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In this paper, we study, for functionals having a minimax geometry on a constraint, the existence of bounded Palais-Smale sequences carrying Morse index type information. [ABSTRACT FROM AUTHOR]

In 1990, Schnyder used a 3-spanning-tree decomposition of a simple triangulation, now known as the Schnyder wood , to give a fundamental grid-embedding algorithm for planar maps. In the framework of mating of trees, a uniformly sampled Schnyder-wood-decorated triangulation can produce a triple of random walks. We show that these three walks converge in the scaling limit to three Brownian motions produced in the mating-of-trees framework by Liouville quantum gravity (LQG) with parameter 1, decorated with a triple of SLE_{16}'s curves. These three SLE_{16}'s curves are coupled such that the angle difference between them is 2\pi /3 in imaginary geometry. Our convergence result provides a description of the continuum limit of Schnyder's embedding algorithm via LQG and SLE. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective variety over the complex numbers and S(d) the scheme parametrizing d-dimensional Lie subalgebras of H^0(X,\mathcal {T}X). This article is dedicated to the study of the geometry of the moduli space \text {Inv} of involutive distributions on X around the points \mathcal {F}\in \text {Inv} which are induced by Lie group actions. For every \mathfrak {g}\in S(d) one can consider the corresponding element \mathcal {F}(\mathfrak {g})\in \text {Inv}, whose generic leaf coincides with an orbit of the action of \exp (\mathfrak {g}) on X. We show that under mild hypotheses, after taking a stratification \coprod _i S(d)_i\to S(d) this assignment yields an isomorphism \phi :\coprod _i S(d)_i\to \text {Inv} locally around \mathfrak {g} and \mathcal {F}(\mathfrak {g}). This gives a common explanation for many results appearing independently in the literature. We also construct new stable families of foliations which are induced by Lie group actions. [ABSTRACT FROM AUTHOR]

We study compactifications of the moduli space of unordered points in the plane via variation of GIT-quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given. [ABSTRACT FROM AUTHOR]

A great sphere fibration is a sphere bundle with total space S^n and fibers which are great k-spheres. Given a smooth great sphere fibration, the central projection to any tangent hyperplane yields a nondegenerate fibration of \mathbb {R}^n by pairwise skew, affine copies of \mathbb {R}^k (though not all nondegenerate fibrations can arise in this way). Here we study the topology and geometry of nondegenerate fibrations, we show that every nondegenerate fibration satisfies a notion of Continuity at Infinity, and we prove several classification results. These results allow us to determine, in certain dimensions, precisely which nondegenerate fibrations correspond to great sphere fibrations via the central projection. We use this correspondence to reprove a number of recent results about sphere fibrations in the simpler, more explicit setting of nondegenerate fibrations. For example, we show that every germ of a nondegenerate fibration extends to a global fibration, and we study the relationship between nondegenerate line fibrations and contact structures in odd-dimensional Euclidean space. We conclude with a number of partial results, in hopes that the continued study of nondegenerate fibrations, together with their correspondence with sphere fibrations, will yield new insights towards the unsolved classification problems for sphere fibrations. [ABSTRACT FROM AUTHOR]

We show that the 1-cusped quotient of the hyperbolic space \mathbb {H}^3 by the tetrahedral Coxeter group \Gamma _*=[5,3,6] has minimal volume among all non-arithmetic cusped hyperbolic 3-orbifolds, and as such it is uniquely determined. Furthermore, the lattice \Gamma _* is incommensurable to any Gromov-Piatetski-Shapiro type lattice. Our methods have their origin in the work of C. Adams [J. Differential Geom. 34 (1991), pp. 115–141; Noncompact hyperbolic 3-orbifolds of small volume , de Gruyter, Berlin, 1992]. We extend considerably this approach via the geometry of the underlying horoball configuration induced by a cusp. [ABSTRACT FROM AUTHOR]

We revisit the Harder-Narasimhan stratification on a minuscule p-adic flag variety, by the theory of modifications of G-bundles on the Fargues-Fontaine curve. We compare the Harder-Narasimhan strata with the Newton strata introduced by Caraiani-Scholze. As a consequence, we get further equivalent conditions in terms of p-adic Hodge-Tate period domains for fully Hodge-Newton decomposable pairs. Moreover, we generalize these results to arbitrary cocharacters case by considering the associated B_{dR}^+-affine Schubert varieties. Applying Hodge-Tate period maps, our constructions give applications to p-adic geometry of Shimura varieties and their local analogues. [ABSTRACT FROM AUTHOR]

Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some technical conditions, that they exhibit Weibull asymptotics with respect to different natural measures on the space of unimodular lattices in \mathbb {R}^d. This follows from very general Poisson approximation results for shrinking targets which should be of independent interest. Furthermore, we show in the appendix that the logarithm laws of Kleinbock-Margulis [Invent. Math. 138 (1999), pp. 451–494], Khinchin and Gallagher [J. London Math. Soc. 37 (1962), pp. 387–390] can be deduced from our distributional results. [ABSTRACT FROM AUTHOR]

We study the topology of a Ricci limit space (X,p), which is the Gromov-Hausdorff limit of a sequence of complete n-manifolds (M_i, p_i) with \mathrm {Ric}\ge -(n-1). Our first result shows that, if M_i has Ricci bounded covering geometry, i.e. the local Riemannian universal cover is non-collapsed, then X is semi-locally simply connected. In the process, we establish a slice theorem for isometric pseudo-group actions on a closed ball in the Ricci limit space. In the second result, we give a description of the universal cover of X if M_i has a uniform diameter bound; this improves a result by Ennis and Wei [Differential Geom. Appl. 24 (2006), pp. 554-562]. [ABSTRACT FROM AUTHOR]

In this paper, we give a geometric construction of the Jacquet-Langlands transfer for automorphic forms of higher weights. Our method is by studying the geometry of the mod p fibres of Hodge type Shimura varieties which satisfy a mild assumption and the cohomological correspondences between them. [ABSTRACT FROM AUTHOR]

We prove that the saddle connection graph associated to any half-translation surface is 4–hyperbolic and uniformly quasi-isometric to the regular countably infinite-valent tree. Consequently, the saddle connection graph is not quasi-isometrically rigid. We also characterise its Gromov boundary as the set of straight foliations with no saddle connections. In our arguments, we give a generalisation of the unicorn paths in the arc graph which may be of independent interest. [ABSTRACT FROM AUTHOR]

We investigate which orbits of an n-dimensional torus action on a 2n-dimensional toric Kähler manifold M are minimal. In other words, we study minimal submanifolds appearing as the fibres of the moment map on a toric Kähler manifold. Amongst other questions we investigate and give partial answers to the following: How many such minimal Lagrangian tori exist? Can their stability, as critical points of the area functional, be characterised just from the ambient geometry? Given a toric symplectic manifold, for which sets of orbits S, is there a compatible toric Kähler metric whose set of minimal Lagrangian orbits is S? [ABSTRACT FROM AUTHOR]

The goal of this paper is the study of simple rank 2 parabolic vector bundles over a 2-punctured elliptic curve C. We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to P1 × P1. We also showcase a special curve Γ isomorphic to C embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over P1 via a modular map which turns out to be the 2:1 cover ramified in Γ. We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles. [ABSTRACT FROM AUTHOR]

Consider a proper, isometric action by a unimodular, locally compact group G on a complete Riemannian manifold M. For equivariant elliptic operators that are invertible outside a cocompact subset of M, we show that a localised index in the K-theory of the maximal group C*-algebra of G is well-defined. The approach is based on the use of maximal versions of equivariant localised Roe algebras, and many of the technical arguments in this paper are used to handle the ways in which they differ from their reduced versions. By using the maximal group C*-algebra instead of its reduced counterpart, we can apply the trace given by integration over G to recover an index defined earlier by the last two authors, and developed further by Braverman, in terms of sections invariant under the group action. This leads to refinements of index-theoretic obstructions to Riemannian metrics of positive scalar curvature on noncompact manifolds, and also on orbifolds and other singular quotients of proper group actions. As a motivating application in another direction, we prove a version of Guillemin and Sternberg's quantisation commutes with reduction principle for equivariant indices of Spinc Callias-type operators. [ABSTRACT FROM AUTHOR]

