We introduce the notion of a Nakajima bundle representation. Given a labelled quiver and a variety or manifold X, such a representation involves an assignment of a complex vector bundle on X to each node of the double quiver; to the edges, we assign sections of, and connections on, associated twisted bundles. We for the most part restrict attention in our development to algebraic curves or Riemann surfaces. Our construction simultaneously generalizes ordinary Nakajima quiver representations on the one hand and quiver bundles on the other hand. These representations admit gauge-theoretic characterizations, analogous to the Atiyah–Drinfel’d–Hitchin–Manin equations in the original work of Nakajima, allowing for the construction of these generalized quiver varieties using a reduction procedure with moment maps. We study the deformation theory of Nakajima bundle representations, prove a Hitchin–Kobayashi correspondence between such representations and stable quiver bundles, examine the natural torus action on the resulting moduli varieties, and comment on scenarios where the variety is hyperkähler. Finally, we produce concrete examples that recover known moduli spaces.

We construct a flat degeneration of the derived moduli stack of Higgs bundles on smooth projective curves, using Jun Li’s stack of bounded expanded degenerations. The degeneration carries a natural relative zero-shifted logarithmic symplectic form over the base, which extends the classical Hitchin symplectic structure from the generic fibre. The induced Hitchin map is shown to be complete (in the valuative sense) and flat, without requiring coprimeness of rank and degree. Finally, we extend the construction globally: the derived moduli stack of Higgs bundles over the universal semistable curve of genus greater than or equal to two carries a relative zero-shifted log-symplectic form over the moduli stack of stable curves.

Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov–Huh. We characterize all choices of minimal kinematics on the moduli space $\mathcal{M}_{0,n}$. These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet–Tevelev.

As is well-known, the conformal class of a surface M with boundary is determined by its Diriclet-to-Neumann (DN) map $\Lambda $. We propose an algorithm for determination of the b-period matrix $\mathbb {B}$ of the (Schottky) double of M via $\Lambda $.

We describe a geometric, stable pair compactification of the moduli space of Enriques surfaces with a numerical polarization of degree $2$, and identify it with a semitoroidal compactification of the period space.

This paper contains a method to prove the existence of smooth curves in positive characteristic whose Jacobians have unusual Newton polygons. Using this method, I give a new proof that there exist supersingular curves of genus $4$ in every prime characteristic. More generally, the main result of the paper is that, for every $g \geq 4$ and prime p, every Newton polygon whose p-rank is at least $g-4$ occurs for a smooth curve of genus g in characteristic p. In addition, this method resolves some cases of Oort’s conjecture about Newton polygons of curves.

We classify quasidiagonals of the $SL(2, R)$ action on products of strata or hyperelliptic loci. We use the technique of diamonds developed by Apisa and Wright in order to use induction on this problem.

For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb {Q}_p$ we give a geometrisation of the $p$-adic Simpson correspondence of rank one in terms of analytic moduli spaces: the $p$-adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$-line bundles. As an application, we study a major open question in $p$-adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the $p$-adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters $\pi ^{{\mathrm {\acute {e}t}}}_1(X)\to K^\times$ in terms of moduli spaces: for projective $X$ over $K=\mathbb {C}_p$, it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group $\operatorname {Pic}(X)$.

In this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative Korteweg-de Vries hierarchy, which we introduced in a previous work. The specialisation of this conjecture to the top degree part of Pixton’s class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we prove that this conjecture follows from the double ramification/Dubrovin–Zhang equivalence conjecture. We also provide several independent computational checks in support of it.

We study the $E_2$-algebra $\Lambda \mathfrak {M}_{*,1}:= \coprod _{g\geqslant 0}\Lambda \mathfrak {M}_{g,1}$ consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion $\Omega B\Lambda \mathfrak {M}_{*,1}$: it is the product of $\Omega ^{\infty }\mathbf {MTSO}(2)$ with a certain free $\Omega ^{\infty }$-space depending on the family of all boundary-irreducible mapping classes in all mapping class groups $\Gamma _{g,n}$ with $g\geqslant 0$ and $n\geqslant 1$.

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus 2}$ lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of $\mathbb {P}^{1}\times \mathbb {P}^{1}$ branched along a specific $(4,\,4)$ curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$ with the symmetric linearization.

We prove a comparison isomorphism between certain moduli spaces of $p$-divisible groups and strict ${\mathcal{O}}_{K}$-modules (RZ-spaces). Both moduli problems are of PEL-type (polarization, endomorphism, level structure) and the difficulty lies in relating polarized $p$-divisible groups and polarized strict ${\mathcal{O}}_{K}$-modules. We use the theory of relative displays and frames, as developed by Ahsendorf, Lau and Zink, to translate this into a problem in linear algebra. As an application of these results, we verify new cases of the arithmetic fundamental lemma (AFL) of Wei Zhang: The comparison isomorphism yields an explicit description of certain cycles that play a role in the AFL. This allows, under certain conditions, to reduce the AFL identity in question to an AFL identity in lower dimension.

Let $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extensions of) the Laplacians on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4\unicode[STIX]{x1D70B}$.

We prove that one-parameter families of real germs of conformal diffeomorphisms tangent to the involution x ↦−x are rigid in the parameter. We establish a connection between the dynamics in the Poincaré and Siegel domains. Although repeatedly employed in the literature, the dynamics in the Siegel domain does not explain the intrinsic real properties of these germs. Rather, these properties are fully elucidated in the Poincaré domain, where the fixed points are linearizable. However, a detailed study of the dynamics in the Siegel domain is of crucial importance. We relate both points of view on the intersection of the Siegel normalization domains.

We prove that the moduli space of smooth primitively polarized $\text{K3}$ surfaces of genus 33 is unirational.

We show that the Craighero–Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero–Gattazzo surface is the only explicit example of a smooth simply connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod $p$ technique and deformation theory.