We study the infimal value of the Hausdorff dimension of spaces that are Hölder equivalent to a given metric space; we call this bi‐Hölder‐invariant ‘Hölder dimension’. This definition and some of our methods are analogous to those used in the study of conformal dimension. We prove that Hölder dimension is bounded above by capacity dimension for compact, doubling metric spaces. As a corollary, we obtain

We prove an analogue for algebraic stacks of Hermite–Minkowski's finiteness theorem from algebraic number theory, and establish a Chevalley–Weil type theorem for integral points on stacks. As an application of our results, we prove analogues of the Shafarevich conjecture for some surfaces of general type.

We study tilting and projective‐injective modules in a parabolic BGG category O for an arbitrary classical Lie superalgebra. We establish a version of Ringel duality for this type of Lie superalgebras which allows to express the characters of tilting modules in terms of those of simple modules in that category. We also obtain a classification of projective‐injective modules in the full BGG category

We study self‐similarity problem for two classes of flows: (1) special flows over circle rotations and under roof functions with symmetric logarithmic singularities (2) special flows over interval exchange transformations and under roof functions which are of two types piecewise constant with one additional discontinuity which is not a discontinuity of the IET; piecewise linear over exchanged intervals

This paper is concerned with time‐decay rates of the weak solutions to the 3D compressible magnetohydrodynamic flows with discontinuous initial data and large oscillations. The global existence of weak solutions to the Cauchy problem of the 3D compressible magnetohydrodynamic flows has been established by Suen–Hoff (Arch. Ration. Mech. Anal. 205 (2012) 27–58) and Suen (J. Differential Equations 268

We study local singularities of holomorphic families of arbitrary square, symmetric and skew‐symmetric matrices, that is, of mappings of smooth manifolds to the matrix spaces. Our main object is the vanishing topology of the pre‐images of the hypersurface Δ of all degenerate matrices in assumption that the dimension of the source is at least the codimension of the singular locus of Δ in the ambient

We classify webs of minimal degree rational curves on surfaces and give a criterion for webs being hexagonal. In addition, we classify Neron–Severi lattices of real weak del Pezzo surfaces. These two classifications are related to root subsystems of E8.

Let p ⩾ 5 be a prime and let n be a natural number. In this article, we describe the irreducible constituents of the induced characters ϕ ↑ S n for arbitrary linear characters ϕ of a Sylow p ‐subgroup P n of the symmetric group S n , generalising results of Giannelli and Law (J. Algebra 506 (2018) 409–428). By doing this, we introduce Sylow branching coefficients for symmetric groups.

We prove that the closure of every Jordan class J in a semisimple simply connected complex algebraic group G at a point x with Jordan decomposition x = r v is smoothly equivalent to the union of closures of those Jordan classes in the centraliser of r that are contained in J and contain x in their closure. For x unipotent, we also show that the closure of J around x is smoothly equivalent to the closure

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with isolated singularities. A concrete application is for surfaces with geometric genus p g = 5 : the canonical model is embedded in P 4 if and only if we have a complete

Every unitary solution of the Yang–Baxter equation (R‐matrix) in dimension d can be viewed as a unitary element of the Cuntz algebra O d and as such defines an endomorphism of O d . These Yang–Baxter endomorphisms restrict and extend to several other C ∗ ‐ and von Neumann algebras, and furthermore define a II 1 factor associated with an extremal character of the infinite braid group. This paper is

In [Abreu and Sena‐Dias, Ann. Global Anal. Geom. 41 (2012) 209–239], the authors construct two distinct families of scalar‐flat Kähler non‐compact toric metrics using Donaldson's rephrasing of Joyce's construction in action‐angle coordinates. In this paper and using the same set‐up, we show that these are the only scalar‐flat Kähler metrics on any given strictly unbounded toric surface. We also show

Let Z be a quadratic hypersurface of P n ( R ) defined over Q containing points whose coordinates are linearly independent over Q . We show that, among these points, the largest exponent of uniform rational approximation is the inverse 1 / ρ of an explicit Pisot number ρ < 2 depending only on n if the Witt index (over Q ) of the quadratic form q defining Z is at most 1, and that it is equal to 1 otherwise

We study the fundamental group of the p ‐subgroup complex of a finite group G . We show first that π 1 ( A 3 ( A 10 ) ) is not a free group (here A 10 is the alternating group on ten letters). This is the first concrete example in the literature of a p ‐subgroup complex with non‐free fundamental group. We prove that, modulo a well‐known conjecture of Aschbacher, π 1 ( A p ( G ) ) = π 1 ( A p ( S G

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements. We show that these motivic zeta functions satisfy a functional equation arising from matroid Poincaré duality in the sense of Adiprasito–Huh–Katz. In the process

In this paper, we consider discrete groups in PGL d ( R ) acting convex co‐compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well‐known flat torus theorem for CAT ( 0 ) spaces.

