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In this work, we present a brief introduction to Symplectic Geometry relating its origin with the Physics. Then we present the formal definition of symplectic manifold and some important results, with this we consider a function AH;N defined in the Cartesian product of the symplectic manifold (ℝ2n; ω0). Here we make an analysis with the fact that the critical points of this function are related in a biunivocal way to the fixed points of the flow Φt of the symplectic manifold (ℝ2n; ω0)in time t = 1 this thanks to the Hamiltonian diferential equations via the generating functions., En este trabajo presentamos una breve introducción a la Geometría Simpléctica mostrando su origen relacionado con la Física. Luego presentamos la definición formal de variedades simplécticas y algunos resultados importantes, para ello consideramos una función AH;N definida en el producto cartesiano de la variedad simpléctica (ℝ2n; ω0) y hacemos un análisis con el comportamiento de esta función de donde concluimos que los puntos críticos de esta función se relacionan de manera biunívoca con los puntos fijos del flujo Φt de la variedad Simpléctica (ℝ2n; ω0) en el tiempo t = 1 esto debido a las ecuaciones diferenciales Hamiltonianas via las funciones generadoras.

We prove that direct limits of Lie algebroids and their prolongations can be endowed with structures of convenient spaces. Probamos que se puede definir estructuras de espacios convenientes sobre límites directos de algebroides de Lie y sus prolongaciones.

The work of Hausel proves that the Bia\l{}ynicki-Birula stratification of the moduli space of rank two Higgs bundles coincides with its Shatz stratification. He uses that to estimate some homotopy groups of the moduli space of $k$-Higgs bundles of rank two. Unfortunately, those two stratifications do not coincide in general. Here, the objective is to present a different proof of the stabilization of the homotopy groups of $\mathcal{M}^k(2,d)$, and generalize it to $\mathcal{M}^k(3,d)$, the moduli spaces of $k$-Higgs bundles of degree $d$, and ranks two and three respectively, using the results from the works of Hausel and Thaddeus, among other tools., Comment: 19 pages, v3 updated version, corrected typos; to appear in Revista Colombiana de Matem\'aticas

A remarkable achievement of algorithmic topology is A.A.Markov’s theorem on the unsolvability of the homeomorphism problem for manifolds. Boone, Haken and Poenaru extended Markov’s original proof to the case of closed smooth manifolds. One of their initial difficulties was the introduction of a natural finite representation of a differentiable and/or combinatorial manifold. In this paper we extend this representation to compact smooth manifolds and propose an extension to smooth manifolds. Keywords : Computability and recursion theory, algorithmic topology, smooth manifolds. To cite this article: C.M Parra, J. Suarez Ramirez, Representacion finita de variedades compactas, Rev. Integr. Temas Mat. 33 (2015), No. 2, 97–105.

In this thesis, we present the fundamental ideas of Morse theory. Namely, how it allows us to study the topology of a smooth manifold by means of the properties of ``special' smooth functions, called Morse functions, whose critical points are all non-degenerate. The core of the theory is to see how the topology of the sublevel sets changes as one passes through each critical point of a Morse function, namely by attaching cells of certain dimensions dictated by the number of negative eigenvalues of the Hessian of the function. We will review the Morse lemma and two fundamental theorems of the theory, as well as show the existence of (many) Morse functions on any smooth manifold. We will give some applications, such as the computation of the homology groups of the spheres and the complex projective spaces, and study Morse functions on knots.

We study the Yamabe Problem in a special class of Riemannian manifolds, the Ricci solitons. We also explore contact and symplectic manifolds admitting a compatible Ricci soliton, where we obtain some observations. With respect to Poissons geometry, we provide local expressions for Poisson bivectors and the corresponding symplectic forms with broken Lefschetz and wrinkled singularities in dimensions 4, and 6, and discuss the higher dimension case. With respect to the Yamabe Problem on compact Ricci solitons the main result that we obtained is: Theorem There exists a unique U(2)-invariant solution to the Yamabe equation on CP2#-CP2 with the Koiso-Cao metric. We also explore the Ricci soliton equation. We use Hamiltonian and Liouville vector fields to derive some results concerning a Ricci soliton with a compatible symplectic structure. The main result obtained: Theorem Let (M, w) be a symplectic manifold of dimension greater than 2. If w has compatible Ricci soliton g determined by a Hamiltonian or a Liouville holomorphic vector field, then g is Einstein. We also constructed singular Poisson structures on manifolds of dimension 4 and 6, where the singularties are given by a broken Lefschetz fibration or a wrinkled fibration. The main results are: Theorem A closed, orientable, smooth 4-manifold equipped with a wrinkled fibration admits a complete Poisson structure. The fibres of the fibration are leaves of the symplectic foliation and both structures share the same singularities. Theorem A generalized broken Lefschetz fibration admits a Poisson structure compatible with the fibration structure. Also, generalized wrinkled fibrations in dimension 6 admit compatible Poisson structures.

We establish the existence of a symmetry within the Gromov-Witten theory of $\mathbb{CP}^n$ and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of the permutohedron. This symmetry expresses some difficult to compute invariants in terms of others less difficult to compute. We focus on enumerative implications; in particular this technique yields a one line proof of the uniqueness of the rational normal curve. Our method involves a study of the toric geometry of the permutohedron, and degeneration of Gromov-Witten invariants., Comment: 15 pages

We define a prequantization in the category of smooth diffeological spaces ( a category which includes finite an infinite dimensional systems), in the case where the symplectic form is exact, extending, for this case, the results of Kirillov, Kostant and Souriau which are devoloped for finite dimensional Physical systems. Definimos una precuantización en la categoría de espacios difeológicos suaves (una categoría que incluye sistemas de dimensión finita e infinita), en el caso en que la forma simpléctica es exacta, para este caso, los resultados de Kirillov, Kostant y Souriau que son desarrollados para sistemas físicos de dimensión finita.