For the open unit disc D in the complex plane, it is well known that if φ ∈ C(D) then its Berezin transform ~ φ also belongs to C(D). We say that D is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in Cn and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform. [ABSTRACT FROM AUTHOR]

Given two planar, conformal, smooth open sets \Omega and \omega , we prove the existence of a sequence of smooth sets (\Omega _\epsilon) which geometrically converges to \Omega and such that the (perimeter normalized) Steklov eigenvalues of (\Omega _\epsilon) converge to the ones of \omega. As a consequence, we answer a question raised by Girouard and Polterovich on the stability of the Weinstock inequality and prove that the inequality is genuinely unstable. However, under some a priori knowledge of the geometry related to the oscillations of the boundaries, stability may occur. [ABSTRACT FROM AUTHOR]

Given two ni-dimensional Alexandrov spaces Xi of curvature ≥ 1, the join of X1 and X2 is an (n1 + n2 + 1)-dimensional Alexandrov space X of curvature ≥ 1, which contains Xi as convex subsets such that their points are π/2 apart. If a group acts isometrically on a join that preserves Xi, then the orbit space is called a quotient of join. We show that an n-dimensional Alexandrov space X with curvature ≥ 1 is isometric to a finite quotient of join, if X contains two compact convex subsets Xi without boundary such that X1 and X2 are at least \frac \pi 2 apart and dim (X1) + dim (X2) = n−1. [ABSTRACT FROM AUTHOR]

We are concerned with a super-Liouville equation on compact surfaces with genus larger than one, obtaining the first non-trivial existence result for this class of problems via min-max methods. In particular we make use of a Nehari manifold and, after showing the validity of the Palais-Smale condition, we exhibit either a mountain pass or linking geometry. [ABSTRACT FROM AUTHOR]

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure of the tangle. Our construction comes from the geometry of traceless SU(2) character varieties associated to resolutions of the tangle diagram, and was inspired by Kronheimer and Mrowka's singular instanton link homology. [ABSTRACT FROM AUTHOR]

We study collapsed manifolds with Ricci bounded covering geometry, i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Applying the techniques in the Ricci flow, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on the collapsed manifolds with (sectional curvature) local bounded covering geometry, to the manifolds with (global) Ricci bounded covering geometry. [ABSTRACT FROM AUTHOR]

In this paper we establish partial structure results on the geometry of compact Hermitian manifolds of semipositive Griffiths curvature. We show that after appropriate arbitrary small deformation of the initial metric, the null spaces of the Chern-Ricci two-form generate a holomorphic, integrable distribution. This distribution induces an isometric, holomorphic, almost free action of a complex Lie group on the universal cover of the manifold. Our proof combines the strong maximum principle for the Hermitian Curvature Flow (HCF), new results on the interplay of the HCF and the torsion-twisted connection, and observations on the geometry of the torsion-twisted connection on a general Hermitian manifold. [ABSTRACT FROM AUTHOR]

Hurwitz numbers enumerate branched genus g covers of the Riemann sphere with fixed ramification data or equivalently certain factorisations of permutations. Double Hurwitz numbers are an important class of Hurwitz numbers, obtained by considering ramification data with a specific structure. They exhibit many fascinating properties, such as a beautiful piecewise polynomial structure, which has been well-studied in the last 15 years. In particular, using methods from tropical geometry, it was possible to derive wall-crossing formulae for double Hurwitz numbers in arbitrary genus. Further, double Hurwitz numbers satisfy an explicit recursive formula. In recent years several related enumerations have appeared in the literature. In this work, we focus on two of those invariants, so-called monotone and strictly monotone double Hurwitz numbers. Monotone double Hurwitz numbers originate from random matrix theory, as they appear as the coefficients in the asymptotic expansion of the famous Harish-Chandra-Itzykson-Zuber integral. Strictly monotone double Hurwitz numbers are known to be equivalent to an enumeration of Grothendieck dessins d'enfants. These new invariants share many structural properties with double Hurwitz numbers, such as piecewise polynomiality. In this work, we enlarge upon this study and derive new explicit wall-crossing and recursive formulae for monotone and strictly monotone double Hurwitz numbers. The key ingredient is a new interpretation of monotone and strictly monotone double Hurwitz numbers in terms of tropical covers, which was recently derived by the authors. An interesting observation is the fact that monotone and strictly monotone double Hurwitz numbers satisfy wall-crossing formulae, which are almost identical to the classical double Hurwitz numbers. [ABSTRACT FROM AUTHOR]

