37 results
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2. Symmetric cubic laminations.
- Author
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Blokh, Alexander, Oversteegen, Lex, Selinger, Nikita, Timorin, Vladlen, and Vejandla, Sandeep Chowdary
- Subjects
- *
LOCUS (Mathematics) , *CIRCLE , *MATHEMATICAL connectedness , *TOPOLOGICAL spaces , *POLYNOMIALS , *GEOMETRY - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
3. A Helmholtz-type decomposition for the space of symmetric matrices.
- Author
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Miller, Evan and Sawyer, Eric
- Subjects
- *
SYMMETRIC matrices , *SYMMETRIC spaces , *VECTOR fields , *HELMHOLTZ equation , *EIGENVALUES , *GEOMETRY - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
4. Helly-type problems.
- Author
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Bárány, Imre and Kalai, Gil
- Subjects
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RADON , *COMBINATORICS , *GEOMETRY - Abstract
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]
- Published
- 2022
- Full Text
- View/download PDF
5. A geometric Jacquet-Langlands transfer for automorphic forms of higher weights.
- Author
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Yu, Jize
- Subjects
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AUTOMORPHIC forms , *GEOMETRICAL constructions , *GEOMETRY - Abstract
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]
- Published
- 2022
- Full Text
- View/download PDF
6. mth roots of the identity operator and the geometry conjecture.
- Author
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Simons, Stephen
- Subjects
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MATHEMATICAL optimization , *MONOTONE operators , *HILBERT space , *GEOMETRY , *LOGICAL prediction , *MATHEMATICAL economics , *CHEBYSHEV approximation - Abstract
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]
- Published
- 2022
- Full Text
- View/download PDF
7. Rouche's theorem and the geometry of rational functions.
- Author
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Richards, Trevor J.
- Subjects
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GEOMETRY , *ARITHMETIC - Abstract
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]
- Published
- 2022
- Full Text
- View/download PDF
8. Hessenberg varieties, intersections of quadrics, and the Springer correspondence.
- Author
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Chen, Tsao-Hsien, Vilonen, Kari, and Xue, Ting
- Subjects
- *
SYMMETRIC spaces , *QUADRICS , *FOURIER transforms , *LETTERS , *GEOMETRY , *MATHEMATICS - Abstract
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]
- Published
- 2020
- Full Text
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9. Maximally algebraic potentially irrational cubic fourfolds.
- Author
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Laza, Radu
- Subjects
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LOGICAL prediction , *IRRATIONAL numbers , *MATHEMATICS , *GEOMETRY - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
10. Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms.
- Author
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Delorme, Patrick, Knop, Friedrich, Krötz, Bernhard, and Schlichtkrull, Henrik
- Subjects
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SYMMETRIC spaces , *INFINITY (Mathematics) , *GENERALIZATION , *GEOMETRY - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
11. Geometry of the moduli of parabolic bundles on elliptic curves.
- Author
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Vargas, Néstor Fernández
- Subjects
- *
ELLIPTIC curves , *VECTOR bundles , *GEOMETRY , *AUTOMORPHISMS , *HYPERELLIPTIC integrals - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
12. Coarse geometry and Callias quantisation.
- Author
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Guo, Hao, Hochs, Peter, and Mathai, Varghese
- Subjects
- *
COMPACT groups , *GEOMETRY , *ORBIFOLDS , *RIEMANNIAN manifolds , *K-theory , *ELLIPTIC operators , *ALGEBRA - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
13. Berezin regularity of domains in Cn and the essential norms of Toeplitz operators.
- Author
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Čučković, Željko and Şahutoğlu, Sönmez
- Subjects
- *
TOEPLITZ operators , *PSEUDOCONVEX domains , *CONVEX domains , *GEOMETRY - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
14. On the geometry of the second fundamental form of the Torelli map.
- Author
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Frediani, Paola and Pirola, Gian Pietro
- Subjects
- *
GEOMETRY , *GEODESICS , *MATHEMATICS , *CURVES , *HYPERELLIPTIC integrals - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
15. QUANTITATIVE VOLUME SPACE FORM RIGIDITY UNDER LOWER RICCI CURVATURE BOUND II.
- Author
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LINA CHEN, XIAOCHUN RONG, and SHICHENG XU
- Subjects
- *
GEOMETRIC surfaces , *GEOMETRY , *CONCAVE surfaces , *LOGICAL prediction , *GEOMETRIC rigidity - Abstract
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]
