We study simply connected Lie groups G for which the hull‐kernel topology of the primitive ideal space Prim ( G ) of the group C ∗ ‐algebra C ∗ ( G ) is T 1 , that is, the finite subsets of Prim ( G ) are closed. Thus, we prove that C ∗ ( G ) is AF‐embeddable. To this end, we show that if G is solvable and its action on the centre of [ G , G ] has at least one imaginary weight, then Prim ( G ) has

Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the ℓ ‐norm of convex bodies whose Löwner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschläger verified the reverse statement;

In this paper, we compute the cohomology class of certain ‘special maximal‐rank loci’ originally defined by Aprodu and Farkas. By showing that such classes are non‐zero, we are able to verify the non‐emptiness portion of the Strong Maximal Rank Conjecture in a wide range of cases. As an application, we obtain new evidence for the existence portion of a well‐known conjecture due to Bertram, Feinberg

We prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity, we prove that the Γ ‐limit of such energy is almost surely a deterministic quadratic Dirichlet‐type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies

In this paper we prove the existence of an optimal set for the minimization of the k th variational eigenvalue of the p ‐Laplacian among p ‐quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the p ‐Laplacian associated with Schrödinger potentials. In order to deal with these nonlinear shape optimization problems, we develop

The purpose of this article is proving the equality of two natural L ‐invariants attached to the adjoint representation of a weight one cusp form, each defined by purely analytic, respectively, algebraic means. The proof departs from Greenberg's definition of the algebraic L ‐invariant as a universal norm of a canonical Z p ‐extension of Q p associated to the representation. We relate it to a certain

We show the optimal C 1 , 1 regularity of geodesics in nef and big cohomology class on Kähler manifolds away from the non‐Kähler locus, assuming sufficiently regular initial data. As a special case, we prove the C 1 , 1 regularity of geodesics of Kähler metrics on compact Kähler varieties away from the singular locus. Our main novelty is an improved boundary estimate for the complex Monge–Ampère equation

We construct a transcendental entire f : C → C such that (1) f has bounded singular set, (2) f has a wandering domain, and (3) each singular value of f escapes to infinity under iteration by f .

A compact set E ⊂ R d is said to be arithmetically thick if there exists a positive integer n so that the n ‐fold arithmetic sum of E has non‐empty interior. We prove the arithmetic thickness of E , if E is uniformly non‐flat, in the sense that there exists ε 0 > 0 such that for x ∈ E and 0 < r ⩽ diam ( E ) , E ∩ B ( x , r ) never stays ε 0 r ‐close to a hyperplane in R d . Moreover, we prove the arithmetic

Given a positive integer s , the s ‐colour size‐Ramsey number of a graph H is the smallest integer m such that there exists a graph G with m edges with the property that, in any colouring of E ( G ) with s colours, there is a monochromatic copy of H . We prove that, for any positive integers k and s , the s ‐colour size‐Ramsey number of the k th power of any n ‐vertex bounded degree tree is linear

We study the moduli space of pairs consisting of a smooth cubic surface and a smooth hyperplane section, via a Hodge theoretic period map due to Laza, Pearlstein, and the second‐named author. The construction associates to such a pair a so‐called Eckardt cubic threefold, admitting an involution, and the period map sends the pair to the anti‐invariant part of the intermediate Jacobian of this cubic

We give a topological description of the two‐row Springer fiber over the real numbers. We show its cohomology ring coincides with the oddification of the cohomology ring of the complex Springer fiber introduced by Lauda–Russell. We also realize Ozsváth–Rasmussen–Szabó's odd TQFT from pullbacks and exceptional pushforwards along inclusion and projection maps between hypertori. Using these results, we

We consider partial sums of a weighted Steinhaus random multiplicative function and view this as a model for the Riemann zeta function. We give a description of the tails and high moments of this object. Using these we determine the likely maximum of T log T independently sampled copies of our sum and find that this is in agreement with a conjecture of Farmer–Gonek–Hughes on the maximum of the Riemann

For integers k ⩾ 1 and n ⩾ 2 k + 1 , the Kneser graph K ( n , k ) is the graph whose vertices are the k ‐element subsets of { 1 , … , n } and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form K ( 2 k + 1 , k ) are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every k ⩾ 3 , the odd graph K

The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller–Segel model. The model we study has the same entropy as the minimal Keller–Segel model, but a different dynamics to minimize this entropy. Consequently, the conditions on the mass for the existence of stationary solutions or blow‐up

Let S ( g ) be the symmetric algebra of a reductive Lie algebra g equipped with the standard Poisson structure. If C ⊂ S ( g ) is a Poisson‐commutative subalgebra, then tr . deg C ⩽ b ( g ) , where b ( g ) = ( dim g + rk g ) / 2 . We present a method for constructing the Poisson‐commutative subalgebra Z ⟨ h , r ⟩ of transcendence degree b ( g ) via a vector space decomposition g = h ⊕ r into a sum

We extend the definition of relative Gromov–Witten invariants with negative contact orders to all genera. Then we show that relative Gromov–Witten theory forms a partial CohFT. Some cycle relations on the moduli space of stable maps are also proved.