If V stands for a subspace of C(R) such that every nonzero function in V attains its maximum at one (and only one) point, then we prove that dim(V) ≤ 2. This provides the final answer to a lineability problem posed by Vladimir I. Gurariy in 2003. Moreover, we generalize the previous result in the following terms: If m ∈ N and Vm stands for a subspace of C(R) such that every nonzero function in V_m attains its maximum at m (and only m) points, then dim(Vm) ≤ 2 for m > 1 as well. Besides being a problem closely related to real analysis, this problem actually needs the use of tools from general topology, geometry, and complex analysis, such as decompositions (or partitions) of manifolds or Moore's theorem, among others. [ABSTRACT FROM AUTHOR]

We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometry of generalisations of alternating links, including weakly generalised alternating links described by the first author. We give diagrammatical properties that determine when such links are hyperbolic, find the geometry of their checkerboard surfaces, bound volume, and exclude exceptional Dehn fillings. [ABSTRACT FROM AUTHOR]

In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [Compos. Math. 154 (2018), pp. 2403-2425]. [ABSTRACT FROM AUTHOR]

We investigate the geometry of étale 4:1 coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic, depending on the values of the Weil pairing restricted to the group defining the covering. We recall from our previous work the results concerning the non-isotropic case and fully describe the isotropic case. We show that the necessary information to construct the Klein coverings is encoded in the 6 points on P1 defining the genus 2 curve. The main result of the paper is the fact that in both cases the Prym map associated to these coverings is injective. Additionally, we provide a concrete description of the closure of the image of the Prym map inside the corresponding moduli space of polarised abelian varieties. [ABSTRACT FROM AUTHOR]

Using Élie Cartan's method of equivalence, we prove an upper bound for the generality of generic rank-1 Bäcklund transformations relating two hyperbolic Monge-Ampère systems. In cases when the Bäcklund transformation admits a symmetry group whose orbits have codimension 1, 2, or 3, we obtain classification results and new examples of auto-Bäcklund transformations. [ABSTRACT FROM AUTHOR]

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary, cornered asymptotically hyperbolic manifolds, and proves a theorem of Cartan-Hadamard-type near infinity for the normal exponential map on the finite boundary. As a main application, a normal form for such manifolds at the corner is then constructed, analogous to the normal form for usual asymptotically hyperbolic manifolds and suited to studying geometry at the corner. The normal form is at the same time a submanifold normal form near the finite boundary and an asymptotically hyperbolic normal form near the infinite boundary. [ABSTRACT FROM AUTHOR]

The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichm üller space and is quasi-isometric to the underlying mapping class group. We study this space in two main directions. We first show that strata corresponding to triangulations containing a same multiarc are strongly convex within the whole space and use this result to deduce properties about the mapping class group. We then focus on the quotient of this space by the mapping class group to obtain a type of combinatorial moduli space. In particular, we are able to identity how the diameters of the resulting spaces grow in terms of the complexity of the underlying surfaces. [ABSTRACT FROM AUTHOR]

We study the moduli space of smooth complete intersections of two quadrics in ℙn by relating it to the geometry of the singular members of the corresponding pencils. By giving an alternative presentation for the moduli space of complete intersections, we compute the Picard group for all n ≥ 3. [ABSTRACT FROM AUTHOR]

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds X that can be expressed as a semidirect product of R² with R endowed with a left invariant metric. For any such compact minimal surface M, we provide an a priori radius estimate which depends only on the maximum distance of points of the boundary ∂M to a vertical geodesic of X. We also give a generalization of the classical Radó theorem in R³ to the context of compact minimal surfaces with graphical boundary over a convex horizontal domain in X, and we study the geometry, existence, and uniqueness of this type of Plateau problem. [ABSTRACT FROM AUTHOR]

