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For a harmonic diffeomorphism between the Poincaré disks, Wan [J. Differential Geom. 35 (1992), pp. 643–657] showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to r-differentials. We study the relationship between bounded holomorphic r-differentials/(r-1)-differential and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space SL(r,\mathbb R)/SO(r) arising from cyclic/subcyclic Higgs bundles. Also, we show the equivalence between the boundedness of holomorphic differentials and having a negative upper bound of the induced curvature on hyperbolic affine spheres in \mathbb {R}^3, maximal surfaces in \mathbb {H}^{2,n} and J-holomorphic curves in \mathbb {H}^{4,2}. Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods. [ABSTRACT FROM AUTHOR]

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseteq \mathbb {P}_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a\leq b. Nondegenerate examples known as grids have been known since 2011. Nondegenerate nongrids were first described in 2018, working in characteristic 0. Almost all of these new examples are of a special kind called half grids. In this paper, based partly on the author's thesis, we use a feature of geometry in positive characteristic to give new methods of producing geproci half grids and non-half grids. [ABSTRACT FROM AUTHOR]

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]

To investigate the degree d connectedness locus, Thurston [ On the geometry and dynamics of iterated rational maps , Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied \sigma _d-invariant laminations , where \sigma _d is the d-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f(z) = z^2 +c. In the spirit of Thurston's work, we consider the space of all cubic symmetric polynomials f_\lambda (z)=z^3+\lambda ^2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C_sCL together with the induced factor space \mathbb {S}/C_sCL of the unit circle \mathbb {S}. As will be verified in the third paper of the series, \mathbb {S}/C_sCL is a monotone model of the cubic symmetric connectedness locus , i.e. the space of all cubic symmetric polynomials with connected Julia sets. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a Helmholtz-type decomposition for the space of square integrable, symmetric-matrix-valued functions analogous to the standard Helmholtz decomposition for vector fields. This decomposition provides a better understanding of the strain constraint space, which is important to the Navier–Stokes regularity problem. In particular, we give a full characterization the orthogonal complement of the strain constraint space and investigate the geometry of the eigenvalue distribution of matrices in the strain constraint space. [ABSTRACT FROM AUTHOR]

In this paper, we give three different new proofs of the validity of the geometry conjecture about cycles of projections onto nonempty closed, convex subsets of a Hilbert space. The first uses a simple minimax theorem, which depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's inequality, which has found many applications in optimization and mathematical economics. The third uses three results on maximally monotone operators on a Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals. [ABSTRACT FROM AUTHOR]

A well known conjecture due to Hassett asserts that a cubic fourfold X whose transcendental cohomology TX cannot be realized as the transcendental cohomology of a K3 surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic 2-cycles on them, it is natural to ask for the most algebraic cubic fourfolds X to which this conjecture is still applicable. In this paper, we show that for an appropriate "algebraicity index" κX ∈ Q+, there exists a unique class of cubics maximizing κX, not having an associated K3 surface; namely, the cubic fourfolds with an Eckardt point (previously investigated in by Laza, Pearlstein, and Zhang [Adv. Math. 340 (2018), pp. 684-722]). Arguably, they are the most algebraic conjecturally irrational cubic fourfolds, and thus a good testing ground for Hassett's irrationality conjecture for cubic fourfolds. [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]

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]

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]

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]

In this paper, we use Rouché's theorem and the pleasant properties of the arithmetic of the logarithmic derivative to establish several new results and bounds regarding the geometry of the zeros, poles, and critical points of a rational function. Included is an improvement on a result by Alexander and Walsh regarding the "exclusion region" around a given zero or pole of a rational function in which no critical point may lie. [ABSTRACT FROM AUTHOR]

This paper lays the foundation for Plancherel theory on real spherical spaces Z=G/H, namely it provides the decomposition of L^2(Z) into different series of representations via Bernstein morphisms. These series are parametrized by subsets of spherical roots which determine the fine geometry of Z at infinity. In particular, we obtain a generalization of the Maass-Selberg relations. As a corollary we obtain a partial geometric characterization of the discrete spectrum: L^2(Z)_{\mathrm {disc}}\neq \emptyset if \mathfrak {h}^\perp contains elliptic elements in its interior. In case Z is a real reductive group or, more generally, a symmetric space our results retrieve the Plancherel formula of Harish-Chandra (for the group) as well as that of Delorme and van den Ban-Schlichtkrull (for symmetric spaces) up to the explicit determination of the discrete series for the inducing datum. [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]

