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(Selected Degree Final Projects written under our supervision)

[UCM18] G. Gallego: Introduction to symplectic geometry and integrable systems.

Abstract: The main goal of this work is to prove the Arnold-Liouville theorem, which gives a sufficient condition for a Hamiltonian mechanical system to be integrable by quadratures. To that end we define and develop the concepts involved in the theorem, giving some elementary notions of symplectic geometry and its application to Classical Mechanics.

Keywords: Symplectic geometry, integrable systems, Arnold-Liouville theorem, hamiltonian flows, Hamilton equations, Lie derivative, time-dependent vector fields.

Mathematical Subject Classification: 37J05, 37J35, 53D05, 58A05, 70H05.

To get a PDF file of the work (Spanish), click here.

[UCM18] M. Jaenada: The Schoenflies theorem.

Abstract: In this work we will prove the Schoenflies Theorem. We will introduce first the Jordan curve theorem and some variations; this is at the basis of Schoenflies theorem. Then we will prove the Shoenflies theorem in the polygonal case, and produce a polygonal approximation of any Jordan curve, which will give way to the final step of the proof. As an application we will study the theorem in the (real) projective plane, that shows how Jordan-Schoenflies fails in compact surfaces other than the sphere, and provides a criterion to distinguish Jordan curves.

Keywords: Jordan curve, Jordan theorem, polygonal Jordan curve, linear accessibility, polygonal approximation, Schoenflies theorem, fundamental group, (real) projective plane, Jordan curves in the projective plane.

Mathematical Subject Classification: 57N05, 57N50.

To get a PDF file of the work (Spanish), click here.

[UCM17] M.A. Berbel: The real symplectic group.

Abstract: In this work we study the real symplectic group consisting of linear endomorphisms that preserve an anti­symmetric bilinear form. A basic algebraic approach shows that its special group is the whole group and provides generators known as sym­plec­tic transvections. From the geometric viewpoint we see that the symplectic group is a Lie group to which a Lie algebra can be associated. Its orthogonal subgroup is a compact Lie subgroup isomorphic to the unitary group and, furthermore, it is a deforma­tion retract of the symplectic group. Analytical tools such as the exponential mapping, the logarithm and real powers of definite positive symmetric matrices are developed in the process.

Keywords: Symplectic group, transvection, Lie group, orthogonal subgroup, exponential mapping, real powers of matrices.

Mathematical Subject Classification: 15A16, 15A60, 22E15, 22E60, 51A50.

To get a PDF file of the work (Spanish), click here.

[UCM16] A. Alonso: The topology of division problems: envy-free fair division and consensus division.

Abstract: We study two division problems and the topology behind their solutions. The problems are fair division and consensus division, and the topo­logical results involved are the Brouwer Fixed Point Theorem and the Borsuk-Ulam Theorem. These theorems are deduced from their discrete versions: the Sperner Lemma and the Tucker Lemma. In fact, we prove a generalization of the latter: a weak versión of the Ky Fan lemma. The proofs involve the use of suitable graphs associated to polyhedra. Also we discuss the formal equivalences among all these results.

Keywords: Envy-free fair division, consensus division, Sperner's lemma, Brouwer's fixed point theorem, weak Ky Fan's lemma, Tucker's lemma, Borsuk-Ulam's theorem.

Mathematical Subject Classification: 52B11, 55M20, 55M25, 91A12, 91B02.

To get a PDF file of the work (Spanish), click here.

[UCM16] F. Coltraro: The Poincaré-Hopf theorem.

Abstract: We explore the relationships between functions —vector fields and real functions— defined on smooth manifolds and the topology of the manifolds themselves. We will mostly use tools from Differential Topology. Main results are the Poincaré-Hopf Index Theorem and the Gauss-Bonnet formula for hypersurfaces of even dimen­sion. Basically both results show that certain geometrical quantities —the total index of a vector field and the integral curvature— are invariants of the manifolds where they are defined. In order to obtain these theorems our main tool will be the Brouwer-Kronecker topological degree; with it we will be able to define the key notion of this article: the index of a vector field at an isolated singularity. Along the way we will also give a short introduction to Morse Theory, which in turn allows us to prove the Reeb Theorem. Finally, we study under which hypothesis we can be certain that non-zero vector fields defined on manifolds exist.

Keywords: Differential topology, topological degree, index of a vector field, Morse theory, non-zero vector fields, Poincaré-Hopf theorem, Gauss-Bonnet formula.

Mathematical Subject Classification: 57R19, 57R25, 54C20.

To get a PDF file of the work (Spanish), click here.

[UCM15] E. Fernández: Paracompactness and metrization.

Abstract: The aim of this report is to study the notions of Point Set Topology stated in the title. The topology of metric spaces is an important topic in different fields: differential topology, riemmanian geometry and Banach spaces, among others. The study of the metrization problem inherently entails an in-depth study of the concepts of normality and paracompactness. One of its most remarkable applications is the construction of continuous special functions (Urysohn functions, partitions of unity) which allow a better treatment of the topo­lo­gical properties of spaces. Among other results proved here we single out E. Michael's characteri­zations of paracompactess and Bing and Nagata-Smirnov metrization theorems; also Dugundji-Borsuk and Rudin theorems on mappings into locally convex real topological vector spaces.

Keywords: Metrization, normality, paracompactness, Stone theorem, Bing metrization theorem, Nagata-Smirnov metrization theorem, partition of unity.