We show that the partial compactification of a stratum of Abelian differentials previously considered by Mirzakhani and Wright is not an algebraic variety. Despite this, we use a combination of algebro‐geometric and other methods to provide a short, unconditional proof of Mirzakhani and Wright's formula for the tangent space to the boundary of a G L + ( 2 , R ) orbit closure, and give new results on

We introduce a new notion of a homogeneous pair for a pseudo‐Riemannian metric g and a positive function f on a manifold M admitting a free R > 0 ‐action. There are many examples admitting this structure. For example, (a) a class of pseudo‐Hessian manifolds admitting a free R > 0 ‐action and a homogeneous potential function such as the moduli space of torsion‐free G 2 ‐structures, (b) the space of

We consider a standard one‐dimensional Brownian motion on the time interval [0,1] conditioned to have vanishing iterated time integrals up to order N . We show that the resulting processes can be expressed explicitly in terms of shifted Legendre polynomials and the original Brownian motion, and we use these representations to prove that the processes converge weakly as N → ∞ to the zero process. This

In recent years, maximizing Gál sums regained interest due to a firm link with large values of L ‐functions. In the present paper, we initiate an investigation of small sums of Gál type, with respect to the L 1 ‐norm. We also consider the intertwined question of minimizing weighted versions of the usual multiplicative energy. We apply our estimates to: (i) a logarithmic refinement of Burgess' bound

We consider the twistor theory of nilconformal harmonic maps from a Riemann surface into the Cayley plane O P 2 = F 4 / Spin ( 9 ) . By exhibiting this symmetric space as a submanifold of the Grassmannian of 10‐dimensional subspaces of the fundamental representation of F 4 , techniques and constructions similar to those used in earlier works on twistor constructions of nilconformal harmonic maps into

For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulae counting the number

Fix a degree d projective curve X ⊂ P r over an algebraically closed field K . Let U ⊂ ( P r ) ∗ be a dense open subvariety such that every hyperplane H ∈ U intersects X in d smooth points. Varying H ∈ U produces the monodromy action φ : π 1 ét ( U ) → S d . Let G X ≔ im ( φ ) . The permutation group G X is called the sectional monodromy group of X . In characteristic 0, G X is always the full symmetric

We use Galois cohomology methods to produce optimal mod p d level lowering congruences to a p ‐adic Galois representation that we construct as a well‐chosen lift of a given residual mod p representation. Using our explicit Galois cohomology methods, for F a number field, Γ F its absolute Galois group and G a reductive group, k a finite field and a suitable representation ρ ¯ : Γ F → G ( k ) , ramified

Let Σ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce wild globally hyperbolic anti‐de Sitter structures on Σ × R and provide two parameterisations of their deformation space: as a quotient of the product of two copies of the Teichmüller space of crowned hyperbolic surfaces and as the bundle over the Teichmüller space of Σ of meromorphic quadratic differentials

For a unital ring R , a Sylvester rank function is a numerical invariant which can be described in three equivalent ways: on finitely presented left R ‐modules, or on rectangular matrices over R , or on maps between finitely generated projective left R ‐modules. We extend each Sylvester rank function to all pairs of left R ‐modules M 1 ⊆ M 2 , and to all maps between left R ‐modules satisfying suitable

The Gasper and Rahman multivariate ( − q ) ‐Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank q ‐Bannai–Ito algebra A n q . Lifting the action of the algebra to the connection coefficients, we find a realization of A n q by means of difference operators. This provides an algebraic interpretation for the

We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category O of representations of the quantum loop algebra introduced by Hernandez–Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type A , we prove that this

In the integral cohomology ring of the classifying space of the projective linear group P G L n (over C ), we find a collection of p ‐torsion classes y p , k of degree 2 ( p k + 1 + 1 ) for any odd prime divisor p of n , and k ⩾ 0 . If, in addition, p 2 ∤ n , there are p ‐torsion classes ρ p , k of degree p k + 1 + 1 in the Chow ring of the classifying stack of P G L n , such that the cycle class map

