Dongryul Kim - BlogDepartment of Mathematics, Imperial College LondonZola2026-09-17T00:00:00+00:00https://dongryul-kim.github.io/blog/atom.xmlGluing manifolds along boundaries2023-12-17T00:00:00+00:002023-12-17T00:00:00+00:00https://dongryul-kim.github.io/blog/gluing-manifolds/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…Looking back on the Problem Selection Committee2023-07-18T00:00:00+00:002023-07-18T00:00:00+00:00https://dongryul-kim.github.io/blog/imo-psc-2023/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, …μ-filtrations2023-06-25T00:00:00+00:002023-06-25T00:00:00+00:00https://dongryul-kim.github.io/blog/mu-filtrations/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 …How to teach cohomology to kiddies2023-01-23T00:00:00+00:002023-01-23T00:00:00+00:00https://dongryul-kim.github.io/blog/eilenberg-maclane/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 char…Linear algebra through pictures2022-12-26T00:00:00+00:002022-12-26T00:00:00+00:00https://dongryul-kim.github.io/blog/string-diagrams/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…Solving Hartshorne exercises2020-04-22T00:00:00+00:002026-08-23T00:00:00+00:00https://dongryul-kim.github.io/blog/hartshorne-exercises/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. (…How to sheafify in one go2020-04-19T00:00:00+00:002020-04-19T00:00:00+00:00https://dongryul-kim.github.io/blog/sheafification/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 corr…Eliminating aperiodicity in tilings2018-10-03T00:00:00+00:002018-10-03T00:00:00+00:00https://dongryul-kim.github.io/blog/translational-tilings/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 t…A rough guide to linear algebra2018-08-27T00:00:00+00:002026-09-17T00:00:00+00:00https://dongryul-kim.github.io/blog/linear-algebra-book/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…Maxwell's equations2018-08-23T00:00:00+00:002018-08-23T00:00:00+00:00https://dongryul-kim.github.io/blog/maxwell-equations/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…Tate's thesis 2: global zeta functions2018-04-21T00:00:00+00:002018-04-21T00:00:00+00:00https://dongryul-kim.github.io/blog/tate-thesis-2/In the previous post, we defined local zeta functions and meromorphically extended them to the complex plane. We now work towards defining the global zeta function. Although local zeta functions are supposed to be local factors of the zeta function, we are going to allow multiplying local zeta funct…Tate's thesis 1: local zeta functions2018-04-15T00:00:00+00:002018-04-15T00:00:00+00:00https://dongryul-kim.github.io/blog/tate-thesis-1/The main goal of this series of posts is to give a brief summary of Tate’s thesis [Tat67]. Given a number field \( K \) and a character \( \chi : \mathrm{Cl} _ \mathfrak{m}(K) \rightarrow S^1 \), one defines the Dirichlet \( L \)-function as
\[ \displaystyle L(s, \chi) = \sum _ {\mathfrak{a} \subset…Dwyer–Kan localization2018-03-20T00:00:00+00:002018-03-20T00:00:00+00:00https://dongryul-kim.github.io/blog/dwyer-kan/Classically, the localization \( \mathcal{C} [\mathcal{W}^{-1}] \) of a (small) category \( \mathcal{C} \) at a subcategory \( \mathcal{W} \) is given by:
the objects of \( \mathcal{C}[\mathcal{W}]^{-1} \) are objects of \( \mathcal{C} \),
the morphisms \( \mathcal{C}[\mathcal{W}^{-1}] (X, Y) \) ar…Faisceaux Algébriques Cohérents 5 – Serre duality2018-01-14T00:00:00+00:002018-01-14T00:00:00+00:00https://dongryul-kim.github.io/blog/fac-5/Earlier, we have computed the dimension of \( H^i(\mathbb{P} _ k^n, \mathscr{O} _ X(m)) \) for all \( i \) and \( m \). It turned out that it is \( 0 \) for all \( i \neq 0, n \), and at \( i = 0, n \), we had
\[ \displaystyle \dim _ k H^0(\mathbb{P} _ k^n, \mathscr{O} _ X(m)) \cong \dim _ k H^n(\ma…Faisceaux Algébriques Cohérents 4 – Coherent sheaves2018-01-13T00:00:00+00:002018-01-13T00:00:00+00:00https://dongryul-kim.github.io/blog/fac-4/It’s quite awkward to be introducing coherent sheaves at this point, when the title of the paper is "Coherent algebraic sheaves". So far, we’ve mostly gotten away with quasi-coherent sheaves. Serre introduces the notion of coherent sheaves, which contain some idea of finite generation in addition to…Faisceaux Algébriques Cohérents 3 – coherent sheaves on projective space2018-01-12T00:00:00+00:002018-01-12T00:00:00+00:00https://dongryul-kim.github.io/blog/fac-3/As we know, cohomology of sheaves on affine schemes is boring. The next simplest case we can consider is sheaves on projective schemes.