In this paper we give a geometric interpretation of the second fundamental form of the period map of curves and we use it to improve the upper bounds on the dimension of a totally geodesic subvariety Y of Ag generically contained in the Torelli locus obtained by Elisabetta Colombo, Paola Frediani, and Alessandro Ghigi [Internat. J. Math. 26 (2015), no. 1, 1550005] and A. Ghigi, P. Pirola, and S. Torelli (to appear on Communications in Contemporary Mathematics, https:// doi.org/10.1142/S0219199720500200). We get dim Y ≤ 2g − 1 if g is even, dim Y ≤ 2g if g is odd. We also study totally geodesic subvarieties Z of Ag generically contained in the hyperelliptic Torelli locus and we show that dim Z ≤ g + 1. [ABSTRACT FROM AUTHOR]

In high-contrast composites, if an inclusion is in close proximity to the matrix boundary, then the stress, which is represented by the gradient of a solution to the Lamé systems of linear elasticity, may exhibit the singularities with respect to the distance \varepsilon between them. In this paper, we establish the asymptotic formulas of the stress concentration for core-shell geometry with C^{1,\alpha } boundaries in all dimensions by precisely capturing all the blow-up factor matrices, as the distance \varepsilon between interfacial boundaries of a core and a surrounding shell goes to zero. Further, a direct application of these blow-up factor matrices gives the optimal gradient estimates. [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]

In this paper a novel contour integral method is proposed for linear convection-diffusion equations. The method is based on the inversion of the Laplace transform and makes use of a contour given by an elliptic arc joined symmetrically to two half-lines. The trapezoidal rule is the chosen integration method for the numerical inversion of the Laplace transform, due to its well-known fast convergence properties when applied to analytic functions. Error estimates are provided as well as careful indications about the choice of several involved parameters. The method selects the elliptic arc in the integration contour by an algorithmic strategy based on the computation of pseudospectral level sets of the discretized differential operator. In this sense the method is general and can be applied to any linear convection-diffusion equation without knowing any a priori information about its pseudospectral geometry. Numerical experiments performed on the Black-Scholes (1D) and Heston (2D) equations show that the method is competitive with other contour integral methods available in the literature. [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]

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]

Let Ω be the complement of a connected, essential hyperplane arrangement. We prove that every dominant endomorphism of Ω extends to an endomorphism of the tropical compactification X of Ω associated to the Bergman fan structure on the tropical variety trop(Ω). This generalizes a result in [Compos. Math. 149 (2013), pp. 1211–1224], which states that every automorphism of Drinfeld’s half-space over a finite field Fq extends to an automorphism of the successive blow-up of projective space at all Fq-rational linear subspaces. This successive blow-up is in fact the minimal wonderful compactification by de Concini and Procesi, which coincides with X by results of Feichtner and Sturmfels. Whereas the proof in [Compos. Math. 149 (2013), pp. 1211–1224] is based on Berkovich analytic geometry over the trivially valued finite ground field, the generalization proved in the present paper relies on matroids and tropical geometry. [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]

In this paper, we will show that the Hesselink stratification of a Hilbert scheme of hypersurfaces is independent of the choice of Plücker coordinate and there is a positive relation between the length of Hesselink's worst virtual 1-parameter subgroup and multiplicity of a projective hypersurface. [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]

A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given by Fomin and Zelevinsky.) The universal objects are closely related to a fan FB called the mutation fan for B. In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except once-punctured surfaces without boundary components, and as a result we obtain a construction of the rational part of FB for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces and use it to construct universal geometric coefficients for these surfaces. [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]

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]

For a finite dimensional vector space equipped with a C-algebra structure, one can define rational maps using the algebraic structure. In this paper, we describe the growth of the degree sequences for this type of rational maps. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class C¹. Moreover, applications of this result are given. [ABSTRACT FROM AUTHOR]

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin [Duke Math. J. 170 (2021), pp. 1291–1375]. We also include some explicit calculations for the projective plane, which confirm some folklore conjectures in this case. [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]

After fixing a marking (V,W) of a quasi-Fuchsian punctured torus group G, the complex length λV and the complex twist τV,W parameters define a holomorphic embedding of the quasi-Fuchsian space QF of punctured tori into C². It is called the complex Fenchel-Nielsen coordinates of QF. For c ∈ C, let Qγ ,c be the affine subspace of C² defined by the linear equation λV = c. Then we can consider the linear slice Lc of QF by QF ∩ Qγ,c which is a holomorphic slice of QF. For any positive real value c, Lc always contains the so-called Bers-Maskit slice BMγ ,c defined in [Topology 43 (2004), no. 2, 447-491]. In this paper we show that if c is sufficiently small, then Lc coincides with BMγ ,c whereas Lc has other components besides BMγ ,c when c is sufficiently large. We also observe the scaling property of Lc. [ABSTRACT FROM AUTHOR]

The Atiyah–Singer index theorem, a landmark achievement of the early 1960s, brings together ideas in analysis, geometry, and topology. We recount some antecedents and motivations, various forms of the theorem, and some of its implications, which extend to the present. [ABSTRACT FROM AUTHOR]