An extension of J. Palis’ λ-lemma is obtained for one case of non-transversal intersection of the stable and unstable manifolds of an hyperbolic fixed point of a C∈-local diffeomorphism of R2. Se demuestra una generalización del λ-lemma de J. Palis para un caso de intersección no transversal de las variedades estable e inestable de un punto fijo hiperbólico de un difeomorfismo local de clase C∈ del plano en sí mismo.

The thesis is divided into two main sections. The first one is a monograph on the Atiyah-Singer theorem. The second part is a research that aims to find a relationship between Morse theory and the heat equation Maestría

We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2,C). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons., 21 pages; v2. Accepted for publication in Internat. J. Math. v3. Mistake in polynomial of Theorem 1.1(3) corrected

In the first part of this paper we give three definitions equivalent of theconcept of k-dimensional foliation on a complex analytic manifold. Below, we provideone quarter equivalent definition for foliations of dimension one, using the concept of holomorphic vector field. En la primera parte del presente trabajo damos tres definicionesequivalentes del concepto de foliación de dimensión k sobre una variedad analíticacompleja. A continuación, brindamos una cuarta definición equivalente para foliacionesde dimensión uno, usando el concepto de campo vectorial holomorfo.

Singular lagrangian, classical mechanics, special relativity.

In this paper, we study involute curve of biharmonic curve in the special three-dimensional o-Ricci symmetric para-Sasakian manifold P. We obtain matrix representation for involute curve of biharmonic curve in terms of curvature and torsion of biharmonic curve in the special three-dimensional Á¡Ricci symmetric para-Sasakian manifold P. 177-185 talatkorpinar@gmail.com essin.turhan@gmail.com vasil@firat.edu.tr

In this paper, we study matrix representations of focal curves in terms of biharmonic curves in the special three-dimensional o−Ricci symmetric para-Sasakian manifold P. We construct new parametric equations of focal curves in terms of matrix representations in the special three-dimensional o −Ricci symmetric para-Sasakian manifold P. 195-201 talatkorpinar@gmail.com essin.turhan@gmail.com

Let pC : C = CN → N be the bundle of linear connections on a smooth manifold N and let pM : M ! N be the bundle of pseudo-Riemannian metrics of a given signature (n₊ ; n-) , n₊ n- = n = dimN on N. The structure of the first-order Lagrangians defined on the bundle M x N C → N that are invariant under the natural action of the diffeomorphisms of N, is determined.

Given a closed, orientable Riemannian manifold, we study the stable norm on the real homology groups. In particular, for $n\geq 2$ we prove that a Riemannian $n$-torus, which has the same stable norms as a flat $n$-torus on the first and $n-1$ homology groups, is in fact isometric to the flat torus.

On moduli of smoothness of Banach spaces. Banas, Josef. 18 págs. semestral Nivel analítico

Let be a complete manifold with a ”parallel” foliation . We show that if this foliation and its orthogonal foliation admit eache one an ”unisection”, then the manifold is riemannian product of transverses leaves.

We prove that 2 is a Lewowicz number of every linear Anosov diffeomorphism on the torus. This result is independent of any linear metric and provides an explicit Lyapounov function for the diffeomorfisms.

The total absolute curvature of manifolds has been discussed several times and some papers about it already belong to the classical literature on that topics. All of them deal with manifolds inmersed in Riemannian spaces.

Para el potencial generalizado de Morse es posible calcular en forma exacta las funciones de onda para los estados acotados y del continuo así como la función de Green asociada con el operador Hamiltonano. For the generalizad. Morse potencial it is possible to calculate the exact wave functions for the bound and continuum states as well as the Green function associated with the Hamiltonian.

In this paper we survey some recent variational principies, which have proved to be very useful in the applications to the theory of differential equations, both ordinary and partial. We start with a basic principle due to Ekeland [4}, which provides new proofs to the well known minimax theorems of Ambrosetti - Rabinowitz [2} and Rabinowitz{7}, {8}. For proofs of these results we refer to{8}. We also mention some applications to semilinear elliptic equations. El artículo no presenta resumen

We extend the work of Sternberg and Weinstein concerning the phase space of a particle in a Yang-Mills field.

A classical theorem in mechanics states that a Hamiltonian which is invariant under a symmetry group admits additional integral s of motion. Thi s paper investigates the converse of the above theorem. If a Hamiltonian admits integrals then a symmetry can be constructed and the flaw studied on a quotient space. The quotient space is shown to be symplectic and the resulting flow Hamiltonian. The constructions used are similar to the recent constructions of Nehoroshev, Marsden and Weinstein and Meyer. The general theory presented is used to give an intrinsic derivation of Hamilton's equations of motion. Al so special local coordinates are given which display the integrals in a simple form.

In this article we prove some properties relatives to inmersions, and almost-Hermitian submersions on W4-manifolds.

In [3] C. Ruiz gave a definition for "fibration" on the category Annb of Banach rings which is closely related to a functorGR : Annb→△ Gr which takes its values on the category of the simplicial groups. It was assumed that the fibration / of Annb are the Banach ring homomorphisms such that the homomorphisms GR(f) of simplicial groups, are Kan fibrations. This notion happened to be equivalent (Cf. [3; p. 169]) to the notion of Serre fibration given by KAROUBI and VILLAMAYOR [4].