Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular, many of the exceptional Lie incidence geometries occur. In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well-known characterizations.

Ginzburg algebras associated to triangulated surfaces provide a means to categorify the cluster algebras of these surfaces. As shown by Ivan Smith, the finite derived category of such a Ginzburg algebra can be embedded into the Fukaya category of the total space of a Lefschetz fibration over the surface. Inspired by this perspective, we provide a description of the unbounded derived category in terms of a perverse schober. The main novelty is a gluing formalism describing the Ginzburg algebra as a colimit of certain local Ginzburg algebras associated to discs. As a first application, we give a new construction of derived equivalences between these Ginzburg algebras associated to flips of an edge of the triangulation. Finally, we note that the perverse schober as well as the resulting gluing construction can also be defined over the sphere spectrum.

Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative nonassociative algebras and also arise naturally in the context of simple affine group schemes of type $\mathsf {F}_4$ , $\mathsf {E}_6$ , or $\mathsf {E}_7$ . We study these objects over an arbitrary base ring R, with particular attention to the case $R = \mathbb {Z}$ . We prove in this generality results previously in the literature in the special case where R is a field of characteristic different from 2 and 3.

We give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves Hermitian lattices over number fields.

We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_{K}$ for a finite extension $K/\mathbb{Q}_{p}$. This is done by considering a moduli space of Breuil–Kisin modules, satisfying an additional Galois condition, over the unrestricted deformation ring. For $K$ unramified over $\mathbb{Q}_{p}$ and Hodge–Tate weights in $[0,p]$, we study the geometry of this space. As a consequence, we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_{p}$, with Hodge–Tate weights in $[0,p]$, are potentially diagonalizable.

Let $F/\mathbf{Q}$ be a totally real field and $K/F$ a complex multiplication (CM) quadratic extension. Let $f$ be a cuspidal Hilbert modular new form over $F$ . Let ${\it\lambda}$ be a Hecke character over $K$ such that the Rankin–Selberg convolution $f$ with the ${\it\theta}$ -series associated with ${\it\lambda}$ is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin–Selberg $L$ -values $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})$ , as ${\it\chi}$ varies over the class group characters of $K$ . Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters ${\it\chi}$ such that $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})\neq 0$ increases with the absolute value of the discriminant of $K$ . We crucially rely on the André–Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan–Zhang and Andreatta–Goren–Howard–Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin–Selberg $L$ -values. The Waldspurger formula plays an underlying role.

Deninger et Werner ont développé un analogue pour les courbes $p$-adiques de la correspondance classique de Narasimhan et Seshadri entre les fibrés vectoriels stables de degré $0$ et les représentations unitaires du groupe fondamental topologique pour une courbe complexe propre et lisse. Par transport parallèle, ils ont associé fonctoriellement à chaque fibré vectoriel sur une courbe $p$-adique, dont la réduction est fortement semi-stable de degré $0$, une représentation $p$-adique du groupe fondamental de la courbe. Ils se sont posé quelques questions : leur foncteur est-il pleinement fidèle ? La cohomologie des systèmes locaux fournis par celui-ci admet-elle une filtration de Hodge-Tate ? Leur construction est-elle compatible avec la correspondance de Simpson $p$-adique développée par Faltings ? Nous répondons à ces questions dans cet article.

We prove a relative Lefschetz–Verdier theorem for locally acyclic objects over a Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal $2$ -category of cohomological correspondences. We show that local acyclicity is equivalent to dualisability and deduce that duality preserves local acyclicity. As another application of the category of cohomological correspondences, we show that the nearby cycle functor over a Henselian valuation ring preserves duals, generalising a theorem of Gabber.

We give an explicit characterization on the singularities of exceptional pairs in any dimension. In particular, we show that any exceptional Fano surface is $\frac {1}{42}$ -lc. As corollaries, we show that any $\mathbb R$ -complementary surface X has an n-complement for some integer $n\leq 192\cdot 84^{128\cdot 42^5}\approx 10^{10^{10.5}}$ , and Tian’s alpha invariant for any surface is $\leq 3\sqrt {2}\cdot 84^{64\cdot 42^5}\approx 10^{10^{10.2}}$ . Although the latter two values are expected to be far from being optimal, they are the first explicit upper bounds of these two algebraic invariants for surfaces.