A lattice d-simplex is the convex hull of d+1 affinely independent integer points in Rd. It is called empty if it contains no lattice point apart from its d+1 vertices. The classification of empty 3-simplices has been known since 1964 (White), based on the fact that they all have width one. But for dimension 4 no complete classification is known. Haase and Ziegler (2000) enumerated all empty 4-simplices up to determinant 1000 and based on their results conjectured that after determinant 179 all empty 4-simplices have width one or two. We prove this conjecture as follows: - We show that no empty 4-simplex of width three or more can have a determinant greater than 5058, by combining the recent classification of hollow 3-polytopes (Averkov, Krumpelmann and Weltge, 2017) with general methods from the geometry of numbers. We continue the computations of Haase and Ziegler up to determinant 7600, and find that no new 4-simplices of width larger than two arise. In particular, we give the whole list of empty 4-simplices of width larger than two, which is as computed by Haase and Ziegler: There is a single empty 4-simplex of width four (of determinant 101), and 178 empty 4-simplices of width three, with determinants ranging from 41 to 179. [ABSTRACT FROM AUTHOR]

This article deals with the irreducible components of the space of codimension one foliations in a projective space defined by logarithmic forms of a certain degree. We study the geometry of the natural parametrization of the logarithmic components and we give a new proof of the stability of logarithmic foliations, obtaining also that these irreducible components are reduced. [ABSTRACT FROM AUTHOR]

We introduce a new notion of an angle between intermediate subfactors and prove various interesting properties of the angle and relate it to the Jones index. We prove a uniform 60 to 90 degree bound for the angle between minimal intermediate subfactors of a finite index irreducible subfactor. From this rigidity we can bound the number of minimal (or maximal) intermediate subfactors by the kissing number in geometry. As a consequence, the number of intermediate subfactors of an irreducible subfactor has at most exponential growth with respect to the Jones index. This answers a question of Longo's published in 2003. [ABSTRACT FROM AUTHOR]

For a separated Noetherian scheme X with an ample family of line bundles and a non-zero-divisor W ∊ Г(X,L) of a line bundle L on X, we classify certain thick subcategories of the derived matrix factorization category DMF(X,L,W) of the Landau-Ginzburg model (X,L,W). Furthermore, by using the classification result and the theory of Balmer's tensor triangular geometry, we show that the spectrum of the tensor triangulated category (DMF(X,L,W),⊗1/2 ) is homeomorphic to the relative singular locus Sing(X0/X), introduced in this paper, of the zero scheme X0 ⊂ X of W. [ABSTRACT FROM AUTHOR]

Given an algebraic variety X defined over an algebraically closed field, we study the space RZ(X, x) consisting of all the valuations of the function field of X which are centered in a closed point x of X. We concentrate on its homeomorphism type. We prove that, when x is a regular point, this homeomorphism type only depends on the dimension of X. If x is a singular point of a normal surface, we show that it only depends on the dual graph of a good resolution of (X, x) up to some precise equivalence. This is done by studying the relation between RZ(X, x) and the normalized non-Archimedean link of x in X coming from the point of view of Berkovich geometry. We prove that their behavior is the same. [ABSTRACT FROM AUTHOR]

We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideals, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear: generalized Poisson submanifolds and generalized Poisson transversals. These are submanifolds for which the geometry normal to the submanifold is complex, respectively symplectic. We show that generalized Poisson submanifolds carry a canonical holomorphic ideal, and we give a necessary and sufficient condition for the corresponding blow-up to be generalized complex. For generalized Poisson transversals we prove a normal form theorem for a neighborhood of the submanifold and use it to define a generalized complex blow-up. [ABSTRACT FROM AUTHOR]

For a curve C of genus 6 or 8 and a torsion bundle η of order ℓ we study the vanishing of the space of global sections of the twist EC ⊗ η of the rank 2 Mukai bundle EC of C. The bundle EC was used in a well-known construction of Mukai which exhibits general canonical curves of low genus as sections of Grassmannians in the Plücker embedding. Globalizing the vanishing condition, we obtain divisors on the moduli spaces R6,ℓ and R8,ℓ of pairs [C, η]. First we characterize these divisors by different conditions on linear series on the level curves, afterwards we calculate the divisor classes. As an application, we are able to prove that R8,3 is of general type. [ABSTRACT FROM AUTHOR]

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de facto cocycles, known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: The definition of equivariant characteristic forms in the Cartan model due to Nicole Berline and Michéle Vergne motivates a refinement of equivariant integral cohomology by all Cartan cocycles. In view of this, we will also review previous definitions critically, in particular the one given in work of Kiyonori Gomi. [ABSTRACT FROM AUTHOR]

