Sam Auyeung

Sam Auyeung

Blog Posts - 2026

2026

2-Dimensional Tilings and Curvature

8 minute read

Published:

In Euclidean geometry, if we want to tile the plane with regular polygons of the same size, we can ask which pairs $(p,q)$ are admissible where $p$ is the number of sides of the regular polygon and $q$ is the number of them that would share the same vertex. The sum of the interior angles of a polygon with $p$ sides is $(p-2)\pi$ and if it’s regular, then each angle is $(p-2)\pi/p$. On the other hand, if we have $q$ of them sharing a vertex, to get total coverage, we need $\dfrac{q(p-2)\pi}{p} = 2\pi$ which is equivalent to $q(p-2)=2p$. Rearranging and adding $4$ to both sides gives $pq -2q-2p+4=4$ which then lets us factor the left: $(p-2)(q-2)=4$. The only solutions where $p,q>0$ to this are $(3,6),(4,4),(6,3)$. We can ask the same question but for other 2D geometries such as spherical or hyperbolic geometry. On the sphere, the sum of angles of a triangle is greater than $\pi$ and so the relationship is $(p-2)(q-2)<4$ whereas on the Poincaré hyperbolic disk, it is $(p-2)(q-2)>4$.

Single Variable Complex Analysis: Riemann Surfaces and Automorphisms

30 minute read

Published:

Much of this note is based on John Milnor’s Dymamics in One Complex Variable. Let’s begin by reviewing some single variable complex analysis on $\mathbb{C}$. A map from $f:\mathbb{C} \to \mathbb{C}$ is complex differentiable if $f$ is a differentiable map when considered as a real map $f:\mathbb{R}^2 \to \mathbb{R}^2$ and moreover, its linearization at each point commutes with the action of multiplication by $i$ which, as a real map, can be written as the matrix $\begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix}$. From this, one derives the Caucy-Riemann equations. Another term used to describe such maps is holomorphic. One can also prove that when using the usual dot product of $\mathbb{R}^2$ to define lengths and angles, a holomorphic map $f$ is conformal whenever $f’(z) \neq 0$; i.e. it preserves angles. The proof is simply to show $\dfrac{\langle df_z(v),df_z(w) \rangle}{|df_z(v)||df_z(w)|}=\dfrac{\langle v,w \rangle}{|v||w|}=:\cos \theta$.

Electromagnetism from a Gauge Theory Perspective

20 minute read

Published:

Let $M:=(\mathbb{R}^4,g)$ be the usual Minkowski space where the Lorentzian metric $g=-dt^2 + dx^2 + dy^2+dz^2$ has signature $(-+++)$ with a special time direction. Being a contractible space, there is only one principal $U(1)$-bundle (up to isomorphism) on $M$ and we may equip the bundle with a $U(1)$ connection $A$ which in physical terms, is a potential. The curvature is $F = dA + \dfrac{1}{2}[A,A]$ where $[A,A]$ is the Lie bracket but since $U(1)$ is abelian, this part is 0 and we only have the usual exterior derivative (but it’s hopefully clear we can generalize to other gauge theories with a choice of Lie group $G$). This 2-form $F$ is called the Faraday 2-form and we should think of it as being the fundamental object from which we’ll observe electric and magnetic fields. We’ll also need to discuss some structure coming from the metric. The metric can be extended to other tensors and of interest, to differential forms. Let $\eta = dt\wedge dx\wedge dy\wedge dz$ be the volume form. The Hodge star operator on differential forms is written as $*\beta$ where $\beta$ is any differential $k$-form and $*\beta$ is a $(n-k)$-form, $n$ being the dimension of the manifold. Let $\alpha,\beta$ be two $k$-forms. This operator is defined by $\alpha \wedge *\beta = g(\alpha,\beta) \eta$. Since $g$ is symmetric, then this also equals $\beta \wedge *\alpha$. For example, $*dt$ is defined by $dt \wedge *dt = g(dt,dt)\eta = -\eta$. So $*dt = -dx\wedge dy \wedge dz$.

$M\text{String}$, Topological Modular Forms, and the Witten Genus

12 minute read

Published:

This is the third in a series about maps from some kind of Thom spectrum to a K-theory of vector bundles. However, we depart from the target being K-theory to replacing it with a different spectrum which is still periodic. Let’s begin with defining a string manifold. Suppose $M$ is a spin $n$-manifold; i.e. $w_2(M)=0$ and we have chosen a choice of lift of the orthonormal frame bundle which is a spin bundle. Then in this case, the 1st Pontryagin class is canonically twice some other class which we denote $\frac{1}{2}p_1(M)$. It’s possible this class is 0 or that it is 2-torsion and so $p_1(M)=0$. In any case, a string manifold has a lift to the $\text{String}(n)$ group which trivializes this class. There are several ways to think of this group. Topologically, it fits into a Whitehead tower $…\to \text{Fivebrane}(n) \to \text{String}(n) \to \text{Spin}(n) \to SO(n) \to O(n)$ where as we move left, the groups become increasingly connected.