- Published
- 2018
- Full Text
- View/download PDF
16. On the structure of Hermitian manifolds with semipositive Griffiths curvature.
- Author
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Ustinovskiy, Yury
- Subjects
- *
HERMITIAN structures , *MANIFOLDS (Mathematics) , *CURVATURE , *LIE groups , *GEOMETRY , *GEOMETRIC connections - Abstract
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]
- Published
- 2020
- Full Text
- View/download PDF
17. Numerical inverse Laplace transform for convection-diffusion equations.
- Author
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Guglielmi, Nicola, López-Fernández, María, and Nino, Giancarlo
- Subjects
- *
TRANSPORT equation , *LAPLACE transformation , *DIFFERENTIAL operators , *ANALYTIC functions , *NUMERICAL integration , *LINEAR equations , *GEOMETRY - Abstract
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]
- Published
- 2020
- Full Text
- View/download PDF
18. Klein coverings of genus 2 curves.
- Author
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Borówka, Paweł and Ortega, Angela
- Subjects
- *
MONODROMY groups , *ABELIAN varieties , *CURVES , *GEOMETRY - Abstract
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]
- Published
- 2020
- Full Text
- View/download PDF
19. EXPONENTIAL MAP AND NORMAL FORM FOR CORNERED ASYMPTOTICALLY HYPERBOLIC METRICS.
- Author
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MCKEOWN, STEPHEN E.
- Subjects
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MANIFOLDS (Mathematics) , *NORMAL forms (Mathematics) , *INFINITY (Mathematics) , *GEOMETRY - Abstract
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]
- Published
- 2019
- Full Text
- View/download PDF
20. ON ENDOMORPHISMS OF ARRANGEMENT COMPLEMENTS.
- Author
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KURUL, ŞEVDA and WERNER, ANNETTE
- Subjects
- *
ANALYTIC geometry , *FINITE fields , *PROJECTIVE spaces , *AUTOMORPHISMS , *ENDOMORPHISMS , *MATHEMATICS , *GEOMETRY - Abstract
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]
- Published
- 2019
- Full Text
- View/download PDF
21. RELATIVE SINGULAR LOCUS AND BALMER SPECTRUM OF MATRIX FACTORIZATIONS.
- Author
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YUKI HIRANO
- Subjects
- *
MATRIX decomposition , *TRIANGULATED categories , *GEOMETRY - Abstract
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]
- Published
- 2019
- Full Text
- View/download PDF
22. INSTABILITY AND SINGULARITY OF PROJECTIVE HYPERSURFACES.
- Author
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CHEOLGYU LEE
- Subjects
- *
HILBERT space , *HYPERSURFACES , *SUBGROUP growth , *GEOMETRY , *POLYNOMIALS - Abstract
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]
- Published
- 2018
- Full Text
- View/download PDF
23. A GAP THEOREM FOR THE COMPLEX GEOMETRY OF CONVEX DOMAINS.
- Author
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ZIMMER, ANDREW
- Subjects
- *
GEOMETRY , *NUMERICAL solutions to boundary value problems , *OPERATOR theory , *MATHEMATICAL constants , *PSEUDOCONVEX domains - Abstract
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]
- Published
- 2018
- Full Text
- View/download PDF
24. REDUCIBILITY IN SASAKIAN GEOMETRY.
- Author
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BOYER, CHARLES P., HONGNIAN HUANG, LEGENDRE, EVELINE, and TØNNESEN-FRIEDMAN, CHRISTINA W.
- Subjects
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SASAKIAN manifolds , *GEOMETRY , *AUTOMORPHISMS , *MATHEMATICAL decomposition , *CLASSIFICATION - Abstract
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]
- Published
- 2018
- Full Text
- View/download PDF
25. UNIVERSAL GEOMETRIC CLUSTER ALGEBRAS FROM SURFACES.
- Author
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READING, NATHAN
- Subjects
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CLUSTER algebras , *TRIANGULATION , *COEFFICIENTS (Statistics) , *GEOMETRIC surfaces , *GEOMETRY - Abstract
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]
- Published
- 2014
- Full Text
- View/download PDF
26. ON THE SPECTRAL NORM OF GAUSSIAN RANDOM MATRICES.
- Author
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VAN HANDEL, RAMON
- Subjects
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MATRICES (Mathematics) , *GAUSSIAN processes , *EUCLIDEAN geometry , *RANDOM matrices , *GEOMETRY - Abstract
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]
