In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface $\unicode[STIX]{x1D6F4}$ via the topological Fukaya category. We prove that the topological Fukaya category of $\unicode[STIX]{x1D6F4}$ is equivalent to the category of matrix factorizations of a certain mirror LG model $(X,W)$. Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which are of independent interest.

We establish the geometric Langlands correspondence for rank-one groups over the projective line with three points of tame ramification.

We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

We resolve parts (A) and (B) of Problem 1.100 from Kirby’s list [Problems in low-dimensional topology, in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 35–473] by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.

A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the stable category, for each homogeneous prime ideal $\mathfrak{p}$ in the cohomology ring of the group scheme.

We generalise Simpson’s nonabelian Hodge correspondence to the context of projective varieties with Kawamata log terminal (klt) singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety $X$ and any resolution of singularities, any vector bundle on the resolution that appears to come from $X$ numerically, does indeed come from $X$. Furthermore, and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established.

We exhibit invariants of smooth projective algebraic varieties with integer values, whose nonvanishing modulo $p$ prevents the existence of an action without fixed points of certain finite $p$-groups. The case of base fields of characteristic $p$ is included. Counterexamples are systematically provided to test the sharpness of our results.

The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser–Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco–Shalen–Johannson decomposition of $M$.

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

Let $p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4$ be a prime number. We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i) $16$-rank of the class group $\text{Cl}(-4p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-4p})$; (ii) $8$-rank of the ordinary class group $\text{Cl}(8p)$ of the real quadratic field $\mathbb{Q}(\sqrt{8p})$; (iii) the solvability of the negative Pell equation $x^{2}-2py^{2}=-1$ over the integers; (iv) $2$-part of the Tate–Šafarevič group $\unicode[STIX]{x0428}(E_{p})$ of the congruent number elliptic curve $E_{p}:y^{2}=x^{3}-p^{2}x$. Our results are conditional on a standard conjecture about short character sums.

If $K$ is a simplicial complex on $m$ vertices, the flagification of $K$ is the minimal flag complex $K^{f}$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\stackrel{}{\longrightarrow }K\stackrel{}{\longrightarrow }K^{f}$. This induces a sequence of maps of polyhedral products $(\text{}\underline{X},\text{}\underline{A})^{L}\stackrel{g}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K}\stackrel{f}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K^{f}}$. We show that $\unicode[STIX]{x1D6FA}f$ and $\unicode[STIX]{x1D6FA}f\circ \unicode[STIX]{x1D6FA}g$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\text{}\underline{CY},\text{}\underline{Y})^{K}$ is a co-$H$-space if and only if the 1-skeleton of $K$ is a chordal graph, and we deduce that the maps $f$ and $f\circ g$ have right homotopy inverses in this case.

We establish a tilting equivalence for rational, homotopy-invariant cohomology theories defined over non-archimedean analytic varieties. More precisely, we prove an equivalence between the categories of motives of rigid analytic varieties over a perfectoid field $K$ of mixed characteristic and over the associated (tilted) perfectoid field $K^{\flat }$ of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of $K$ and $K^{\flat }$ are isomorphic.

We introduce a new method for computing triply graded link homology, which is particularly well adapted to torus links. Our main application is to the $(n,n)$-torus links, for which we give an exact answer for all $n$. In several cases, our computations verify conjectures of Gorsky et al. relating homology of torus links with Hilbert schemes.