We explicitly compute a monoidal subcategory of the monoidal center of Deligne's interpolation category Re ̲ p ( S t ) , for t not necessarily a natural number, and we show that this subcategory is a ribbon category. For t = n , a natural number, there exists a functor onto the braided monoidal category of modules over the Drinfeld double of S n which is essentially surjective and full. Hence the new

The image of the branch set of a piecewise linear (PL)‐branched cover between PL n ‐manifolds is a simplicial ( n − 2 ) ‐complex. We demonstrate that the reverse implication also holds: an open and discrete map f : S n → S n with the image of the branch set contained in a simplicial ( n − 2 ) ‐complex is equivalent up to homeomorphism to a PL‐branched cover.

This paper is concerned with the existence and stability of phase transition steady states to a quasi‐linear hyperbolic–parabolic system of chemotactic aggregation, which was proposed in [Ambrosi, Bussolino and Preziosi, J. Theoret. Med. 6 (2005) 1–19; Gamba et al., Phys. Rev. Lett. 90 (2003) 118101.] to describe the coherent vascular network formation observed in vitro experiment. Considering the

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

The Kauffman bracket skein module K ( M ) of a 3‐manifold M is the quotient of the Q ( A ) ‐vector space spanned by isotopy classes of links in M by the Kauffman relations. A conjecture of Witten states that if M is closed, then K ( M ) is finite dimensional. We introduce a version of this conjecture for manifolds with boundary and prove a stability property for generic Dehn filling of knots. As a

We show that if a Sylow p ‐subgroup of a finite group G is of order p , then the normalized unit group of the integral group ring of G contains a normalized unit of order p q if and only if G contains an element of order p q , where q is any prime. We use this result to answer the Prime Graph Question for most sporadic simple groups and some simple groups of Lie type, including seven new infinite series'

Khovanov–Lauda–Rouquier algebras R θ of finite Lie type are affine quasi‐hereditary with standard modules Δ ( π ) labeled by Kostant partitions π of θ . In type A , we construct explicit projective resolutions of standard modules Δ ( π ) .

A parabolic representation of the free group F 2 is one in which the images of both generators are parabolic elements of P S L ( 2 , C ) . Roughly the Riley slice is an open subset R ⊂ C which is a model for the parabolic, discrete and faithful characters of F 2 . The complement of the closure of the Riley slice is a bounded Jordan domain within which there are isolated points, accumulating only at

Let K be a field of characteristic zero, X and Y be smooth K ‐varieties, and let G be an algebraic K ‐group. Given two algebraic morphisms φ : X → G and ψ : Y → G , we define their convolution φ ∗ ψ : X × Y → G by φ ∗ ψ ( x , y ) = φ ( x ) · ψ ( y ) . We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism φ : X → G which

It has been known that several objects such as cluster variables, coefficients, seeds, and Y ‐seeds in different cluster patterns with common exchange matrices share the same periodicity under mutations. We call it synchronicity phenomenon in cluster patterns. In this expository note we explain the mechanism of synchronicity based on several fundamental results on cluster algebra theory such as separation

In this note, we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p , and use these results to study the distribution of the rank of random matrices over F p and the equi‐distribution behavior of normal vectors of random hyperplanes. We also study the probability that a random square matrix

We prove that non‐compact finite volume hyperbolic 3‐manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling which types of cusps occur. Using these methods we produce the first known example

We prove various extensions of the Loomis–Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors w i of a not necessarily orthonormal basis of R n , or their orthogonal complements. In order to prove such inequalities, we estimate the constant in the Brascamp–Lieb inequality in terms of the vectors w i . Restricted and

We prove an almost everywhere convergence result for Bochner–Riesz means of L p functions on Heisenberg‐type groups, yielding the existence of a p > 2 for which convergence holds for means of arbitrarily small order. The proof hinges on a reduction of weighted L 2 estimates for the maximal Bochner–Riesz operator to corresponding estimates for the non‐maximal operator, and a ‘dual Sobolev trace lemma’

Let ε 1 , … , ε n be independent and identically distributed Rademacher random variables taking values ± 1 with probability 1 / 2 each. Given an integer vector a = ( a 1 , … , a n ) , its concentration probability is the quantity ρ ( a ) : = sup x ∈ Z Pr ( ε 1 a 1 + ⋯ + ε n a n = x ) . The Littlewood–Offord problem asks for bounds on ρ ( a ) under various hypotheses on a , whereas the inverse Littlewood–Offord

This paper concerns the Lipschitz equivalence of dust‐like self‐similar sets with commensurable ratios. We obtain an algebraic criterion of Falconer–Marsh type to check whether such two self‐similar sets are Lipschitz equivalent, and prove that it is a sufficient and necessary condition.

The ‘moduli continuity method’ permits an explicit algebraisation of the Gromov–Hausdorff compactification of Kähler–Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the ‘log setting’ to describe explicitly some compact moduli spaces of K‐polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval

We introduce the notion of joint spectrum of a compact set of matrices S ⊂ GL d ( C ) , which is a multi‐dimensional generalization of the joint spectral radius. We begin with a thorough study of its properties (under various assumptions: irreducibility, Zariski‐density, and domination). Several classical properties of the joint spectral radius are shown to hold in this generalized setting and an analogue

We characterize the maximal entropy measures of partially hyperbolic C 2 diffeomorphisms whose center foliations form circle bundles, by means of suitable finite sets of saddle points, that we call skeletons.

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

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

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

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

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