pramana's blog

A presheaf which is not a sheaf

Thu, 12 Jun 2025 00:00:00 +0000

I’ve been reading Ravi Vakil’s Rising Sea: Foundations of Algebraic Geometry this semester, specifically the second chapter which gives an introduction to sheaf theory. While the first half of the chapter serves to introduce the idea of a sheaf and to convince you that it makes sense geometrically to define it, I would say the meat of the chapter is defining the sheafification of a presheaf and explain how it helps us do homology on sheaves.

Can a machine learn mathematical structure?

Tue, 23 Jul 2024 00:00:00 +0000

When I think of machine learning, I tend to think about estimation; a machine learning algorithm is a method of estimating a function given a (usually large) set of data. As a result, I’ve primarily seen machine learning used when the function we are trying to estimate is assumed to be continuous (or differentiable, smooth, etc.), because then you can use nice results from analysis to prove convergence. But algebraic structures like groups can be discrete objects, where there is no way to smoothly interpolate between two group elements. Moreover, groups adhere to very strict rules that dictate how they can be studied, which makes this continuous way of thinking hard to apply – I mean, it’s hard to imagine that a machine learning model, just armed with data about groups, functions between them, and some backpropagation, could reach the isomorphism theorems. Despite this, the research I did last semester opened my mind more to using classical machine learning methods to get structure of discrete groups, given that we ask train the machine on the right questions.

Why homology is weaker than homotopy

Fri, 15 Mar 2024 00:00:00 +0000

The main two flavors of invariants in an introductory algebraic topology class are the homology groups associated with a space and the homotopy groups (typically just the first homotopy group, $\pi_1$). The fundamental group is formed by looking at loops starting at a certain point. Similarly, the homotopy groups look at cycles, which can be thought of as loops that commute. Because of these similarities, I wondered about which one was “better” at differentiating topological spaces.

Understanding exact sequences

Thu, 28 Dec 2023 00:00:00 +0000

Let’s start with a motivating question I had from group theory: if $N \trianglelefteq G$ is a normal subgroup, is it necessarily true that $G\cong G/N\times N$? Looking at non-abelian groups quickly shows that the answer is no (take the dihedral group $D_3=\langle r,t \mid r^3=t^2=1, rt=tr^{-1}\rangle$ and the normal subgroup $\langle r\rangle\cong \mathbb{Z}/3$).

I later learned about semidirect products $\rtimes_\phi$, which generalize the direct product $\times$. But even this was insufficient to describe group decompositions.

My first CTF: HackMIT 2023's puzzle

Sat, 30 Sep 2023 00:00:00 +0000

On July 4th, MIT emailed all applicants to their hackathon, HackMIT, that they have started a puzzle, where the top 50 competitors would be granted admission into HackMIT.

Taking their advice, I went to checking the website. In the FAQ section, I noticed something new:

What does it mean?

The strange placement of upper/lowercase letters and the length of all words being the same made me think some message was being hidden here in binary. If we let uppercase letter $= 1$ and lowercase letter $=0$, then we get the strings

Now

Wed, 06 Sep 2023 00:00:00 +0000

Following the ideas behind a now page, I am…

Going through my second year of undergrad.
Building React apps in preparation for hackathons.
Getting better at digital art, starting with anatomy.
Learning a new program for video editing, Premiere Pro.

Visualizing prime spectrums with graphs

Wed, 07 Jun 2023 00:00:00 +0000

In Atiyah and Macdonald’s book we construct a certain topological space from a ring called the prime spectrum of a ring. While working through the exercises, I saw an interesting way of viewing closed and open subsets of the prime spectrum that borrowed ideals from combinatorics.

What is the prime spectrum of a ring?

Definition (Prime ideal).
An ideal $\mathfrak{a}$ of a ring $A$ is a subring (a subset of $A$ that is still a ring under $A$’s operations) that has the property that any $a\in A$ times $p\in \mathfrak{a}$ has $ap\in \mathfrak{a}$ as well. That is, the elements in the ideal stay in the ideal after multiplying by any element of $A$.

Visualizing horospheres on hyperbolic groups

Tue, 02 May 2023 00:00:00 +0000

This semester I was able to participate in our undergradute research lab, and, relevant to my past work, our project was in the field of geometric group theory. I plan on introducing our project and go over some topics in geometric group theory I haven’t covered before. This project was joint work with Noah Jillson and Katerina Stuopis with our advisors Daniel Levitin and Tullia Dymarz.

If you want to skip the mathematical jargon and see the results, skip to the end.

Weak solutions to dirichlet boundary problems

Sat, 18 Feb 2023 00:00:00 +0000

For the final project in Analysis I, my professor asked us to write a paper on a topic adjacent to analysis. Being an applied mathematician, he recommended I study a very specific inequality called Gårding’s inequality.