We give complexity analysis for the class of short generating functions. Assuming #P$\not \subseteq$FP/poly, we show that this class is not closed under taking many intersections, unions or projections of generating functions, in the sense that these operations can increase the bit length of coefficients of generating functions by a super-polynomial factor. We also prove that truncated theta functions are hard for this class.

We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees and so on. The notion of point degree spectrum creates a connection among various areas of mathematics, including computability theory, descriptive set theory, infinite-dimensional topology and Banach space theory. Through this new connection, for instance, we construct a family of continuum many infinite-dimensional Cantor manifolds with property C whose Borel structures at an arbitrary finite rank are mutually nonisomorphic. This resolves a long-standing question by Jayne and strengthens various theorems in infinite-dimensional topology such as Pol’s solution to Alexandrov’s old problem.

The elliptic algebras in the title are connected graded $\mathbb {C}$-algebras, denoted $Q_{n,k}(E,\tau )$, depending on a pair of relatively prime integers $n>k\ge 1$, an elliptic curve E and a point $\tau \in E$. This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$. When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$, we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$, respectively. When $X_{n/k}=E^g$ and $\tau =0$, the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set $S \subset \mathbb {R}^k$ by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ , is semialgebraically homeomorphic to S. In this paper, we consider a weaker version of this question. We prove that for any $\ell \geq 0$ , there exists an algorithm which takes as input a description of a semialgebraic subset $S \subset \mathbb {R}^k$ given by a quantifier-free first-order formula $\phi $ in the language of the reals and produces as output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ is $\ell $ -equivalent to S. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell )}}$ , where s is the number of polynomials appearing in the formula $\phi $ , and d a bound on their degrees. For fixed $\ell $ , this bound is singly exponential in k. In particular, since $\ell $ -equivalence implies that the homotopy groups up to dimension $\ell $ of $|\Delta |$ are isomorphic to those of S, we obtain a reduction (having singly exponential complexity) of the problem of computing the first $\ell $ homotopy groups of S to the combinatorial problem of computing the first $\ell $ homotopy groups of a finite simplicial complex of size bounded by $(sd)^{k^{O(\ell )}}$ .

We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions.

Let k be an algebraically closed field of positive characteristic. For any integer $m\ge 2$ , we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs.

Let ${\mathbf {G}}$ be a complex algebraic reductive group and ${\mathbf {H}}\subset {\mathbf {G}}$ be a spherical algebraic subgroup. Let ${\mathfrak {g}},{\mathfrak {h}}$ denote the Lie algebras of ${\mathbf {G}}$ and ${\mathbf {H}}$ , and let ${\mathfrak {h}}^{\bot }$ denote the orthogonal complement to ${\mathfrak {h}}$ in ${\mathfrak {g}}^*$ . A ${\mathfrak {g}}$ -module is called ${\mathfrak {h}}$ -distinguished if it admits a nonzero ${\mathfrak {h}}$ -invariant functional. We show that the maximal ${\mathbf {G}}$ -orbit in the annihilator variety of any irreducible ${\mathfrak {h}}$ -distinguished ${\mathfrak {g}}$ -module intersects ${\mathfrak {h}}^{\bot }$ . This generalises a result of Vogan [Vog91].

We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G.

Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup $\bf H$ , and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules over W-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20].

Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models.

We prove, in $\text{ZF}+\boldsymbol{{\it\Sigma}}_{2}^{1}$-determinacy, that, for any analytic equivalence relation $E$, the following three statements are equivalent: (1) $E$ does not have perfectly many classes, (2) $E$ satisfies hyperarithmetic-is-recursive on a cone, and (3) relative to some oracle, for every equivalence class $[Y]_{E}$ we have that a real $X$ computes a member of the equivalence class if and only if ${\it\omega}_{1}^{X}\geqslant {\it\omega}_{1}^{[Y]}$. We also show that the implication from (1) to (2) is equivalent to the existence of sharps over $ZF$.

We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis, Inter. Math. Res. Not. 2014 (2014), 1289–1311 relating to work of Arveson, Acta Math. 118 (1967), 95–109 from the 1960s, and extends related work of Kakariadis and Katsoulis, J. Noncommut. Geom. 8 (2014), 771–787.

We revisit regularity of SLE trace, for all $\unicode[STIX]{x1D705}\neq 8$, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia–Rodemich–Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index $\min (1+\unicode[STIX]{x1D705}/8,2)$, improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler.

A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of $p'$-order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$-subgroups. We present two applications of this last result, one to tame fusion systems.