- Published
- 2017
- Full Text
- View/download PDF
27. EIGENVALUES AND EIGENFUNCTIONS OF DOUBLE LAYER POTENTIALS.
- Author
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YOSHIHISA MIYANISHI and TAKASHI SUZUKI
- Subjects
- *
EIGENVALUES , *EIGENFUNCTIONS , *GEOMETRY , *EIGENANALYSIS , *MATHEMATICS - Abstract
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]
- Published
- 2017
- Full Text
- View/download PDF
28. DEGREE GROWTH OF RATIONAL MAPS INDUCED FROM ALGEBRAIC STRUCTURES.
- Author
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FAVRE, CHARLES and JAN-LI LIN
- Subjects
- *
ORDERED algebraic structures , *VECTOR spaces , *GEOMETRY , *ALGEBRA , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2017
- Full Text
- View/download PDF
29. STRICTLY CONVEX WULFF SHAPES AND C¹ CONVEX INTEGRANDS.
- Author
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HUHE HAN and TAKASHI NISHIMURA
- Subjects
- *
WULFF construction (Statistical physics) , *GEOMETRIC shapes , *CONVEX surfaces , *CONVEX functions , *GEOMETRY - Abstract
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]
- Published
- 2017
- Full Text
- View/download PDF
30. QUANTITATIVE DARBOUX THEOREMS IN CONTACT GEOMETRY.
- Author
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ETNYRE, JOHN B., KOMENDARCZYK, RAFAL, and MASSOT, PATRICK
- Subjects
- *
DARBOUX transformations , *GEOMETRY , *RIEMANNIAN geometry , *RIEMANNIAN metric , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2016
- Full Text
- View/download PDF
31. ANALYTIC AND GEOMETRIC PROPERTIES OF GENERIC RICCI SOLITONS.
- Author
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CATINO, G., MASTROLIA, P., MONTICELLI, D. D., and RIGOLI, M.
- Subjects
- *
GEOMETRY , *VECTOR fields , *MAXIMUM principles (Mathematics) , *LAPLACIAN matrices , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2016
- Full Text
- View/download PDF
32. Z-GRADED SIMPLE RINGS.
- Author
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BELL, J. and ROGALSKI, D.
- Subjects
- *
WEYL groups , *WEYL space , *GEOMETRY , *INTEGERS , *MATHEMATICS - Abstract
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]
- Published
- 2016
- Full Text
- View/download PDF
33. QUASI-QUANTUM PLANES AND QUASI-QUANTUM GROUPS OF DIMENSION p³ AND p4.
- Author
-
HUA-LIN HUANG and YUPING YANG
- Subjects
- *
ALGEBRA , *GEOMETRY , *ABELIAN varieties , *PRIME numbers , *DIMENSIONS - Abstract
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]
- Published
- 2015
- Full Text
- View/download PDF
34. CLASSIFICATION OF SUBDIVISION RULES FOR GEOMETRIC GROUPS OF LOW DIMENSION.
- Author
-
RUSHTON, BRIAN
- Subjects
- *
GEOMETRY , *SUBDIVISION surfaces (Geometry) , *CW complexes , *HYPERBOLIC spaces , *HYPERBOLIC groups , *GEOMETRIC group theory , *MATHEMATICAL models - Abstract
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]
- Published
- 2014
- Full Text
- View/download PDF
35. OPTIMAL TRANSPORT AND THE GEOMETRY OF L1(ℝd).
- Author
-
EKELAND, IVAR and SCHACHERMAYER, WALTER
- Subjects
- *
GEOMETRY , *BANACH spaces , *TOPOLOGY , *RANDOM variables , *PROBABILITY theory - Abstract
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]
- Published
- 2014
- Full Text
- View/download PDF
36. ASYMPTOTIC GEOMETRY OF BANACH SPACES AND UNIFORM QUOTIENT MAPS.
- Author
-
DILWORTH, S. J., KUTZAROVA, DENKA, LANCIEN, G., and RANDRIANARIVONY, N. L.
- Subjects
- *
METRIC spaces , *LIPSCHITZ spaces , *BANACH spaces , *GEOMETRY , *SUBSPACES (Mathematics) - Abstract
Recently, Lima and Randrianarivony pointed out the role of the property (ß) of Rolewicz in nonlinear quotient problems and answered a tenyear- old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. In the present paper, we prove that the modulus of asymptotic uniform smoothness of the range space of a uniform quotient map can be compared with the modulus of (ß) of the domain space. We also provide conditions under which this comparison can be improved. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
37. ERRATUM TO "SOME HYPERBOLIC THREE-MANIFOLDS THAT BOUND GEOMETRICALLY".
- Author
-
KOLPAKOV, ALEXANDER, MARTELLI, BRUNO, and TSCHANTZ, STEVEN
- Subjects
- *
HYPERBOLIC processes , *GEOMETRY - Abstract
We indicate a non-fatal, but annoying mistake in our paper entitled "Some hyperbolic three-manifolds that bound geometrically" [4]. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
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