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the following: for any dimension there exists some ε > 0 so that if the squeezing function on a smoothly bounded convex domain is greater than 1 -- ε outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another consequence is the following: for any dimension d there exists some ε > 0 so that if the holomorphic sectional curvature of the Bergman metric on a smoothly bounded convex domain is within e of --4/(d+1) outside a compact set, then the domain is strongly pseudoconvex (and hence the holomorphic sectional curvature limits to --4/(d + 1) on the boundary). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study reducibility properties in Sasakian geometry. First we give the Sasaki version of the de Rham decomposition theorem; however, we need a mild technical assumption on the Sasaki automorphism group which includes the toric case. Next we introduce the concept of cone reducible and consider S3 bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2-torus of Reeb type. In particular, we can classify all such Sasakian structures up to contact isotopy on S3 bundles over a Riemann surface of genus greater than zero. Finally, we show that in the toric case an extremal Sasaki metric on a Sasaki join always splits. [ABSTRACT FROM AUTHOR]

Graphons are analytic objects representing limits of convergent sequences of graphs. Lovász and Szegedy conjectured that every finitely forcible graphon, i.e., any graphon determined by finitely many graph densities, has a simple structure. In particular, one of their conjectures would imply that every finitely forcible graphon has a weak ε-regular partition with the number of parts bounded by a polynomial in ε−1. We construct a finitely forcible graphon W such that the number of parts in any weak ε-regular partition of W is at least exponential in ε−2 /25 log∗ε−2. This bound almost matches the known upper bound for graphs and, in a certain sense, is the best possible for graphons. [ABSTRACT FROM AUTHOR]

This is the second paper of two in a series under the same title; both study the quantitative volume space form rigidity conjecture: a closed n-manifold of Ricci curvature at least (n−1)H, H = ±1 or 0 is diffeomorphic to an H-space form if for every ball of definite size on M, the lifting ball on the Riemannian universal covering space of the ball achieves an almost maximal volume, provided the diameter of M is bounded for H ≠ 1. In the first paper, we verified the conjecture for the case that the Riemannian universal covering space M̃ is not collapsed. In the present paper, we will verify this conjecture for the case that Ricci curvature is also bounded above, while the above non-collapsing condition on M̃ is not required. [ABSTRACT FROM AUTHOR]

The “dancing metric” is a pseudo-Riemannian metric g of signature (2,2) on the spaceM4 of non-incident point-line pairs in the real projective plane RP2. The null curves of (M4, g) are given by the “dancing condition”: at each moment, the point is moving towards or away from the point on the line about which the line is turning. This is the standard homogeneous metric on the pseudo-Riemannian symmetric space SL3(R)/GL2(R), also known as the “para-K¨ahler Fubini-Study metric”, introduced by P. Libermann. We establish a dictionary between classical projective geometry (incidence, cross ratio, projective duality, projective invariants of plane curves, etc.) and pseudo- Riemannian 4-dimensional conformal geometry (null curves and geodesics, parallel transport, self-dual null 2-planes, the Weyl curvature, etc.). Then, applying a twistor construction to (M4, g), a G2-symmetry is revealed, hidden deep in classical projective geometry. To uncover this symmetry, one needs to refine the “dancing condition” to a higher-order condition. The outcome is a correspondence between curves in the real projective plane and its dual, a projective geometric analog of the more familiar “rolling without slipping and twisting” for a pair of Riemannian surfaces. [ABSTRACT FROM AUTHOR]