Let p be a rational prime and q a power of p . Let ℘ be a monic irreducible polynomial of degree d in F q [ t ] . In this paper, we define an analogue of the Hodge–Tate map which is suitable for the study of Drinfeld modules over F q [ t ] and, using it, develop a geometric theory of ℘ ‐adic Drinfeld modular forms similar to Katz's theory in the case of elliptic modular forms. In particular, we show

We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3‐manifolds and their Heegaard genus. We give many examples including translations in the affine group of the line and provide upper bounds for the overcommutation length in SL 2 , related to the Steinberg relation.

We classify del Pezzo non‐commutative surfaces that are finite over their centres and have no worse than canonical singularities. Using the minimal model program, we introduce the minimal model of such surfaces. We first classify the minimal models and then give the classification of these surfaces in general. This presents a complementary result and method to the classification of del Pezzo orders

Let J be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on J where F n → F if the critical structures of F n converge to the critical structure of F . We show that this occurs precisely when the critical structures of the F n are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic

We generalize the notion of tight geodesics in the curve complex to tight trees. We then use tight trees to construct model geometries for certain surface bundles over graphs. This extends some aspects of the combinatorial model for doubly degenerate hyperbolic 3‐manifolds developed by Brock, Canary and Minsky during the course of their proof of the Ending Lamination Theorem. Thus we obtain uniformly

For spherical and parabolic averages of the Fourier transform of fractal measures, we obtain new upper bounds on rates of decay by an ‘intermediate dimension’ trick.

We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A 1 and A 2 , vanishes in any (co)homological degree p > 2 . Moreover, its (higher) cohomological calculus is isomorphic as a bimodule to its (higher) homological calculus, by exchanging degrees p and 2 − p , and we prove a generalised version of the 2‐Calabi–Yau property. For the ADE Dynkin graphs, the preprojective

The classical completeness problem raised by Beurling and independently by Wintner asks for which ψ ∈ L 2 ( 0 , 1 ) , the dilation system { ψ ( k x ) : k = 1 , 2 , … } is complete in L 2 ( 0 , 1 ) , where ψ is identified with its extension to an odd 2‐periodic function on R . This difficult problem is nowadays commonly called as the periodic dilation completeness problem (PDCP). By Beurling's idea

Nielsen realization problem for the mapping class group Mod ( S g ) asks whether the natural projection p g : Homeo + ( S g ) → Mod ( S g ) has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of torsion‐free subgroups of Mod ( S g ) . The main result of this paper is that the Torelli group

In this paper, we study the evolution problem u t ( x , t ) − λ j ( D 2 u ( x , t ) ) = 0 , in Ω × ( 0 , + ∞ ) , u ( x , t ) = g ( x , t ) , on ∂ Ω × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , in Ω , where Ω is a bounded domain in R N (which verifies a suitable geometric condition on its boundary) and λ j ( D 2 u ) stands for the j th eigenvalue of the Hessian matrix D 2 u . We assume that u 0 and g are

We derive a fully explicit version of the Selberg trace formula for twist‐minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2‐dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the

We consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle

Let O be an order in the imaginary quadratic field K . For positive integers M ∣ N , we determine the least degree of an O ‐CM point on the modular curve X ( M , N ) / K ( ζ M ) and also on the modular curve X ( M , N ) / Q ( ζ M ) : that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally

We initiate a study of distortion elements in the Polish groups Diff + k ( S 1 ) ( 1 ⩽ k < ∞ ), as well as Diff + 1 + A C ( S 1 ) , in terms of maximal metrics on these groups. We classify distortion in the k = 1 case: a C 1 circle diffeomorphism is C 1 ‐undistorted if and only if it has a hyperbolic periodic point. On the other hand, answering a question of Navas, we exhibit analytic circle diffeomorphisms

We show that for any subgroup H of Out ( F N ) , either H contains an atoroidal element or a finite index subgroup H ′ of H fixes a nontrivial conjugacy class in F N . This result is an analog of Ivanov's subgroup theorem for mapping class groups and Handel–Mosher's subgroup theorem for Out ( F N ) in the setting of irreducible elements.

