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**DFP's**

(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 theArnold-Liouville theorem,which gives a sufficient condition for aHamiltonian mechanical systemto beintegrable 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 theSchoenflies Theorem. We will introduce first theJordan curve theoremand 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 realsymplectic groupconsisting of linear endomorphisms that preserve an antisymmetric bilinear form. A basic algebraic approach shows that itsspecial groupis the whole group and provides generators known assymplectic transvections.From the geometric viewpoint we see that the symplectic group is aLie groupto which aLie algebracan be associated. Itsorthogonal subgroupis a compact Lie subgroup isomorphic to the unitary group and, furthermore, it is adeformation retractof 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 twodivision problemsand the topology behind their solutions. The problems arefair divisionandconsensus division,and the topological results involved are theBrouwer Fixed Point Theoremand theBorsuk-Ulam Theorem. These theorems are deduced from their discrete versions: theSperner Lemmaand theTucker Lemma. In fact, we prove a generalization of the latter: a weak versión of theKy 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 thePoincaré-Hopf Index Theoremand theGauss-Bonnet formulafor hypersurfaces of even dimension. Basically both results show that certain geometrical quantities —thetotal index of a vector fieldand theintegral curvature— are invariants of the manifolds where they are defined. In order to obtain these theorems our main tool will be theBrouwer-Kronecker topological degree;with it we will be able to define the key notion of this article: the index of a vector field at anisolated singularity. Along the way we will also give a short introduction toMorse Theory,which in turn allows us to prove theReeb Theorem. Finally, we study under which hypothesis we can be certain thatnon-zero vector fieldsdefined 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 themetrization probleminherently entails an in-depth study of the concepts ofnormalityandparacompactness. 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 outE. Michael'scharacterizations of paracompactess andBingandNagata-Smirnov metrization theorems; alsoDugundji-BorsukandRudin theoremson mappings intolocally 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 theJordan curve theorem,with special emphasis on some aspects connected with Graph Theory, namelyplanarity. He explains how the famous non-planar graphsK_{5}andK_{3,3}contain a relevant part of the topological essence of the Jordan Theorem. This is half of the famousKuratowski theorem,the other half is that those graphs are theminimal 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 withpolygonal edges,theirfacesand the boundaries of the latter.

Keywords:Jordan curve theorem, Brouwer fixed point theorem, planar graph, complete graphK_{5}, complete bipartite graphK_{3,3}, Kuratowski theorem.

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

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

Main Tutor:

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

Abstract:The author proposes a uniparametric algebraic model of hyperbolic, euclidean and elliptic geometries considering the real curvaturekas 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 theBrouwer fixed point theorem, local invariance of dimensionandinvariance of domain,or theJordan-Brouwer separation theorem. He also compares Singular Homology with Simplicial Homology. He briefly introduces the notion ofmapping degree,presenting its main properties and using them to prove theBrouwer 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 theBrouwer-Kronecker degree,using de Rham cohomology and integration of forms on manifolds. Two first important applications are: thefixed point theoremand the so-calledhairy 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 theHopf invariant. After defining this invariant and obtaining its basic properties, the author computes the Hopf invariant of the famousHopf 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 mappingswhose 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 anddifferentiable retractionsare 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 finddifferentiable pseudoretracts. In addition, one must embed any given manifold with boundary into a boundaryless manifold of the same dimension. This is easyup 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.