Topological manifolds

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  • In this book I present differential geometry and related mathematical topics with the help of examples from physics. It is well known that there is something strikingly mathematical about the physical universe as it is conceived of in the physical sciences. The convergence of physics with mathematics, especially differential geometry, topology and global analysis is even more pronounced in the newer quantum theories such as gauge field theory and string theory. The amount of mathematical sophistication required for a good understanding of modern physics is astounding.

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  • Lecture notes on topology and geometry present on: General Topology, Algebraic Topology, Differential Topology, Differential Geometry, Differentiable manifolds,... Invite you to refer to the lecture content more learning materials and research.

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  • Without much exaggeration, it can be said that only one important topological concept came to light before Poincar¶e. This was the Euler characteristic of surfaces, whose name stems from the paper of Euler (1752) on what we now call the Euler polyhedron formula. When writing in English, one usually expresses the formula as

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  • Annals of Mathematics This paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. In [CM3]–[CM5] we describe the case where the surfaces are topologically disks on any fixed small scale. Although the focus of this paper, general planar domains, is more in line with [CM6], we will prove a result here (namely, Corollary III.

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  • Classical differential geometry is the approach to geometry that takes full advantage of the introduction of numerical coordinates into a geometric space. This use of coordinates in geometry was the essential insight of Rene Descartes that allowed the invention of analytic geometry and paved the way for modern differential geometry. The basic object in differential geometry (and differential topology) is the smooth manifold. This is a topological space on which a sufficiently nice family of coordinate systems or "charts" is defined.

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  • Holomorphic disks and topological invariants for closed three-manifolds ´ ´ ´ By Peter Ozsvath and Zoltan Szabo* Abstract The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y , equipped with a Spinc structure. Given a Heegaard splitting of Y = U0 ∪Σ U1 , these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Σ relative to certain totally real subspaces associated to U0 and U1 . 1. Introduction Let Y be a connected, closed, oriented three-manifold, equipped with a Spin structure s. ...

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  • Let X be a projective manifold and f : X → X a rational mapping with large topological degree, dt λk−1 (f ) := the (k − 1)th dynamical degree of f . We give an elementary construction of a probability measure µf such that d−n (f n )∗ Θ → µf for every smooth probability measure Θ on X. We show t that every quasiplurisubharmonic function is µf -integrable. In particular µf does not charge either points of indeterminacy or pluripolar sets, hence µf is f -invariant with constant jacobian f ∗ µf = dt µf...

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  • Let M be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group G. If m0 , n0 are the dimensions of the maximal lightlike subspaces tangent to M and G, respectively, where G carries any bi-invariant metric, then we have n0 ≤ m0 . We study G-actions that satisfy the condition n0 = m0 .

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  • SAMUEL EILENBERG. Automata, Languages, and Machines: Volumes A and B MORRIS HIRSCH N D STEPHEN A SMALE. Differential Equations, Dynamical Systems, and Linear Algebra WILHELM MAGNUS. Noneuclidean Tesselations and Their Groups FRANCOIS TREVES. Linear Partial Differential Equations Basic WILLIAM BOOTHBY. Introduction to Differentiable Manifolds and Riemannian M. An Geometry ~ A Y T O N GRAY. Homotopy Theory : An Introduction to Algebraic Topology ROBERT ADAMS. A. Sobolev Spaces JOHN BENEDETTO. J. Spectral Synthesis D. V. WIDDER.

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  • The phenomenon of Mirror Symmetry, in its “classical” version, was first observed for Calabi-Yau manifolds, and mathematicians were introduced to it through a series of remarkable papers [20], [13], [38], [40], [15], [30]. Some very strong conjectures have been made about its topological interpretation – e.g. the Strominger-Yau-Zaslow conjecture. In a different direction, the framework of mirror symmetry was extended by Batyrev, Givental, Hori, Vafa, etc. to the case of Fano manifolds.

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  • Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and min-max schemes, jointly with the compactness result of [35]. 1.

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  • Characteristic cohomology classes, defined in modulo 2 coefficients by Stiefel [26] and Whitney [28] and with integral coefficients by Pontrjagin [24], make up the primary source of first-order invariants of smooth manifolds. When their utility was first recognized, it became an obvious goal to study the ways in which they admitted extensions to other categories, such as the categories of topological or PL manifolds; perhaps a clean description of characteristic classes for simplicial complexes could even give useful computational techniques.

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  • R´sum´ anglais e e For a hyperbolic metric on a 3-dimensional manifold, the boundary of its convex core is a surface which is almost everywhere totally geodesic, but which is bent along a family of disjoint geodesics. The locus and intensity of this bending is described by a measured geodesic lamination, which is a topological object.

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  • For diffeomorphisms of smooth compact finite-dimensional manifolds, we consider the problem of how fast the number of periodic points with period n grows as a function of n. In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily fast growth is possible; in fact, the first author has shown that arbitrarily fast growth is topologically (Baire) generic for C 2 or smoother diffeomorphisms.

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