Mathematical Subject Classification: 54D15, 54D20, 54E35, 54E30.

To get a PDF file of the work (Spanish), click here.

[UCM15] J. Porras: The Jordan curve theorem and planar graphs.

Abstract: The author studies the Jordan curve theorem, with special emphasis on some aspects connected with Graph Theory, namely planarity. He explains how the famous non-planar graphs K5 and K3,3 contain a relevant part of the topological essence of the Jordan Theorem. This is half of the famous Kuratowski theorem, the other half is that those graphs are the minimal non-planar exam­ples. There, he discusses carefully the usual­ly understated constructions behind the proof of that second half, providing rigurous arguments for some facts that are usually taken for granted. In particular, extreme care is given to the topological properties of plane graphs with polygonal edges, their faces and the boundaries of the latter.

Keywords: Jordan curve theorem, Brouwer fixed point theorem, planar graph, complete graph K5, complete bipartite graph K3,3, Kuratowski theorem.

Mathematical Subject Classification: 05C10, 57M15, 57N05.

To get a PDF file of the work (Spanish), click here.

[UCM15] F. Criado: Azimuth: design and development of a non-euclidean video game.

Main Tutor: Marco Antonio Gómez.

Abstract: The author proposes a uniparametric algebraic model of hyperbolic, euclidean and elliptic geometries considering the real curvature k as the parameter. This will express the intuition that these models are similar for small values of |k|. He studies with detail the geometric invariants of the models: first fundamental form, cur­vature, geodesics and parallel transport. Then, the model is applied to define some polyhedral surfaces, an extended notion of geodesics in them and some comput­ation­ally efficient methods for their simulation involving computations of geodesic distances.

Keywords: Differential geometry, non-euclidean geometry, video game development, Unity, computational geometry, computer graphics.

Mathematical Subject Classification: 53A05, 53A35, 53B20, 53C23.
Computing Classification System: Geometric topology, Computational geometry, Algorithmic game theory, Software design engineering.

To get a PDF file of the work (Spanish), click here.

[UCM15] M. Pulido: Singular homology.

Abstract: The author gives a direct and self­contained presentation of Singular Ho­mo­lo­gy: construction, relative homology, exact se­quen­ces, Mayer-Vietoris. Then, he applies it to deduce some important classical results as the Brouwer fixed point theorem, local in­va­riance of dimension and in­va­riance of domain, or the Jordan-Brouwer separation theorem. He also compares Singular Homology with Simplicial Homology. He briefly introduces the notion of mapping degree, presenting its main properties and using them to prove the Brouwer hairy ball theorem. As a final point, he presents a gen­er­al­iza­tion of the Jordan-Brouwer theorem that ex­plains the difference between homology and homotopy: the Alexander horned sphere shows that Schoenflies theorem fails in dimension bigger than 2, but the failure is homotopic, not homologic.

Keywords: Singular homology, exact sequence of a pair, Mayer-Vietoris, degree, invariance theorems, Brouwer theorems.

Mathematical Subject Classification: 55N10, 57N65, 55Q99, 55M25.

To get a PDF file of the work (Spanish), click here.

[UCM13] I. González: The Brouwer-Kronecker degree.

Abstract: The author studies the Brouwer-Kronecker degree, using de Rham co­ho­mo­lo­gy and integration of forms on manifolds. Two first important applications are: the fixed point theorem and the so-called hairy ball theorem, both due to Brouwer. But the main motivation are the homotopy groups of spheres. These homotopy groups are all trivial for or­der smaller than dimension, as is ex­plained by using the Sard-Brown theorem. For order equal to dimension the Brouwer-Kronecker degree gives the solution: the group is cyclic infinite, a famous theorem by Hopf which is the central result here. For order bigger than dimension the situation becomes much more complicated, except the case of the circle (all homotopy trivial for order bigger that 1, the proof of which is included for completeness). Still, degree gives a method to understand something more through the Hopf invariant. After defining this invariant and obtaining its basic properties, the author computes the Hopf invariant of the famous Hopf fibration, to conclude that the third homotopy group of the 2-sphere is infinite.

Keywords: Homotopy groups, de Rham cohomology, Brouwer-Kronecker degree, Hopf invariant, Hopf fibration.

Mathematical Subject Classification: 57R19, 57R35, 58A12.

To get a PDF file of the work (Spanish), click here.

[UCM13] J.A. Rojo: Approximation and homotopy.

Abstract: In this paper the author deals with differentiable aproximation and ho­mo­to­py of continuous and proper mappings whose target is a manifold with boundary. The initial motivation is the direct com­pu­ta­tion by differentiable methods of the de Rham cohomology (with or without compact supports) for such a manifold, as is usually done for boundaryless manifolds. When there is no boundary, tubular neigh­bor­hoods and differentiable retractions are the key tool, but as is well known, there cannot be differentiable retracts in the presence of a boundary. To amend this requires a careful examination of the construction of collars in manifolds with boundary, which will provide the means to find differentiable pseudoretracts. In addition, one must embed any given manifold with boundary into a boundaryless manifold of the same dimension. This is easy up to diffeo­mor­phism, but it is achieved here without any modification of either the manifold or the ambiente space.

Keywords: Boundary, approximation of mappings, homotopy, tubular neighborhood, retraction, collar, pseudoretraction.

Mathematical Subject Classification: 57R19, 57R35, 58A12.

To get a PDF file of the work (Spanish), click here.