This paper begins the study of relations between Riemannian geometry and contact topology on (2n + 1)-manifolds and continues this study on 3-manifolds. Specifically we provide a lower bound for the radius of a geodesic ball in a contact (2n+ 1)-manifold (M, ξ) that can be embedded in the standard contact structure on R2n+1, that is, on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form a for ξ. In dimension 3, this further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curve techniques to provide a lower bound for the radius of a PS-tight ball. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three- dimensional generic shrinking Ricci soliton is given by quotients of either S³, RS² or R³ under some very weak conditions on the vector field X generating the soliton structure. In doing so we introduce analytical tools that could be useful in other settings; for instance we prove that the Omori-Yau maximum principle holds for the X-Laplacian on every generic Ricci soliton without any assumption on X. [ABSTRACT FROM AUTHOR]

A locally conformally Kähler (LCK) manifold is a complex manifold M which has a Kähler structure on its cover, such that the deck transform group acts on it by homotheties. Assume that the Kähler form is exact on the minimal Kähler cover of M. We prove that any bimeromorphic map M'\rightarrow M is in fact holomorphic; in other words, M has a unique minimal model. This can be applied to a wide class of LCK manifolds, such as the Hopf manifolds, their complex submanifolds and to OT manifolds. [ABSTRACT FROM AUTHOR]

We prove that the Ricci flow for complete metrics with bounded geometry depends continuously on initial conditions for finite time with no loss of regularity. This relies on recent work of Bahuaud, Guenther, Isenberg and Mazzeo where sectoriality for the generator of the Ricci-DeTurck flow is proved. We use this to prove that for initial metrics sufficiently close in Hölder norm to a rotationally symmetric asymptotically hyperbolic metric and satisfying a simple curvature condition, but a priori distant from the hyperbolic metric, Ricci flow converges to the hyperbolic metric. [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]

The Weyl algebra over a field k of characteristic 0 is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all Z-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study Z-graded simple rings A of any dimension which have a graded quotient ring of the form K[t, t-1; σ] for a field K. Under some further hypotheses, we classify all such A in terms of a new construction of simple rings which we introduce in this paper. In the important special case that GKdimA = tr. deg(K/k) + 1, we show that K and σ must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry. [ABSTRACT FROM AUTHOR]

The aim of this paper is to contribute more examples and classification results of finite pointed quasi-quantum groups within the quiver framework initiated by the first author. The focus is put on finite dimensional graded Majid algebras generated by group-like elements and two skew-primitive elements which are mutually skew-commutative. Such quasi-quantum groups are associated to quasi-quantum planes in the sense of nonassociative geometry. As an application, we obtain an explicit classification of graded pointed Majid algebras with abelian coradical of dimension p³ and p4 for any prime number p. [ABSTRACT FROM AUTHOR]

Subdivision rules create sequences of nested cell structures on CW-complexes, and they frequently arise from groups. In this paper, we develop several tools for classifying subdivision rules. We give a criterion for a subdivision rule to represent a Gromov hyperbolic space, and show that a subdivision rule for a hyperbolic group determines the Gromov boundary. We give a criterion for a subdivision rule to represent a Euclidean space of dimension less than 4. We also show that Nil and Sol geometries cannot be modeled by subdivision rules. We use these tools and previous theorems to classify the geometry of subdivision rules for low-dimensional geometric groups by the combinatorial properties of their subdivision rules. [ABSTRACT FROM AUTHOR]

A classical theorem due to R. Phelps states that if C is a weakly compact set in a Banach space E, the strongly exposing functionals form a dense subset of the dual space E'. In this paper, we look at the concrete situation where C ⊂ L1(ℝd) is the closed convex hull of the set of random variables Y ∊ L1(ℝd) having a given law ν. Using the theory of optimal transport, we show that every random variable X ∊ L∞(ℝd), the law of which is absolutely continuous with respect to the Lebesgue measure, strongly exposes the set C. Of course these random variables are dense in L∞(ℝd). [ABSTRACT FROM AUTHOR]

Let G = (V,E,w) be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset W ⊂ V of vertices and weights aw such that 1/|V| ∑v∈Vƒ(v)~ ∑w∈Waw ƒ(w) for functions ƒ : V → R that are ''smooth'' with respect to the geometry of the graph; here ~ indicates that we want the right-hand side to be as close to the left-hand side as possible. The main application comprises problems where ƒ is known to vary smoothly over the underlying graph but is expensive to evaluate on even a single vertex. We prove an inequality showing that the integration problem can be rewritten as a geometric problem (''the optimal packing of heat balls''). We discuss how one would construct approximate solutions of the heat ball packing problem; numerical examples demonstrate the efficiency of the method. [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]

This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology. [ABSTRACT FROM AUTHOR]