My colleagues and I were discussing the following probability brain teaser at lunch: Suppose I have a a fair coin and I use it to write down a 100-character sequence of heads and tails. I don’t show this process to you at all nor tell you any part of the sequence but you are allowed to ask me one yes/no question which I’ll answer truthfully. Then, you will write down your own 100-character sequence with the goal of maximizing the number of matches between our sequences. e.g. if my sequence is $THH…$ and you submit $HTH…$, the first two are not a match but the third is. What yes/no question would you ask and what would you then submit (and why)? What is the expected value of your strategy?
This is a third in a series about direct uses of moduli spaces, in this case, of pseudo-holomorphic curves. The paper in the title is by Tobias Ekholm and Ivan Smith, posted in 2011 and published 2016. We’ll go over some of the main arguments for their theorem but first, I want to outline some classical results.
This is a second post in a series on direct uses of moduli spaces. It is mainly notes on the 2011 paper mentioned in the title by Mohammed Abouzaid. Recall that every smooth manifold $M$ embeds as an exact Lagrangian submanifold into its own cotangent bundle $T^*M$; just use the zero section or the differential of any smooth function. However, an interesting question we may ask is whether a homotopy sphere $\Sigma^n$ embeds as a Lagrangian (may be non-exact) into the cotangent bundle for the standard sphere $T^* S^n$. By the Whitney trick, there is no obstruction to a smooth embedding of exotic $n$-spheres into $T^*S^n$.
This is a follow-up on Donaldson’s work in gauge theory about the failure of the h-bordism theorem for 4-manifolds. The main example utilizes a K3 surface. Since all K3 surfaces are diffeomorphic by Kodaira’s work, we can use any K3 for our purposes. Recall that $(b^+_2,b^-_2) = (3,19)$ since the intersection form is $3H\oplus 2(-E_8)$.
For the month of May, I planned to write a series of posts about some spectacular applications of moduli spaces. Of course, there are many applications of moduli spaces which amount to constructing the space and then integrating in order to produce certain counts or algebro-topological invariants like certain cohomology classes. However, I wanted to write about direct uses of moduli spaces. Alas, I was delayed but here is the first in the series, about Donaldon’s work on instantons and the Diagonalization Theorem which he proved in his thesis and was awarded the Fields Medal. See Freed-Uhlenbeck or Nabler’s survey on the subject. I will also discuss some corollaries such as the existence of exotic $\mathbb{R}^4$s and further aspects of Donaldson theory.
In physics, there are many phenomenon which seem to exist on multiple scales or even every scale. For example, if we look at how electrons align in a ferromagnetic material, at certain critical temperatures, the electrons will align with each other in patches. This means that within a patch of material, all the electrons are basically oriented in the same direction whereas in a neighboring patch, they’ll align in a different direction. You can model this type of behavior using statistical mechanics and surprisingly, the sorts of patterns one sees at these critical temperatures seem to appear at all scales. One can “renormalize” which is sort of like “zooming in and out” and find these phenomena. This type of behavior also happens for phase diagrams. For example, if you follow the boundary between gaseous and liquid water, eventually the boundary disappears and there is no hard line between gas and liquid phase. Instead, you will see patterns that persist even as you renormalize. So despite the fact that individual particles behave randomly, macroscopic behavior emerges.
At the Simons Collaboration 2026 meeting on quantum field theory, Yilin Wang gave a talk with the title above. She focused on the 1+1 and 2+1 dimension cases. Broadly, the holographic principle gives a correspondence between quantum gravity on $M$ and conformal/quantum field theory on $\partial M$. $M$ is typically a conformally compact Einstein manifold.
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 $(p-2)(q-2)>4$.
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 $\dfrac{\langle df_z(v),df_z(w) \rangle}{|df_z(v)||df_z(w)|}=\dfrac{\langle v,w \rangle}{|v||w|}=:\cos \theta$.
A couple of weeks ago, I was in a book club where we read Ryan Kemp’sWhat We Are in the Light. It’s a collection of 16 essays which can mostly be read independently of each other. They are thoughtful, genuine, and challenging. One of them is about technology (another about AI), namely the internet, and how, in Kemp’s estimation, it makes life worse. I didn’t think this was a controversial position but my peers in the book club bristled and closed their minds to it, saying that technology isn’t all bad but it is impractical to cut the internet and other related technologies from our lives without drastic negative effects. I don’t necessarily disagree with the latter two points: there are benefits to technology and the way most societies are structured now, it is difficult to opt out without drastic inconveniences. However, there are inconveniences and then there are harmful effects. I want to at least spell out the issues with technology (as I see it) so that I must acknowledge its drawbacks. Obviously, this will be a different post than my usual.
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 $*dt = -dx\wedge dy \wedge dz$.
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.
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.
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 $H^1(M,\mathbb{Z}/2)$.
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.
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 $\deg f = k$.
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.
