Showing papers on "Geometry and topology published in 2014"

Chow Rings, Decomposition of the Diagonal, and the Topology of Families

Abstract: Preface vii 1Introduction 1 1.1 Decomposition of the diagonal and spread 3 1.2 The generalized Bloch conjecture 7 1.3 Decomposition of the small diagonal and application to the topology of families 9 1.4 Integral coefficients and birational invariants 11 1.5 Organization of the text 13 2Review of Hodge theory and algebraic cycles 15 2.1 Chow groups 15 2.2 Hodge structures 24 3Decomposition of the diagonal 36 3.1 A general principle 36 3.2 Varieties with small Chow groups 44 4Chow groups of large coniveau complete intersections 55 4.1 Hodge coniveau of complete intersections 55 4.2 Coniveau 2 complete intersections 64 4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections 67 4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces 86 5On the Chow ring of K3 surfaces and hyper-Kahler manifolds 88 5.1 Tautological ring of a K3 surface 88 5.2 A decomposition of the small diagonal 96 5.3 Deligne's decomposition theorem for families of K3 surfaces 106 6Integral coefficients 123 6.1 Integral Hodge classes and birational invariants 123 6.2 Rationally connected varieties and the rationality problem 127 6.3 Integral decomposition of the diagonal and the structure of the Abel-Jacobi map 139 Bibliography 155 Index 163

Spectral geometry of the steklov problem

Abstract: The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry. While this problem shares some common properties with its more familiar Dirichlet and Neumann cousins, its eigenval- ues and eigenfunctions have a number of distinctive geometric features, which makes the subject especially appealing. In this survey we discuss some recent advances and open questions, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry.

Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems

Abstract: The book is devoted to the study of the geometrical and topological structure of gauge theories. It consists of the following three building blocks: - Geometry and topology of fibre bundles, - Clifford algebras, spin structures and Dirac operators, - Gauge theory. Written in the style of a mathematical textbook, it combines a comprehensive presentation of the mathematical foundations with a discussion of a variety of advanced topics in gauge theory. The first building block includes a number of specific topics, like invariant connections, universal connections, H-structures and the Postnikov approximation of classifying spaces. Given the great importance of Dirac operators in gauge theory, a complete proof of the Atiyah-Singer Index Theorem is presented. The gauge theory part contains the study of Yang-Mills equations (including the theory of instantons and the classical stability analysis), the discussion of various models with matter fields (including magnetic monopoles, the Seiberg-Witten model and dimensional reduction) and the investigation of the structure of the gauge orbit space. The final chapter is devoted to elements of quantum gauge theory including the discussion of the Gribov problem, anomalies and the implementation of the non-generic gauge orbit strata in the framework of Hamiltonian lattice gauge theory. The book is addressed both to physicists and mathematicians. It is intended to be accessible to students starting from a graduate level.

Function Theory on Symplectic Manifolds

Abstract: This is a book on symplectic topology, a rapidly developing field of mathematics which originated as a geometric tool for problems of classical mechanics. Since the 1980s, powerful methods such as Gromov's pseudo-holomorphic curves and Morse-Floer theory on loop spaces gave rise to the discovery of unexpected symplectic phenomena. The present book focuses on function spaces associated with a symplectic manifold. A number of recent advances show that these spaces exhibit intriguing properties and structures, giving rise to an alternative intuition and new tools in symplectic topology. The book provides an essentially self-contained introduction into these developments along with applications to symplectic topology, algebra and geometry of symplectomorphism groups, Hamiltonian dynamics and quantum mechanics. It will appeal to researchers and students from the graduate level onwards.

Buchsbaum Rings and Applications: An Interaction Between Algebra, Geometry and Topology

Abstract: and some examples.- 0 Some foundations of commutative and homological algebra.- I Characterizations of Buchsbaum modules.- II Hochster-Reisner theory for monomial ideals. An interaction between algebraic geometry, algebraic topology and combinatorics.- III On liaison among curves in projective three space.- IV Rees modules and associated graded modules of a Buchsbaum module.- V Further applications and examples.- Appendix On generalizations of Buchsbaum modules.- Notations.

Non-holonomic dynamics and Poisson geometry

Abstract: This is a survey of basic facts presently known about non-linear Poisson structures in the analysis of integrable systems in non-holonomic mechanics. It is shown that by using the theory of Poisson deformations it is possible to reduce various non-holonomic systems to dynamical systems on well-understood phase spaces equipped with linear Lie-Poisson brackets. As a result, not only can different non-holonomic systems be compared, but also fairly advanced methods of Poisson geometry and topology can be used for investigating them. Bibliography: 95 titles.