We study smooth bundles over surfaces with highly connected almost parallelizable fiber M of even dimension, providing necessary conditions for a manifold to be bordant to the total space of such a bundle and showing that, in most cases, these conditions are also sufficient. Using this, we determine the characteristic numbers realized by total spaces of bundles of this type, deduce divisibility constraints

We construct an integrable colored five‐vertex model whose partition function is a Lascoux atom based on the five‐vertex model of Motegi and Sakai and the colored five‐vertex model of Brubaker, the first author, Bump and Gustafsson. We then modify this model in two different ways to construct a Lascoux polynomial, yielding the first proven combinatorial interpretation of a Lascoux polynomial and atom

We calculate the Masur–Veech volume of the gothic locus G in the stratum H ( 2 3 ) of genus 4. Our method is based on the use of the formulae for the Euler characteristics of gothic Teichmüller curves to determine the number of lattice points of given area. We also use this method to recalculate the Masur–Veech volumes of the Prym loci P 3 ⊂ H ( 4 ) and P 4 ⊂ H ( 6 ) in genus 3 and 4.

We give a complete characterization of the forking independence relation over any set of parameters in the free groups of finite rank, in terms of the J S J decompositions relative to those parameters.

Let C be the category of finite‐dimensional modules over the quantum affine algebra U q ( g ̂ ) of a simple complex Lie algebra g . Let C − be the subcategory introduced by Hernandez and Leclerc. We prove the geometric q ‐character formula conjectured by Hernandez and Leclerc in types A and B for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial

We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of C ∗ ‐algebras agrees with the (ultra)product of the scaled Cuntz semigroups of the involved C ∗ ‐algebras.

This paper considers the propagation of TE‐modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model

In this paper, we show an equi‐distributed property in 2‐dimensional finite abelian groups Z p n × Z p m , where p is a prime number. By using this equi‐distributed property, we prove that Fuglede's spectral set conjecture holds on groups Z p 2 × Z p , namely, a set in Z p 2 × Z p is a spectral set if and only if it is a translational tile.

A group word w is said to be strongly concise in a class C of profinite groups if, for every group G in C such that w takes less than 2 ℵ 0 values in G , the verbal subgroup w ( G ) is finite. Detomi, Morigi and Shumyatsky established that multilinear commutator words — and the particular words x 2 and [ x 2 , y ] — have the property that the corresponding verbal subgroup is finite in a profinite group

In this paper, we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form x 2 + y 2 with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive positive definite binary quadratic form. In particular, for any positive definite binary quadratic form F and binary linear form G , there exist infinitely many ℓ , m ∈

We essentially determine the saturated 2‐fusion systems of J‐component type in which the centralizer of some fully centralized involution of maximal 2‐rank contains a component that is the 2‐fusion system of an alternating group A n for some n ⩾ 8 .

The extremal index (EI) is a parameter that measures the intensity of clustering of rare events and is usually equal to the reciprocal of the mean of the limiting cluster size distribution. We show how to build dynamically generated stochastic processes with an EI for which that equality does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised

In this paper, we study power series having a fixed point of multiplier 1. First, we give a closed formula for the residue fixed point index, in terms of the first coefficients of the power series. Then, we use this formula to study wildly ramified power series in positive characteristic. Among power series having a multiple fixed point of small multiplicity, we characterize those having the smallest

We prove sharp ℓ p L p decoupling inequalities for two quadratic forms in four variables. We also recover several previous results (Bourgain and Demeter, J. Funct. Anal. 270 (2016) 1299–1318; Bourgain and Demeter, J. Anal. Math. 133 (2017) 79–311; Demeter, Guo and Shi, Rev. Mat. Iberoam 35 (2019) 423–460; Oh, Math. Z. 290 (2018) 389–419) in a unified way.

We consider the question of whether solutions of variants of Teichmüller harmonic map flow from surfaces M to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least 2, as well as the flow from cylinders, we prove that such a finite‐time degeneration must occur in situations where the image of thin collars is ‘stretching out’ at a rate of at least

Slow convergence of cyclic projections implies divergence of random projections and vice versa. Let L 1 , L 2 , ⋯ , L K be a family of K closed subspaces of a Hilbert space. It is well known that although the cyclic product of the orthogonal projections on these spaces always converges in norm, random products might diverge. Moreover, in the cyclic case there is a dichotomy: the convergence is fast