We relate two classical dualities in low-dimensional quantum field theory: Kramers–Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/’t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to nonabelian groups; to finite, semisimple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.

Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov and Rozansky.

We prove that $L$–space knots do not have essential Conway spheres with the technology of peculiar modules, a Floer-theoretic invariant for tangles.

Let Sol be the $3$–dimensional solvable Lie group whose underlying space is ${\mathit{R}}^3$ and whose left-invariant Riemannian metric is given by

$$e^{-2z}d{x}^2+e^{2z}d{y}^2+d{z}^2.$$

Let $E:{\mathit{R}}^3\to \text{Sol}$ be the Riemannian exponential map. Given $V=(x,y,z)\in {\mathit{R}}^3$, let ${\gamma }_V=\{E(tV)\mid t\in [0,1]\}$ be the corresponding geodesic segment. Let AGM stand for the arithmetic–geometric mean. We prove that ${\gamma }_V$ is a distance-minimizing segment in Sol if and only if $$\text{AGM}\left(\sqrt{\left|xy\right|},\frac{1}{2}\sqrt{{(|x|+|y|)}^2+z^2}\right)\le \pi .$$

We use this inequality to precisely characterize the cut locus in Sol, prove that the metric spheres in Sol are topological spheres, and almost exactly characterize their singular sets.

We study log Calabi–Yau varieties obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary. Mirrors to such varieties are constructed by Gross and Siebert from a canonical scattering diagram built by using punctured Gromov–Witten invariants of Abramovich, Chen, Gross and Siebert. We show that there is a piecewise-linear isomorphism between the canonical scattering diagram and a scattering diagram defined algorithmically, following a higher-dimensional generalization of the Kontsevich–Soibelman construction. We deduce that the punctured Gromov–Witten invariants of the log Calabi–Yau variety can be captured from this algorithmic construction. This generalizes previous results of Gross, Pandharipande and Siebert on “the tropical vertex” to higher dimensions. As a particular example, we compute these invariants for a nontoric blow-up of the three-dimensional projective space along two lines.

We show that if $Y$ is the boundary of an almost-rational plumbing, then the framed instanton Floer homology $I^{\mathrm{\#}}(Y)$ is isomorphic to the Heegaard Floer homology $\widehat{HF}(Y;ℂ)$. This class of $3$–manifolds includes all Seifert fibered rational homology spheres with base orbifold $S^2$ (we establish the isomorphism for the remaining Seifert fibered rational homology spheres — with base ${ℝ\mathbb{P}}^2$ — directly). Our proof utilizes lattice homology, and relies on a decomposition theorem for instanton Floer cobordism maps recently established by Baldwin and Sivek.

We study unimodular measures on the space ${\mathcal{ℳ}}^d$ of all pointed Riemannian $d$–manifolds. Examples can be constructed from finite-volume manifolds, from measured foliations with Riemannian leaves, and from invariant random subgroups of Lie groups. Unimodularity is preserved under weak${}^{\ast }$ limits, and under certain geometric constraints (eg bounded geometry) unimodular measures can be used to compactify sets of finite-volume manifolds. One can then understand the geometry of manifolds $M$ with large, finite volume by passing to unimodular limits.

We develop a structure theory for unimodular measures on ${\mathcal{ℳ}}^d$, characterizing them via invariance under a certain geodesic flow, and showing that they correspond to transverse measures on a foliated “desingularization” of ${\mathcal{ℳ}}^d$. We also give a geometric proof of a compactness theorem for unimodular measures on the space of pointed manifolds with pinched negative curvature, and characterize unimodular measures supported on hyperbolic $3$–manifolds with finitely generated fundamental group.