When symplectic topology meets Banach space geometry

Abstract: In this paper we survey some recent works that take the first steps toward establishing bilateral connections between symplectic geometry and several other fields, namely, asymptotic geometric analysis, classical convex geometry, and the theory of normed spaces.

Lecture Notes on Gradient Flows and Optimal Transport

Abstract: We present a short overview on the strongest variational formulation for gradient flows of geodesically λ-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures These notes are based on a series of lectures given by the second author for the Summer School “Optimal transportation: Theory and applications” in Grenoble during the week of June 22-26, 2009

Topological Dynamical Systems: An Introduction to the Dynamics of Continuous Mappings

Abstract: There is no recent elementary introduction to the theory of discrete dynamical systems that stresses the topological background of the topic. This book fills this gap: it deals with this theory as 'applied general topology'. We treat all important concepts needed to understand recent literature. The book is addressed primarily to graduate students. The prerequisites for understanding this book are modest: a certain mathematical maturity and course in General Topology are sufficient.

Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth

Abstract: Abstract. We establish local Calderón–Zygmund estimates for solutions to certain parabolic problems with irregular obstacles and nonstandard p(x,t)${p(x,t)}$ -growth. More precisely, we will show that the spatial gradient Du${Du}$ of the solution to the obstacle problem is as integrable as the obstacle ψ${\\psi }$ , i.e. |Dψ| p(·) ,|∂ t ψ| γ 1 ' ∈L loc q ⇒|Du| p(·) ∈L loc q ,foranyq>1,$ |D\\psi |^{p(\\,\\cdot \\,)},|\\partial _t\\psi |^{\\gamma _1^{\\prime }}\\in L^q_\\mathrm {loc}\\Rightarrow |Du|^{p(\\,\\cdot \\,)}\\in L^q_\\mathrm {loc},\\quad \\text{for any}~q>1, $ where γ 1 ' =γ 1 γ 1 -1${\\gamma _1^{\\prime }=\\frac{\\gamma _1}{\\gamma _1-1}}$ and γ 1 ${\\gamma _1}$ is the lower bound for p(·)${p(\\,\\cdot \\,)}$ .

Mathematical Tools for Shape Analysis and Description

Abstract: This book is a guide for researchers and practitioners to the new frontiers of 3D shape analysis and the complex mathematical tools most methods rely on. The target reader includes students, researchers and professionals with an undergraduate mathematics background, who wish to understand the mathematics behind shape analysis. The authors begin with a quick review of basic concepts in geometry, topology, differential geometry, and proceed to advanced notions of algebraic topology, always keeping an eye on the application of the theory, through examples of shape analysis methods such as 3D segmentation, correspondence, and retrieval. A number of research solutions in the field come from advances in pure and applied mathematics, as well as from the re-reading of classical theories and their adaptation to the discrete setting. In a world where disciplines (fortunately) have blurred boundaries, the authors believe that this guide will help to bridge the distance between theory and practice. Table of Contents: Acknowledgments / Figure Credits / About this Book / 3D Shape Analysis in a Nutshell / Geometry, Topology, and Shape Representation / Differential Geometry and Shape Analysis / Spectral Methods for Shape Analysis / Maps and Distances between Spaces / Algebraic Topology and Topology Invariants / Differential Topology and Shape Analysis / Reeb Graphs / Morse and Morse-Smale Complexes / Topological Persistence / Beyond Geometry and Topology / Resources / Bibliography / Authors' Biographies

Categorification of Lagrangian intersections on complex symplectic manifolds using perverse sheaves of vanishing cycles

Abstract: We study intersections of complex Lagrangian in complex symplectic manifolds, proving two main results. First, we construct global canonical perverse sheaves on complex Lagrangian intersections in complex symplectic manifolds for any pair of {\it oriented} Lagrangian submanifolds in the complex analytic topology. Our method uses classical results in complex symplectic geometry and some results from arXiv:1304.4508. The algebraic version of our result has already been obtained by the author et al. in arXiv:1211.3259 using different methods, where we used, in particular, the recent new theory of algebraic d-critical loci introduced by Joyce in arXiv:1304.4508. This resolves a long-standing question in the categorification of Lagrangian intersection numbers and it may have important consequences in symplectic geometry and topological field theory. Our second main result proves that (oriented) complex Lagrangian intersections in complex symplectic manifolds for any pair of Lagrangian submanifolds in the complex analytic topology carry the structure of (oriented) analytic d-critical loci in the sense of arXiv:1304.4508.

