This paper studies the virtual χ − y ‐genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea–Pandharipande. We first prove a structural result expressing the equivariant virtual χ − y ‐genera of Quot schemes universally in terms of the Seiberg–Witten invariants. The formula is simpler for curve classes of Seiberg–Witten length

In this paper we prove that certain points in the Hecke orbit of a CM point in the Siegel modular variety do not lie in the open Torelli locus under suitable conditions on the field of definition and the CM factors that arise in the corresponding CM abelian variety. The proof is a combination of properties of stable Faltings height and known cases of the Sato–Tate equidistribution, and is motivated

We prove two theorems about Goodwillie calculus and use those theorems to describe new models for Goodwillie derivatives of functors between pointed compactly generated ∞ ‐categories. The first theorem says that the construction of higher derivatives for spectrum‐valued functors is a Day convolution of copies of the first derivative construction. The second theorem says that the derivatives of any

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so‐called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of ‘universal’ quasi‐isomorphism and the behaviour of the decomposition under tensor product and compute the Grothendieck

We prove that synthetic lower Ricci bounds for metric measure spaces — both in the sense of Bakry–Émery and in the sense of Lott–Sturm–Villani — can be characterized by various functional inequalities including local Poincaré inequalities, local logarithmic Sobolev inequalities, dimension independent Harnack inequality, and logarithmic Harnack inequality.

We consider the evolution of a small rigid body in an incompressible viscous fluid filling the whole space R 3 . The motion of the fluid is modeled by the Navier–Stokes equations, whereas the motion of the rigid body is described by the conservation law of linear and angular momentum. Under the assumption that the diameter of the rigid body tends to zero and that the density of the rigid body goes

In this paper, we develop the theory of abstract crystals for quantum Borcherds–Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of B ( ∞ ) and B ( λ ) as its application, where B ( ∞ ) and B ( λ ) are the crystals of the negative half part of the quantum Borcherds–Bozec algebra U q ( g ) and its

This paper considers the following fundamental problem about intersections in projective space: When is the intersection of a (varying) curve with a (fixed) hypersurface a general set of points on the hypersurface?

We consider topological dynamical systems ( X , T ) , where X is a compact metrizable space and T denotes an action of a countable amenable group G on X by homeomorphisms. For two such systems ( X , T ) and ( Y , S ) and a factor map π : X → Y , an intermediate factor is a topological dynamical system ( Z , R ) for which π can be written as a composition of factor maps ψ : X → Z and φ : Z → Y . In

In this paper, we study transition density functions for pure jump unimodal Lévy processes killed upon leaving an open set D . Under some mild assumptions on the Lévy density, we establish two‐sided Dirichlet heat kernel estimates when the open set D is C 1 , 1 . Our result covers the case that the Lévy densities of unimodal Lévy processes are regularly varying functions whose indices are equal to

This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the Monge–Kantorovich optimal transport problem. The associated

The deformed Hermitian Yang‐Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper, we introduce and study dHYM equations with variable Kähler metric. These are coupled equations involving both

The affine diffeomorphism group Aff ( S , q ) of a half‐translation surface ( S , q ) comprise the self‐diffeomorphisms with constant differential away from the singularities. This group coincides with the stabiliser of the associated Teichmüller disc under the action of the mapping class group on Teichmüller space. We prove that any finitely generated subgroup of Aff ( S , q ) is undistorted in the

In this paper, we construct incomplete versions of the equivariant stable category; that is, equivariant stabilizations of the category of G ‐spaces with respect to incomplete systems of transfers encoded by an N ∞ operad O . These categories are built from the categories of O ‐algebras in G ‐spaces. Using this operadic formulation, we establish incomplete versions of the usual structural properties

Using a variant of the Boardman–Vogt tensor product, we construct an action of the Grothendieck–Teichmüller group on the completion of the little n ‐disks operad E n . This action is used to establish a partial formality theorem for E n with mod p coefficients and to give a new proof of the formality theorem in characteristic zero.

We consider the inverse problem of determining the shape of a general nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold with boundary ( M , g ) of dimension n = 2 , 3 . We prove results of unique recovery of the nonlinear term F ( t , x , u ) , appearing in the equation ∂ t 2 u − Δ g u + F ( t , x , u ) = 0 on ( 0 , T ) × M with T > 0 , from partial knowledge of

Concordance invariants of knots are derived from the instanton homology groups with local coefficients, as introduced in earlier work of the authors. These concordance invariants include a 1‐parameter family of homomorphisms f r , from the knot concordance group to R . Prima facie, these concordance invariants have the potential to provide independent bounds on the genus and number of double points

We study the compactness properties of metrics of prescribed fractional Q ‐curvature of order 3 in R 3 . We will use an approach inspired from conformal geometry, seeing a metric on a subset of R 3 as the restriction of a metric on R + 4 with vanishing fourth‐order Q ‐curvature. We will show that a sequence of such metrics with uniformly bounded fractional Q ‐curvature can blow up on a large set (roughly

For convex domains with C 1 , ε boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is the almost direct product of a compact group and a non‐compact connected simple Lie group

Let f be a Bianchi modular form, that is, an automorphic form for GL ( 2 ) over an imaginary quadratic field F , and let p be a prime of F at which f is new. Let K be a quadratic extension of F , and L ( f / K , s ) the L ‐function of the base‐change of f to K . Under certain hypotheses on f and K , the functional equation of L ( f / K , s ) ensures that it vanishes at the central point. The Bloch–Kato

We prove maximal regularity results in Hölder and Zygmund spaces for linear stationary and evolution equations driven by a class of differential and pseudo‐differential operators L , both in finite and in infinite dimension. The assumptions are given in terms of the semigroup generated by L . We cover the cases of fractional Laplacians and Ornstein–Uhlenbeck operators with fractional diffusion in finite

We prove that the generating functions for the one row/column colored HOMFLY‐PT invariants of arborescent links are specializations of the generating functions of the motivic Donaldson–Thomas invariants of appropriate quivers that we naturally associate with these links. Our approach extends the previously established tangles‐quivers correspondence for rational tangles to algebraic tangles by developing

We develop the theory of weighted Ricci curvature in a weighted Lorentz–Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. We also show a weighted Lorentz–Finsler

We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, that is, U ⊕ ⟨ − 2 k ⟩ ‐polarized K3 surfaces. Such moduli spaces are proved to be of general type for k ⩾ 220 . The proof relies on the low‐weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli

We study the behaviour of quasi‐geodesics in Out ( F n ) . Given an element ϕ in Out ( F n ) , there are several natural paths connecting the origin to ϕ in Out ( F n ) ; for example, paths given by Stallings' folding algorithm and paths induced by the shadow of greedy folding paths in Outer Space. We show that none of these paths is, in general, a quasi‐geodesic in Out ( F n ) . In fact, in contrast

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

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 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.

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'

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

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 ) .

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

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

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’

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