If you would like to learn more about Gårding’s inequality itself, my paper has all the details about it, the prerequisites to understand it, and an application. In fact, most of the paper covers the necessary material needed to understand what the inequality is even saying (PDE analysis seems to be a lot less abstract than other fields, considering all the machinery I needed). However, for a blog post, I don’t want to bog down any actual interesting results by trying to describe how this inequality functions. This will only focus on the broader subject of weak solutions to PDEs.

Group structures

Wed, 09 Nov 2022 00:00:00 +0000

I’ve been working on Beachy and Blair’s Abstract Algebra Book, specifically their chapter on Group structure (7), and I wanted to summarize what I’ve learned about group structures.

Finite groups and structure

The problems and results derived from this chapter almost all concern with finite groups. We know a lot about finite groups. For example, we have this result:

Theorem (Burnside, 1904). Every group of order $p^a q^b$, where $p$ and $q$ are (not necessarily distinct) primes is solvable.

Graph ends and their purpose in geometric group theory

Wed, 06 Jul 2022 00:00:00 +0000

I’ll give an intuition for how mathematicians define what graph ends are, and explain their use in geometric group theory. However, before I can do that, I’ll explain what ends are in a graph theory perspective.

What are ends?

The word ends is a little confusing, because they don’t really represent the “end” of a graph, at least not how we would define an end in real life. Think of a rope. If two teams were playing tug of war on that rope, we know that it has two ends that the opposite teams would tug on. Now if this rope is of finite length (it probably is), then we would typically call the two points at the end of the rope the ends (obviously).

Tue, 28 May 2019 00:00:00 +0000

About this blog

Mon, 01 Jan 0001 00:00:00 +0000

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👋🏽 Hello, and welcome to my website! My name is Pramana Saldin and I will be a PhD student at UC-Berkeley in Fall 2026. Before this, I recieved my B.S. in Mathematics from UW-Madison (my home state!) and took part in a study abroad semester at the National University of Singapore.

course guide

Mon, 01 Jan 0001 00:00:00 +0000

Some thoughts on UW courses. Inspired by this.

ANTHRO 105: Principles of biological anthropology

bio credit ftw

MATH 475: Introduction to combinatorics

Very tedious course. Know how to use basic wolfram/python code and you will pass.

Friends

Mon, 01 Jan 0001 00:00:00 +0000

My friends (who have a website $\cap$ do mathematics).

Notes

Mon, 01 Jan 0001 00:00:00 +0000

MATH 741 and 742: Abstract algebra I and II

Notes (PDF)

MATH 721: A first course in real analysis

Notes (PDF)

Modular forms

I gave a talk for my Analytic Number Theory course at the National University of Singapore on Ramanujan’s tau function, with a proof of some properties using Hecke operators. (Slides)
I gave a talk at UW’s Undergraduate Math Club on modular forms. The talk is largely based on Professor Keith Conrad’s CTNT summer school. (Slides)

Algebraic geometry

Resultants and Elimination Theory (PDF)
- Expository paper from a directed study in algebraic geometry under Professor Brian Lawrence. The resultant allow us to check if the polynomials $f$ and $g$ share a root just using their coefficients. Using these ideas, we can prove a more general theorem in algebraic geometry called the fundamental theorem of elimination theory.

MATH 531: Probability theory

Notes (PDF)

MATH 621: Introduction to manifolds

Final paper - Swan’s theorem (PDF)
- Abstract. Swan’s theorem establishes a module structure on the set of smooth sections over a vector bundle $\pi\colon E\to M$. This allows us to study algebra by means of vector bundles and vice-versa. We conclude with an introductory result in K-theory that follows from this theorem

MATH 542: Modern algebra II

Notes (PDF)

MATH 522: Analysis II

Notes (PDF)

MATH 514: Numerical analysis

Notes (PDF)

MATH 521: Analysis I

Notes (PDF) (TeX)
Final paper – Gårding’s Inequality (PDF) (TeX)
- Abstract. This is an expository paper on the prerequisites for Gårding’s inequality. Proposed by Lars Gårding, this inequality has applications in the study of weak solutions to elliptic partial differential equations. We will give the prerequisites to state Gårding’s inequality and give one application.

MATH 475: Introduction to combinatorics

Notes (Chapter 14: Burnside’s Lemma and Polya Counting) (PDF)

MATH 531 (UWM): Modern Algebra

Notes (PDF)

Research

Mon, 01 Jan 0001 00:00:00 +0000

Papers

Classification of Finite Groups With Equal Left and Right Quotient Sets
with Haran Mouli.
Preprint.
- We have been made aware of the paper by Marcel Herzog, Gil Kaplan, Patrizia Longobardi, and Mercede Maj: “Products of subsets of groups by their inverses,” which agrees with our main result and extends it to infinite groups. We are grateful to Liubomir Chiriac for showing us this result.
Comparing Left and Right Quotient Sets in Groups
with June Duvivier, Xiaoyao Huang, Ava Kennon, Say-Yeon Kwon, Steven J. Miller, Arman Rysmakhanov, and Ren Watson.
Submitted to Proceedings of the Integers Conference 2025.