Cylindric algebras, or concept algebras as another name, form an interface between algebra, geometry and logic; they were invented by Alfred Tarski around 1947. We prove that there are 2|α| many varieties of geometric (i.e., representable) α-dimensional cylindric algebras, which means that 2|α| properties of definable relations of (possibly infinitary) models of first order theories can be expressed by formula schemes using α variables, where α is infinite. This solves Problem 4.2 in the 1985 Henkin-Monk-Tarski monograph [Cylindric algebras. Part II, Studies in Logic and the Foundations of Mathematics, vol. 115, North-Holland, Amsterdam, 1985]; the problem is restated by Németi [On varieties of cylindric algebras with applications to logic, Ann. Pure Appl. Logic 36 (1987), no. 3, 235-277] and Andréka, Monk, and Németi [Algebraic logic, Colloq. Math. Soc. János Bolyai, Vol. 54, North-Holland, Amsterdam, 1991]. For solving this problem, we devise a new construction, which we then use to solve Problem 2.13 of the 1971 Henkin-Monk-Tarski monograph [Cylindric algebras. Part I, Studies in Logic and the Foundations of Mathematics, vol. 64, North-Holland, Amsterdam, 1971] which concerns the structural description of geometric cylindric algebras. There are fewer varieties generated by locally finite-dimensional cylindric algebras, and we get a characterization of these among all the 2|α| varieties. As a by-product, we get a simple recursive enumeration of all the equations true of geometric cylindric algebras, and this can serve as a solution to Problem 4.1 of the 1985 Henkin-Monk-Tarski mono- graph. All of this has logical content and implications concerning ordinary first order logic with a countable number of variables. [ABSTRACT FROM AUTHOR]

We study the connection between amenability, Følner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Følner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Følner condition is a left quasi-isometry invariant of finitely generated semigroups, and hence that left amenability is a left quasi-isometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Følner condition in terms of the existence of weak Følner sets satisfying a local injectivity condition on the relevant translation action of the semigroup. [ABSTRACT FROM AUTHOR]

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Г) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Г). In addition, we identify a milder condition for a finitely generated subgroup of A(Г) that guarantees it is free, undistorted, and retains finite generation when intersected with A(Λ) for subgraphs Λ of Г. These results have applications to both the study of convex cocompactness in Mod(S) and the way in which certain groups can embed in right-angled Artin groups. [ABSTRACT FROM AUTHOR]

Let X be a d × d symmetric random matrix with independent but nonidentically distributed Gaussian entries. It has been conjectured by Latala that the spectral norm of X is always of the same order as the largest Euclidean norm of its rows. A positive resolution of this conjecture would provide a sharp understanding of the probabilistic mechanisms that control the spectral norm of inhomogeneous Gaussian random matrices. This paper establishes the conjecture up to a dimensional factor of order √log log d. Moreover, dimensionfree bounds are developed that are optimal to leading order and that establish the conjecture in special cases. The proofs of these results shed significant light on the geometry of the underlying Gaussian processes. [ABSTRACT FROM AUTHOR]

Eigenvalues and eigenfunctions of two- and three-dimensional double layer potentials are considered. Let Ω be a C2 bounded region in Rn (n = 2, 3). The double layer potential K : L2(∂Ω) → L2(∂Ω) is defined by (Kψ)(x) ≡ ∫ ∂ Ω ψ(y)·vyE(x, y) dsy, where E(x, y) = ∫1/2π log1/∣x-y∣ , if n = 2, 1/π log1/∣x-y∣ , if n = 3, dsy is the line or surface element and vy is the outer normal derivative on ∂Ω. It is known that K is a compact operator on L2(∂Ω) and consists of at most a countable number of eigenvalues, with 0 as the only possible limit point. This paper aims to establish some relationships among the eigenvalues, the eigenfunctions, and the geometry of ∂Ω. [ABSTRACT FROM AUTHOR]

The statement of Lemma 3.1 in Maps between non-commutative spaces (Trans. Amer. Math. Soc. 356 (2004), no. 7, 2927-2944) is not correct. Lemma 3.1 is needed for the proof of Theorem 3.2. Theorem 3.2 as originally stated is true but its "proof" is not correct. Here we change the statements and proofs of Lemma 3.1 and Theorem 3.2. We also prove a new result. Let k be a field, A a left and right noetherian N-graded k-algebra such that dimk(An) < 8 for all n, and J a graded two-sided ideal of A. If the noncommutative scheme Projnc(A) is isomorphic to a projective scheme X, then there is a closed subscheme Z × X such that Projnc(A/J) is isomorphic to Z. This result is a geometric translation of what we actually prove: if the category QGr(A) is equivalent to Qcoh(X), then QGr(A/J) is equivalent to Qcoh(Z) for some closed subscheme Z × X. [ABSTRACT FROM AUTHOR]