Complex Cobordism, Landweber Exactness, and the Todd Genus

14 minute read

Published:

In my previous blog, I wrote about the $\widehat{A}$-genus and its refinement $\alpha: \Omega^{\text{Spin}}_* \to KO^{-*}$ which is a graded ring morphism. This map comes from a $\mathbb{E}_\infty$ ring morphism which I’ll also call $\alpha:M\text{Spin} \to KO$ of ring spectra. Since we see real K-theory featured and spin manifolds, we can ask if there’s any ring spectrum map which features complex K-theory and complex manifolds that then gives rise to a map of graded rings and a genus. The short answer is yes, there’s a map from $MU \to KU$ which is complex cobordism to complex K-theory and the genus is the so-called Todd genus $td: \Omega^U_* \to KU^*$. Here, we’re dealing with stably almost complex manifolds. Complex K-theory also exhibits Bott periodicity and so $KU^* \cong \mathbb{Z}[x,x^{-1}]$ where $\deg x=2$. On the other hand, $\Omega^U_* \cong \mathbb{Z}[x_2,x_4,x_6,…,x_{2i},…]$. I’ll try to explain some of the ideas behind this but in an anachronistic way.

Spinors, Dirac Operators, the $\widehat{A}$-Genus, and its Refinement

22 minute read

Published:

Let $M$ be an oriented smooth $n$-manifold; then there is a a coherent way of choosing oriented bases on $T_x M$ for every $x \in M$. If we choose a metric, then we can make these orthonormal bases and hence, we get a principal $SO(n)$-bundle called the orthonormal frame bundle (the isomorphism class is independent of metric). A spin structure is a choice of principal $\text{Spin}(n)$-bundle which is a lift of the orthonormal frame bundle. This structure group can be thought of as the even units in the Clifford algebra $Cl_n$ and provides us with an action. The obstruction to a spin structure is the cohomology class $w_2(M)$ so we say $M$ is spinnable if $w_2(M)=0$. It turns out that spin structures are parametrized by $H^1(M,\mathbb{Z}/2)$.

Brieskorn Spheres

13 minute read

Published:

Let $a = (a_1,…,a_n) \in \mathbb{N}^n$ be an $n$-tuple of positive integers; we typically want $a_k > 1$. Let $V_a = {z_1^{a_1}+…+z_n^{a_n}=0} \subset \mathbb{C}^n$ be the complex algebraic hypersurface with an isolated singularity at the origin and let $M_a = {z_1^{a_1}+…+z_n^{a_n}=\delta}$ be the Milnor fiber which is a smooth manifold for small $\delta>0$. Actually, we’ll also call $M_a$ the Milnor fiber which is what I just wrote but intersected with a small ball $B^{2n}$ centered as 0. We also have $\Sigma_a = S^{2n-1}_\epsilon \cap V_a$ which is called the link of the singularity, obtained by intersecting the singular variety with a small sphere centered at 0. This link is diffeomorphic to the boundary of $M_a$. We will explain some of the ideas for showing how $\Sigma_a$ is a homotopy sphere in many cases and how one finds their oriented diffeotype. We follow this post, these slides, and Brieskorn’s original paper.

The Hopf Degree Theorem and Framed Cobordism

6 minute read

Published:

Let $M,N$ closed topological manifolds of dimension $n$. Then their top integral homology are generated by their respective fundamental classes $[M],[N]$. The degree of a continuous map $f:M \to N$ is defined by what the induced map $f_*:H_n(M,\mathbb{Z}) \to H_n(N,\mathbb{Z})$ does which can only be multiplication by a some integer $k$; $f_*([M])=k[N]$. We’ll denote this as $\deg f = k$.

Noether’s First Theorem

15 minute read

Published:

Let’s begin with an example from electromagnetism; actually, it’s really just a kinematic observation. Say we have a solid region $W \subset \mathbb{R}^3$ with a closed, piecewise smooth boundary $S$. There may be charged particles in $W$ and we note that if $\rho$ is the time-dependent charge density (which has units charge/volume), then the total charge is $Q(t) = \iiint_W \rho \, dV$. If we take a time derivative, we’d have $\frac{dQ}{dt} = \iiint_W \partial_t \rho\, dV$. Now, if the charges inside $W$ were to teleport around, the total charge would be unaffected but it’s unclassical for particles to teleport around instantly; i.e. we assume locality. So we’d like to stipulate that the only way for the total charge to change is if charged particles enter/leave through $S$ in a continuous way. So we’d have moving charges which create current $I$. Set $\frac{dQ}{dt} = -I$; the minus sign is just to align with the outward pointing orientation $\hat{n}$ of $S$; so if charges leave, then the total charge is decreasing and we’ll say the current is positive in the $\hat{n}$ direction.

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