A presentation on Group Cohomology and Cochain Complexes, with applications to Class Field Theory. Talk given at UChicago Math REU-2025.
This is a short story that I initially wrote for my UChicago supplemental essay for the following prompt:
“If there’s a limited amount of matter in the universe, how can Olive Garden (along with other restaurants and their concepts of food infinity) offer truly unlimited soup, salad, and breadsticks? Explain this using any method of analysis you wish—physics, biology, economics, history, theology… the options, as you can tell, are endless.” – Inspired by Yoonseo Lee, Class of 2023.
...In this paper, we explore the $p$-adic system, by defining it in multiple ways: as an extension of the $p$-adic integers, as well as an extension of the rationals. We then proceed to perform analysis in the $p$-adics, by defining convergence, continuity and discs. We then describe exponentiation and logarithmic functions over the $p$-adics as functions derived from power series. We explore the radius of convergence and other properties of these functions. We then explore the Artin-Hasse exponential, which, though seemingly random, turns out to be an integral power series.
...Card shuffling is an important part of playing any card game. When a card deck isn’t shuffled properly, it leads to uneven and unfair distribution of cards. It would certainly help if you knew what cards other players had!
Suppose we have a specified shuffling algorithm. Is it possible for us to guess the outcome of the card shuffling? Is it possible that the card deck will at some point return to its original configuration? In a previous article we explored ways of trying to guess the outcome of a specific type of card shuffling, known as the riffle shuffle. In this article, we modify the shuffling algorithm to introduce a flip that introduces some interesting complexities.
...Take a square piece of paper. Take a toothpick of a given length. At intervals equal to the length of the toothpick, draw lines on the piece of paper. Now, randomly toss a bunch of such toothpicks so that they fall over the paper. What fraction of the toothpicks will fall in a way so that they intersect one of the lines drawn? In other words, what is the probability of a single toothpick falling over a line?
...Consider a prime $p$. For what integers $n$ does $p$ divide all of $$\binom{n}{1}, \binom n 2, \binom n 3, \dots, \binom{n}{n-1}?$$ Can we characterize all such $n$ given a value of $p$? It turns out that this happens if and only if $n$ is a perfect power of $p$. How do we prove this? In fact, there exists a simple proof using elementary methods. But as always, it is both fun as well as good to prove everything twice. In this article I present an interesting way to characterize all such $n$ using concepts of finite fields.
...A well known theorem about finite fields states the following.
$F$ is a finite field if and only if $|F| = p^k$ for some prime $p$ and positive integer $k$.
In this write-up, we prove the first part of the above theorem, i.e. if $F$ is a finite field, then $|F| = p^k$ using a different approach.
We first list down some basic definitions that will be used as we move forward in this write-up.
...The problem of balancing centrifuges is equivalent to finding sets of $k$ $n$-th roots of unity that sum to $0$. We can represent each slot in the centrifuge as an $n$-th root of unity as the slots in the centrifuge are also equally spaced and equidistant from the origin.
Accepted for publication at the Joint Mathematics Meetings, Boston, January 4 – 7, 2023.
This work was carried out as part of a research project at PROMYS-2022. I am a grateful recipient of the Mehta Fellowship to the PROMYS programs in 2021 and 2022.
...Suppose you have a deck of $52$ cards. You perform a riffle shuffle on these cards. Can you say what the new position of the first card in the deck is, after the shuffle? What about the second card in the deck? The third? In general, is it possible to predict the new position of the $k$th card in the original deck after a new deck is produced through a riffle shuffle?
...A presentation on the mathematics of cryptography, with an emphasis on RSA. A link to the writeup is here.