We study topological aspects of supersolvable abelian arrangements, toric arrangements in particular. The complement of such an arrangement sits atop a tower of fiber bundles, and we investigate the relationship between these bundles and bundles involving classical configuration spaces. In the toric case, we show that the monodromy of a supersolvable arrangement bundle factors through the Artin braid group, and that of a strictly supersolvable arrangement bundle factors further through the Artin pure braid group. The latter factorization is particularly informative – we use it to determine a number of invariants of the complement of a strictly supersolvable arrangement, including the cohomology ring and the lower central series Lie algebra of the fundamental group.

We prove the $S=T$ conjecture proposed by Xiao–Zhu in [XZ17], making use of Scholze’s theory of diamonds and v-stacks and Fargues–Scholze’s geometric Satake equivalence. Following [XZ18], we deduce the Eichler–Shimura relation for Shimura varieties of Hodge type.

We use the Selberg zeta function to analyze the limiting behavior of resonances in a degenerating family of Kleinian Schottky groups. We prove that, after a suitable rescaling, the Selberg zeta functions converge to the Ihara zeta function of a limiting finite graph associated with the relevant non-Archimedean Schottky group acting on the Berkovich projective line.

To reveal more interesting and intricate structures of resonances, we introduce intermediate zeta functions that encapsulate both Archimedean and non-Archimedean information – whereas the classical Selberg and Ihara zeta functions reflect only Archimedean and non-Archimedean properties, respectively. As an application of our techniques, we derive an exponential error term in a result of McMullen (recently extended by Dang–Mehmeti) concerning the asymptotic vanishing rate of the Hausdorff dimension of limit sets of certain degenerating Schottky groups generating symmetric three-funnel surfaces.

We prove a conjecture due to Sturmfels and Uhler concerning the degree of the projective variety associated to the Gaussian graphical model of the cycle. We involve new methods based on the intersection theory on the space of complete quadrics.

Let ${\hat {R}}_{}$ be the I-adic completion of a commutative ring R with respect to a finitely generated ideal I. We give a necessary and sufficient criterion for the category of perfect complexes over ${\hat {R}}_{}$ to be equivalent to the subcategory of dualizable objects in the derived category of I-complete complexes of R-modules. Our criterion is always satisfied when R is noetherian. When specialized to R local and noetherian and to I the maximal ideal, our theorem recovers a recent result of Benson, Iyengar, Krause, and Pevtsova.

We explore connections between existence of $\Bbbk $-rational points for Fano varieties defined over $\Bbbk $, a subfield of $\mathbb {C}$, and existence of Kähler-Einstein metrics on their geometric models. First, we show that geometric models of del Pezzo surfaces with at worst quotient singularities defined over $\Bbbk \subset \mathbb {C}$ admit (orbifold) Kähler–Einstein metrics if they do not have $\Bbbk $-rational points. Then we prove the same result for smooth Fano 3-folds with 8 exceptions. Consequently, we explicitly describe several families of pointless Fano 3-folds whose geometric models admit Kähler-Einstein metrics. In particular, we obtain new examples of prime Fano 3-folds of genus $12$ that admit Kähler–Einstein metrics. Our result can also be used to prove existence of rational points for certain Fano varieties, for example for any smooth Fano 3-fold over $\Bbbk \subset \mathbb {C}$ whose geometric model is strictly K-semistable.

We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. In particular, we generalize and prove a conjecture of C. M. Ringel on cluster-additive functions associated to arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of positive spaces of finite type, we use tropical friezes and cluster-additive functions to explicitly express the Fock-Goncharov pairing between their tropical points and the bijections between global monomials on one and tropical points of the other.

We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset V of affine or projective space. The bounds depend only on $\mathrm {card}(\mathcal P)$ and the degrees of the polynomials in $\mathcal P$, together with $\deg (V)$ and $\dim (V)$, and not on the dimension of the space in which V is embedded. This feature is particularly useful when V has small intrinsic dimension but is presented in a very high-dimensional ambient space.

We describe several applications. First, we extend existing results on bounding the $\varepsilon $-entropy of real algebraic varieties. Second, we derive lower bounds (in terms of the number of connected components) for membership testing in semi-algebraic sets in the algebraic computation tree model. Finally, motivated by quantum complexity theory, we introduce additive and multiplicative notions of relative rank in finite-dimensional vector spaces and algebras with respect to a fixed algebraic subset, generalizing the classical notion of tensor rank. We prove a general lower bound on the maximum relative rank of finite subsets with respect to algebraic sets of bounded degree and dimension that is again independent of the ambient dimension. As an illustration, we obtain a quantum analog of Shannon’s classical lower bound. We prove that if $\Delta _n$ is an algebraic family of allowed gates with

$$ \begin{align*}\dim \Delta_n= \mathrm{poly}(n), \qquad \log\deg\Delta_n= \text{ poly}(n),\end{align*} $$

then some Boolean functions require $2^n/\text {poly}(n)$ gates from $\Delta _n$, even when individual gates in $\Delta _n$ may be highly nonlocal.

Let k be a field endowed with a complex embedding, and let$\mathcal {G}_{\mathrm {mot}}$ be the associated motivic Galois group. The goal of this paper is to show that$\mathcal {G}_{\mathrm {mot}}$ admits a natural manifestation in anabelian geometry. Roughly speaking, we prove that$\mathcal {G}_{\mathrm {mot}}$ coincides with the automorphism group of the functor$X\mapsto \pi _1^{\mathrm {geo}}(X)$ sending a k-variety X to the geometric completion of its topological fundamental group. This can be considered as a motivic version of the Ihara–Matsumoto–Oda Conjecture, greatly generalised and proven by Pop. In a sequel to this paper, we will develop the parallel story for the$\ell $-adic realisation. For this reason, many of the intermediate results are developed at a greater level of generality than strictly necessary for the main results proven here.

