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 example, we would want to construct the space of maps \( \mathsf{Top}(X, Y) \) and give a nice topology, so that the natural adjunction \( \displaystyle \mathsf{Top}(X, \mathsf{Top}(Y, Z)) \cong \mathsf{To…
July 30, 2017Mathematics20 minutes
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 given a graph with \( n \) vertices and no edges. Paul and Carole play a game with this graph. In each round, Paul first picks an independent set \( S \) with \( r \) vertices, and then Carole draws so…
July 13, 2017Mathematics5 minutes
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 structure, given by the direct sum, and also a multiplicative structure, given by the tensor product. Indeed, to each compact Hausdorff space \( X \) can be associated a ring \( K(X) \) coming from the vec…
March 21, 2017Mathematics25 minutes
There is a surprisingly simple, hundred-years-old conjecture in geometry, called the square peg problem. Conjecture 1 (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 Borsuk–Ulam theorem; given an arbitrary map, we want to find something satisfying some given condition. But the situation is not that simple, and I would like to talk about why this problem is hard, un…
January 18, 2017Mathematics8 minutes
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 1. Suppose \( X \) is a space such that all \( H _ k(X) \) are finitely generated abelian groups, and \( H _ k(X) = 0 \) for all sufficiently large \( X \). For a continuous map \( f \), define its Lefschetz number as \( \displaystyle \begin{aligned} \Lambda(f) = \sum _ {k \ge 0}^{} (-1)^k \mathrm{Tr}( f _ {\ast k} : H _ k(X) \rightarrow H…
December 17, 2016Mathematics47 minutes
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 formalized. Definition 1. For subset \( A, B \subseteq \mathbb{R}^3 \), we write \( A \cong B \) if \( A \) can decomposed into finitely many pieces and rearranged to make \( B \). Formally, \( A \cong B \…
November 16, 2016Mathematics6 minutes
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 \(\pi _ 1(X, x _ 0) \simeq \pi _ 1(X, x _ 1)\) for any two \(x _ 0, x _ 1 \in X\) given that \(X\) is path connected. But this isomorphism is not canonical; it again involves the choice of a path \(\gamma : [0,1] \rightarrow X\) which starts at \(x…
December 12, 2015Mathematics19 minutes
