This is a talk I gave for the Kiddie colloquium on November 8th, 2023. Introduction# Let $M, N$ be manifolds with boundary, and let $B \subseteq \partial M$ and $C \subseteq \partial N$ be some union of connected components. If we are given a diffeomorphism $\varphi \colon B \cong C$, how do we glue $M$ and $N$ along this diffeomorphism to obtain $ M \amalg_{B,\varphi} N $ as a smooth manifold? What one usually does is first find a “collar neighborhood,” a neighborhood $U$ of $B$ inside $N$ tha…
December 17, 2023Mathematics27 minutes
As I’ve told some people, I was in Japan for the past month, working as a member of the IMO 2023 Problem Selection Committee. It was a unique experience, very much different from participating as a contestant, and I’d like to share it a little. From left to right: Arnaud Maret, Paul Vaderlind, Ivan Guo, Hiroki Kodama, Elisa Lorenzo Garcia, Sam Bealing, Tetsuya Ando, Yuya Matsumoto, Yuta Takaya, Takuma Kitamura, Genki Shimizu, Géza Kós, <…
July 18, 2023Announcements7 minutes
Say $G / \mathbb{Q}$ is a connected reductive group, and let $\lbrace \mu \rbrace$ be a conjugacy class of geometric cocharacters $\mathbb{G}_ {m,\bar{\mathbb{Q}}} \to G_{\bar{\mathbb{Q}}}$, with reflex field $E$. Say there is a faithful representation $G \hookrightarrow \GL(V)$ and this is cut out by tensors $s_\alpha \in V^\otimes$. Given $X / E$ a scheme, a $G$-torsor on $X$ can then by identified with the data of a vector bundle $M$ on $X$, sections $s_{\alpha,M} \in \Gamma(X, M^\otimes)$, …
June 24, 2023Mathematics18 minutes
This is a talk I gave on January 23rd, 2023 at the Kiddie colloquium. Here is an easy way to teach cohomology in three easy steps. Teach that a doughnut is the same thing as a coffee mug. Define the configuration space of charged particles in an $n$-disc. Define cohomology as a parametrization charged particles. We already know that a doughnut is the same thing as a coffee mug, so let’s skip the first step. Disclaimer: when we speak of a topological space, we imagine it to be a “nice” topologi…
January 23, 2023Mathematics23 minutes
This is a write-up of a talk I gave on May 27th, 2022 at the Kiddie colloquium. The beginning# In the beginning, the world was a formless void, and darkness was upon the face of the deep. And God said, “Let there be finite-dimensional vector spaces!” We fix a base field $k$ throughout the talk. Each finite-dimensional vector space has its own unique color. <use…
December 26, 2022Mathematics43 minutes
Introduction# Shortly after I entered graduate school, I was advised by a number of professors to go through Chapters II and III of Hartshorne’s Algebraic Geometry thoroughly, solving all the exercises within. As it turned out, there are some absurdly difficult results that are given as exercises. (Seriously, openness of the flat locus is an exercise?) It was a laborious and somewhat painful process, but going through those problems did force me to learn all the technical commutative algebra, a…
April 22, 2020Announcements7 minutes
How does sheafification work? If we are working on a topological space, the way Hartshorne and Vakil do it is to first define stalks, and then define a section of the sheafification \( \mathscr{F}^\# \) as a locally compatible collection of germs. Of course, this is relying on the fact that the corresponding topos has enough points. Let us do this in a more general setting. 1. Sites and sheaves# In the general setup, we work on a site \( \mathcal{C} \), i.e., a category with a Grothendieck topo…
April 19, 2020Mathematics25 minutes
Given a set of tiles, can you use them to cover the entire plane without any overlaps? The attempt to answer this question led to the discovery of various tilings exhibiting intricate structures, such as the Wang tilings, the Penrose tilings, and the Socolar–Taylor tiling. In this talk, I will try to eliminate these beautiful tilings and make everything periodic and orderly. This is a talk given on October 2nd, 2018, for the Harvard Math Table. I’ve been meaning to write a post on this subject, …
October 2, 2018Mathematics11 minutes
Around last December, I embarked on a project of writing an introductory linear algebra textbook, starting from the definition of vector spaces and leading up to the spectral theorem. Currently I have a draft that is about 150 pages long, and I plan on improving it when I have time. The book will be available for free from the following link: A rough guide to linear algebra. The motivation for the project was a sort of academic elitism. I wanted there to be a book that would be most useful for a…
August 27, 2018Announcements3 minutes
I learned this while talking to a physics friend. It is unfortunate that the audience aren’t exposed to too much math in most physics classes, so that such elegant mathematical formalisms are rarely mentioned in class. Anyways, here is the interpretation of Maxwell’s equations in terms of \( \mathrm{U}(1) \)-bundles. 1. Maxwell’s equations# When we first learn Maxwell’s equations, we learn a set of four equations, up to some constants: \( \displaystyle \begin{cases} \nabla \cdot E = \rho, \\ \n…
August 22, 2018Mathematics7 minutes