We construct irrational irreducible components of the Hilbert scheme of points of affine space $\mathbb {A}^n$ for $n \geq 12$. We start with irrational components of the Hilbert scheme of curves in $\mathbb {CP}^3$ and use methods developed by Jelisiejew to relate these to irreducible components of the Hilbert schemes of points of $\mathbb {A}^n$. The result solves Problem XX of the Hilbert scheme of points problem list [J24].

We investigate local entropy theory, particularly the property of having completely positive entropy (CPE), from a descriptive set-theoretic point of view. We aim to determine the descriptive complexity of different families of dynamical systems with CPE. For a large class of compact metrizable spaces (including manifolds and topological graphs), we show that the family of dynamical systems with CPE is complete coanalytic and hence not Borel. When we restrict our attention to dynamical systems with additional properties, we obtain contrasting behavior. Specifically, CPE and uniformly positive entropy (UPE) – the latter being a Borel property – coincide for mixing maps on topological graphs. However, the family of mixing maps with CPE on Cantor spaces is coanalytic and not Borel. For dynamical systems with the shadowing property, CPE and UPE coincide regardless of the phase space.

Tight $2 s$-designs are the $2 s$-$(v, k, \lambda )$ designs whose sizes achieve the Fisher type lower bound ${v \choose s}$. Symmetric $2$-designs, the Witt $4$-$(23, 7, 1)$ design and the Witt $4$-$(23, 16, 52)$ design are tight designs. It has been widely conjectured since the 1970s that there are no other nontrivial tight designs. In this paper, we give a proof of this conjecture. In the proof, an upper bound $v \ll s$ is shown by analyzing the parameters of the designs and the coefficients of the Wilson polynomials, and a lower bound $v \gg s (\ln s)^2$ is shown by using estimates on prime gaps.

In this paper, we prove that a compact Kähler manifold X with semi-positive holomorphic sectional curvature admits a locally trivial fibration $\phi \colon X \to Y$, where the fiber F is a rationally connected projective manifold and the base Y is a finite étale quotient of a (compact complex) torus. This result extends the structure theorem, previously established for projective manifolds, to compact Kähler manifolds. A key part of the proof involves analyzing the foliation generated by truly flat tangent vectors and showing the abelianness of the topological fundamental group $\pi _{1}(X)$, with a focus on varieties of special type.

We study the sublevel sets of the Willmore energy on the space of smoothly immersed $ 2 $-spheres in Euclidean $ 3 $-space. We show that the subset of immersions with energy at most $ 12\pi $ consists of four regular homotopy classes. Moreover, we show that, in certain regular homotopy classes, all singularities of the Willmore flow are avoidable – that is, the initial surface admits a regular homotopy to a round sphere whose Willmore energy does not exceed that of the initial surface. This yields a classification of initial surfaces with energy at most $ 12\pi $ that lead to unavoidable singularities. As a further consequence, we obtain an extension of the Li–Yau inequality at $ 12\pi $ for a large class of immersed spheres without triple points. To prove these results, we glue together different instances of the Willmore flow and employ an invariant for triple-point-free immersed spheres.

We classify the complementary vectors of doubly Cohen–Macaulay complexes. This proves a conjecture of Swartz, negatively answers a question of Athanasiadis and Tzanaki, and gives new bounds on the number of independent sets in a matroid. Our technique works more generally for certain level quotients of Stanley–Reisner rings, giving new bounds on the face numbers of Buchsbaum* complexes.

We propose a definition of expander representations of quivers, generalizing dimension (or linear-algebra) expanders, as a qualitative refinement of slope stability. We prove existence of uniform expander representations for any wild quiver over an algebraically closed base field, using the concept of general subrepresentations and spectral properties of Cartan matrices.

For a finite group A with normal subgroup G, a subgroup U of G is an A-prime-power-covering subgroup if U meets every A-conjugacy-class of elements of G of prime-power order. It is conjectured that $|G:U|$ is bounded by some function of $|A:G|$. We prove the conjecture in the case that the action of G on the set of right cosets of U in G is innately transitive. This includes the special case where U is a maximal subgroup of G. It turns out that each prime-power-covering subgroup U for which this G-action is innately transitive must be a maximal subgroup of G. We derive the following number-theoretic consequence: if K is a finite Galois extension of a global field k and L is a finite innate extension of K such that K and L are weakly Kronecker equivalent over k, then the extension $L/K$ must be atomic of degree bounded in terms of $[K:k]$. Our proofs use both a recently proved bound on the order of a finite nonabelian simple group in terms of the number of its classes of elements of prime-power order (depending on the Finite Simple Group Classification) and also new Galois-theoretic concepts for field extensions.

We prove that the Grothendieck ring of the monoidal category of finitely generated graded projective modules for the nil-Brauer category is isomorphic to an integral form of the split iquantum group of rank one. Under this isomorphism, the indecomposable graded projective modules correspond to the icanonical basis. We also derive character formulae for irreducible graded modules and deduce various branching rules.

For a morphism $f : X \to Y$ of schemes, we give a tropical criterion for which points of Y (valued in a field, discrete valuation ring, number ring, or Dedekind domain) lift to X. Our criterion extends the firmaments of Abramovich [Abr09] to a wide range of morphisms, even logarithmic stable maps.