Chebyshev proved the Bertrand’s postulate:

Cross Diagonal Cover - V

1 minute read

Published:

This has been an exciting week! Prof. Sukanta Pati proved an interesting theorem that enables us to get decomposition of $2m-1\times n$ grids into simpler grids, hence simplifying counting to large extent (note that $m=n$ is also allowed). It enables us to surpass the difficulty posed by “more than two crosses in one square”, thus supporting the idea of colouring (i.e. not giving importance to two crosses in a square).

Cross Diagonal Cover - IV

5 minute read

Published:

While discussing this problem with Dr. Shailesh Shirali, he commented that there has to be a way to phrase the problem in terms of a ray of light being reflected off the walls of the rectangle, bouncing around, proceeding from one corner to some other corner.

Cross Diagonal Cover - III

1 minute read

Published:

I found many counterexamples to my conjecture, like

  • for Case 2 in 7 by 5 grid we have 12, 11 by 5 grid we have 19 and 15 by 5 grid we have 26 filled squares
  • for Case 3 in 4 by 10 grid we have 10 filled squares

Also, grant93jr made the following comment:

Cross Diagonal Cover - I

1 minute read

Published:

While doodling in my college classes, I designed an algorithm which I called Cross Diagonal Cover Algorithm:

Spot The Flaw

less than 1 minute read

Published:

Many of you must have seen numerous examples of such situations where we get an absurd result and have to spot flaw in proof.

Hello World

less than 1 minute read

Published:

Here is my first post.

computer

lecture

Celebrating 110th Birthday of D. R. Kaprekar

Published:

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.

Beautiful Repetitions: 5-minute introduction to Iterations & Fractals

Published:

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.

Sperner’s Theorem

Published:

An application of concept of chain, anti-chain and posets.

Continued fractions in disguise

Published:

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.

Celebrating Uncle Paul’s 103rd Birthday

Published:

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.

Geometry Around Us: An Introduction to Non-Euclidean Geometry

Published:

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.

First Case of Fermat’s Last Theorem

Published:

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.

Bachet 1

Published:

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.

Bachet 2

Published:

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 3

Published:

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.

Math-O-Trick

Published:

This was a 45-minutes card magic trick performance prepared and performed by 7 students from my college (alphabetic order):

Tangram

Published:

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.

Lebesgue Differentiation Theorem

Published:

Lebesgue differentiation theorem is an analogue, and a generalization, of the fundamental theorem of calculus in higher dimensions.

Diophantine Equations

Published:

The concepts like greatest common divisor and Euclidean algorithm were introduced. These were used to provide a method for solving linear Diophantine equations in two variables.

Supersingular isogeny Diffie-Hellman

Published:

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).

Engaged in Applications

Published:

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.

EdDSA: Not just ECDSA with a twist

Published:

The objective of this presentation is to educate about the design of contemporary digital signature schemes. It’s for an audience familiar with cryptographic hash functions, digital signatures, and elliptic curve cryptography.

pencil

Substitution ciphers

Posted on August 29, 2021

Substitution is a function which uses a set of rules to transform elements of a sequence into a new sequence using a set of rules which “translate” from the original sequence to its transformation. The easiest substitution is given when each character is replaced by exactly one other character. This encryption can be broken with statistical methods because in every language characters appear with a particular probability.

Transposition ciphers

Posted on September 19, 2021

Permutation of a set $X$ is a bijective function $\sigma : X \to X$ that for each element $x \in X$ assigns a unique value $\sigma(x) \in X$. A transposition is a permutation of two elements and any permutation is also a product of transpositions.

quantum

Isogeny based cryptography

Published:

Isogeny-based cryptography is a kind of elliptic-curve cryptography, whose security relies on (various incarnations of) the problem of finding an explicit isogeny between two given isogenous supersingular elliptic curves over a finite field $\mathbb F_q$. However, given an elliptic curve $E$ in Weierstrass form over a finite field $\mathbb F_q$ and a point $P$ on $E$ of order $n$, one can compute a cyclic separable isogeny of degree $n$ using Velu’s formulas in SageMath (implemented by D. Shumow in 2009).

reading

scholarly

teaching

Math 112 - College Algebra Concepts and Applications

Lower Division Undergraduate Course, University of Arizona, 2020

The following is some information regarding the 15 week iCourse section (103/203/403) of the course Math 112 that I will be assisting with during Fall 2020.

Math 112 - College Algebra Concepts and Applications

Lower Division Undergraduate Course, University of Arizona, 2021

The following is some information regarding the 15 week iCourse section (103/203/403) of the course Math 112 that I will be assisting with during Spring 2021.

Math 511B - Algebra

Core Graduate Course, University of Arizona, 2021

The following is some information regarding the core graduate abstract algebra course (the second half) that I will be assisting with during Spring 2021.

technical

Diophantine Equations

Supervised by S. A. Katre (S. P. Pune University & Bhaskaracharaya Pratishthana, Pune)

Work done during summer 2015 (May 18, 2015 to June 16, 2015). Supported by INSPIRE SHE.

Enigma Cryptanalysis

Supervised by Geetha Venkataraman (Ambedkar University Delhi)

Work done during summer 2015 (July 06, 2015 to July 26, 2015). Supported by INSPIRE SHE.

Number Fields

Supervised by Ramesh Sreekantan (Indian Statistical Institute, Bangalore)

Work done during summer 2016 (June 01, 2016 to July 31, 2016). Supported by INSPIRE SHE.

Arithmetic Geometry - I

Supervised by B. Sahu (National Institute of Science Education and Research, Bhubaneswar)

Work done during fall 2017 (August 01, 2017 to November 17, 2017). Requirement for NISER’s course M498.

Modular Forms

Supervised by M. Manickam (Kerala School of Mathematics, Kozhikode)

Work done during winter 2017 (December 08, 2017 to December 30, 2017). Supported by INSPIRE SHE.

Arithmetic Geometry - II

Supervised by B. Sahu (National Institute of Science Education and Research, Bhubaneswar)

Work done during spring 2018 (January 07, 2018 to April 20, 2018). Requirement for NISER’s course M499.

Sheaf-theoretic de Rham isomorphism

Supervised by Ritwik Mukherjee (National Institute of Science Education and Research, Bhubaneswar)

Work done during fall 2018 (August 01, 2018 to November 20, 2018). Requirement for NISER’s course M598.

The Weil Conjectures for Elliptic Curves

Supervised by Brandon Levin (The University of Arizona, Tucson)

Work done during fall 2020 (August 24, 2020 to December 09, 2020). Requirement for UArizona’s course Math 596G.

Lang-Nishimura theorem

Supervised by Brandon Levin (The University of Arizona, Tucson)

Work done during parts of spring and summer 2021 (March 03, 2021 to August 02, 2021). MS Thesis as an assessment option for the PhD Qualifying Exams at UArizona.