On solutions to the exterior Dirichlet problem for the minimal surfaceequation with catenoidal ends

Abstract: Abstract. In this paper we investigate the Dirichlet problem for the minimal surface equation on certain nonconvex domains of the plane. In our first result, we give, by an independent proof, a numerically explicit version of Williams' existence theorem. Our main result concerns the Dirichlet problem on exterior domains. It was shown by Krust (1989) and Kuwert (1993) that between two different solutions with the same normal at infinity there is a continuum of solutions foliating the space in between. We investigate the space of solutions further and show that, unless it is empty, it contains a maximal and a minimal solution if the boundary data is rectifiable. In the case of sufficiently smooth data we parametrize the set of solutions in terms of the extremal inclinations which the normal of the graph of a solution reaches at the boundary. We show that all theoretically possible values are realized including the horizontal position of the normal for the minimal and maximal solutions. We moreover give an example where the maximal and the minimal solution coincide so that there is exactly one with given normal at infinity. This answers a natural question which has not been touched in the previous papers.

Large Scale Versus Small Scale

Abstract: Recent research in coarse geometry revealed similarities between certain concepts of large scale geometry and topology It is less known that a small scale analog of topology has been developed much earlier in the form of the uniform category This paper is devoted to an exposition of analogies between basic concepts of topology (paracompactness, covering dimension), important ideas of coarse geometry (Property A of G Yu, asymptotic dimension of M Gromov), and notions from the uniform category (l 1-property, the uniform dimension)

Introduction to algebraic stacks

Abstract: These are lecture notes based on a short course on stacks given at the Isaac Newton Institute in Cambridge in January 2011. They form a self-contained

Characteristic classes on Grassmannians

Abstract: In this paper, we study the geometry and topology on the oriented Grassmann manifolds. In particular, we use characteristic classes and the Poincare duality to study the homology groups of Grassmann manifolds. We show that for k=2 or n \leq 8, the cohomology groups H*(G(k,n), R) are generated by the first Pontrjagin class, the Euler classes of the canonical vector bundles. In these cases, the Poincare duality: Hq(G(k,n), R) \to Hk(n-k)-q(G(k,n), R) can be expressed explicitly.

Generalized John disks

Abstract: We establish the basic properties of the class of generalized simply connected John domains.

Normal Crossings Singularities for Symplectic Topology

Abstract: In recent work, we introduced topological notions of simple normal crossings symplectic divisor and variety, showed that they are equivalent, in a suitable sense, to the corresponding geometric notions, and established a topological smoothability criterion for them. The present paper extends these notions to arbitrary normal crossings singularities, from both local and global perspectives, and shows that they are also equivalent to the corresponding geometric notions. In subsequent papers, we extend our smoothability criterion to arbitrary normal crossings symplectic varieties and construct a variety of geometric structures associated with normal crossings singularities in algebraic geometry.

A Note on Obata’s Rigidity Theorem

Abstract: In this note we present various extensions of Obata’s rigidity theorem concerning the Hessian of a function on a Riemannian manifold. They include general rigidity theorems for the generalized Obata equation, and hyperbolic and Euclidean analogs of Obata’s theorem. Besides analyzing the full rigidity case, we also characterize the geometry and topology of the underlying manifold in more general situations.

Nonexistence of positive radial solutions for a problem with singular potential

Abstract: Abstract. This article completes the picture in the study of positive radial solutions in the function space 𝒟 1,2 (ℝ N )∩L 2 (ℝ N ,|x| -α dx)∩L p (ℝ N )${{\\mathcal {D}^{1,2}({\\mathbb {R}^N}) \\cap L^2({{\\mathbb {R}^N}, | x |^{-\\alpha } dx})\\cap L^p({\\mathbb {R}^N})}}$ for the equation -Δu+A |x| α u=u p-1 inℝ N ∖{0}withN≥3,A>0,α>0,p>2.$- \\Delta u + \\frac{A}{| x |^\\alpha } u = u^{p-1} \\quad \\mbox{in } {\\mathbb {R}^N}\\setminus \\lbrace 0\\rbrace \\mbox{ with } N\\ge 3, A> 0, \\alpha > 0, p>2. $ An energy balance identity is employed to prove nonexistence of such solutions in the last remaining open region in the (α,p)${{(\\alpha , p)}}$ plane.

Geometry and topology of CC and CQ states

Abstract: We show that mixed bipartite CC and CQ states are geometrically and topologically distinguished in the space of states. They are characterized by non-vanishing Euler-Poincare characteristics on the topological side and by the existence of symplectic structures on the geometric side.

Some triviality results for quasi-einstein manifolds and einstein warped products

Abstract: In this paper we prove a number of triviality results for Einstein warped products and quasi-Einstein manifolds using different techniques and under assumptions of various nature. In particular we obtain and exploit gradient estimates for solutions of weighted Poisson-type equations and adaptations to the weighted setting of some Liouville-type theorems.