7. The \( \operatorname{Proj} \) construction
We can define projective space by gluing affine space together, but there is a more canonical construction. By a grad…Faisceaux Algébriques Cohérents 2 – Čech cohomology2018-01-11T00:00:00+00:002018-01-11T00:00:00+00:00https://dongryul-kim.github.io/blog/fac-2/In the last post, we have defined sheaf cohomology as derived functors of the global sections. We showed that if \( X \) is a Noetherian scheme, it doesn’t matter if we derive from the category of \( \mathscr{O} _ X \)-modules or from quasi-coherent \( \mathscr{O} _ X \)-modules. This allowed us to …Faisceaux Algébriques Cohérents 1 – sheaf cohomology2018-01-09T00:00:00+00:002018-01-09T00:00:00+00:00https://dongryul-kim.github.io/blog/fac-1/Recently I’ve been reading Serre’s Faisceaux Algébriques Cohérents [Ser55]. The main point of the article is to develop the theory of sheaf cohomology and prove some properties of cohomology of coherent sheaves. Serre also computes the cohomology some of coherent sheaves on projective space. I don’t…Witt vectors2017-12-07T00:00:00+00:002017-12-07T00:00:00+00:00https://dongryul-kim.github.io/blog/witt-vectors/The ring of Witt vectors is a functor that takes in a ring \( R \) and constructs another ring \( W _ p(R) \). If we take a finite field \( \mathbb{F} _ q \) with \( q = p^n \), it outputs a ring \( W _ p(\mathbb{F} _ q) = \mathcal{O} _ K \) where \( K \) is the unique unramified extension of \( \ma…Torus actions on symplectic manifolds I2017-08-18T00:00:00+00:002017-08-18T00:00:00+00:00https://dongryul-kim.github.io/blog/toric-manifold-1/I’ve recently been reading Torus actions on symplectic manifolds [Aud04] by Michèle Audin. I’d like to write down some of the facts I learned from this book.
The main object the book covers is a symplectic manifold with a torus action. Why look at torus actions, not Lie group actions? This is not a …The CGWH category2017-07-31T00:00:00+00:002025-05-19T00:00:00+00:00https://dongryul-kim.github.io/blog/cgwh-spaces/In algebraic topology, people work with topological spaces. This entails doing a lot of operations with topological spaces, i.e., taking products, quotients, pushout, limits, colimits, exponentials, etc. But the category \( \mathsf{Top} \) of topological spaces is not so good to work with. For examp…A Ramsey game2017-07-14T00:00:00+00:002017-07-14T00:00:00+00:00https://dongryul-kim.github.io/blog/ramsey-game/Here is an interesting idea I had while at a summer school in Daejeon. This is possibly well-known, since a lot of people seem to be interested in Ramsey numbers.
1. The Ramsey game
Let \( k \) and \( l \) be positive integers, and consider the following game.
\( \mathsf{RG}[n,r,s] \): There is gi…What on earth is... topological K-theory?2017-03-21T00:00:00+00:002017-03-21T00:00:00+00:00https://dongryul-kim.github.io/blog/topological-k-theory/For some reasons, people are interested in studying vector bundles over spaces. At first consideration, it might seem as if there cannot be anything particularly interesting about vector bundles, since they are a special instance of fiber bundles. But the set of vector bundles have an additive struc…Squares on curves2017-01-19T00:00:00+00:002017-01-19T00:00:00+00:00https://dongryul-kim.github.io/blog/squares-on-curves/There is a surprisingly simple, hundred-years-old conjecture in geometry, called the square peg problem.
Conjecture (Toeplitz, 1911).
Every Jordan curve in \( \mathbb{R}^2 \) has an inscribed square, i.e., a square that has vertices on the curve.
In a sense, this problem is in the spirit of the Bor…The Lefschetz fixed-point formula2016-12-18T00:00:00+00:002016-12-18T00:00:00+00:00https://dongryul-kim.github.io/blog/lefschetz-formula/1. Introduction
In 1926, Solomon Lefschetz Lef26 gave a formula that relates the number of fixed points of a map to the induced maps on homology.
Definition.
Suppose \( X \) is a space such that all \( H _ k(X) \) are finitely generated abelian groups, and \( H _ k(X) = 0 \) for all sufficiently la…The Banach–Tarski paradox2016-11-17T00:00:00+00:002016-11-17T00:00:00+00:00https://dongryul-kim.github.io/blog/banach-tarski/Recently, in a set theory class, I learned the proof of the Banach-Tarski paradox. Contrary to what I expected, the construction of the decomposition is quite clean. I am going to try and be as concise as possible in this post, so excuse me for being a bit hand-wavy in a way that can be easily forma…Base-free interpretation of the fundamental group via groupoids2015-12-12T00:00:00+00:002015-12-12T00:00:00+00:00https://dongryul-kim.github.io/blog/fundamental-groupoid/The very first algebraic invariant one learns in topology is probably the fundamental group of a space. But it always bothers me that the construction of the fundamental group requires the choice of a base point. Yes, of course there is an isomorphism \(\pi _ 1(X, x _ 0) \simeq \pi _ 1(X, x _ 1)\) f…