Extended Abstracts GEOMVAP 2019: Geometry, Topology, Algebra, and Applications; Women in Geometry and Topology (Trends in Mathematics)

This book comprises an overview of twelve months of intense activity of the research group Geometry, Topology, Algebra, and Applications (GEOMVAP) at the Universitat Politècnica de Catalunya (UPC). Namely, it contains extended abstracts of the group meeting in Cardona and of the international Workshop of Women in Geometry and Topology aligned with a series of workshops in the topic. As such, it includes a panoramic view of the main research interests of the group which focus on varieties and manifolds from the algebraic, topological and differential perspective with a view towards applications. The GEOMVAP group has a long tradition working on various interfaces of algebra, geometry and topology. In the last decade, the group has become active contributor in interdisciplinary science and it is now focused on both a theoretical point of view and the transversal applications to several disciplines including Robotics, Machine Learning, Phylogenetics, Physics and Celestial Mechanics. The increasing interdisciplinarity of modern research and the fact that the boundaries between different areas of mathematics are vanishing, with a constant transfer of problems and techniques between them, makes it difficult to progress without a multidisciplinary approach. GEOMVAP gathers together experts in Algebraic, Symplectic and Arithmetic Geometry to stimulate the interaction between them and to allow the study of each object from different points of view. The book aims at established researchers, as well as at PhD and postdoctoral students who want to learn more about the latest advances in pure and applied Geometry and Topology. Preface Contents mathbbQ-Hilbert Functions of Multiplier and Test Ideals 1 Introduction 2 mathbbQ-Good Filtrations 3 Poincaré Series of Multiplier and Test Ideals 3.1 Multiplier Ideals 3.2 Test Ideals References Up-to-Homotopy Algebras with Strict Units 1 Introduction and Main Result 2 Ingredients 2.1 Restriction Operations 2.2 A Kan-Like Structure References Multisymplectic Lagrangian Models in Gravitation 1 Introduction 2 Geometric Structures: Jet Bundles and Multivector Fields 3 Einstein–Hilbert Model (Without Sources) 4 Einstein–Palatini (Metric-Affine) Model (Without Sources) References Computing the Distance to the Stochastic Part of Phylogenetic Varieties 1 Introduction 2 Phylogenetic Varieties 3 Distance to the Stochastic Phylogenetic Regions 3.1 Computations and Conclusions References Generating Embeddable Matrices Whose Principal Logarithm is Not a Markov Generator 1 Introduction and Preliminaries 2 SS Embeddable Matrices Whose Principal Logarithm is Not a Generator References Hamilton-Jacobi Theory and Geometric Mechanics 1 The Perspective of Geometric Mechanics 2 Hamiltonian Mechanics and Hamilton–Jacobi Theory 3 A General Framework for Hamilton–Jacobi Equation 4 Slicing of Hamiltonian Systems 5 Slicings in Fibered Manifolds. Lagrangian and Hamiltonian Mechanics References Legendrian Knots in Contact 3-Manifolds 1 Definitions, Examples and Classification Results 2 Questions and Open Problems References Topological Degree and Periodic Orbits of Semi-dynamical Systems 1 Introduction 2 Periodic Orbits References Presentation of Symplectic Mapping Class Group of Rational 4-Manifolds 1 Preliminaries and Previous Results 2 Main Results 3 Sketch of the Proof 4 Some Remarks References On Several Classes of Ricci Tensor 1 Introduction 2 Geometry 3 Examples References Rank Conditions on Phylogenetic Networks 1 Introduction and Preliminaries 2 Invariants for the General Markov Model 3 Invariants for Equivariant Models References A Contact Geometry Approach to Symmetries in Systems with Dissipation 1 Introduction 2 Contact Manifolds and Contact Hamiltonian Systems 3 Symmetries and Dissipation Laws for Contact Hamiltonian Systems 4 Example: Motion in a Gravitational Field with Friction References Dimension Formulas for the Cohomology of Arithmetic Groups 1 Introduction 2 Borel-Serre Compactification 3 Eisenstein and Cuspidal Cohomology 4 Euler Characteristic References Do Overtwisted Contact Manifolds Admit Infinitely Many Periodic Reeb Orbits? 1 Introduction 2 Contact Geometry 3 b-Contact Manifolds 4 It Is a Trap 5 Overtwisted Disk in b-Contact Manifolds References Topological Radiomics (TOPiomics): Early Detection of Genetic Abnormalities in Cancer Treatment Evolution 1 Introduction 2 Methods 3 Experiments 4 Conclusions References Mixed Multiplier Ideals and the Topological Type of a Plane Curve 1 Introduction 2 Some Definitions 3 Topological Type References Geometry of Non-holonomic Distributions 1 Introduction 2 Notation, Natural Connection and Fundamental Forms 2.1 The Metric gmathcalD and the Connection mathcalD 2.2 The Second Fundamental Form (Gauss, sim1830, Reinhart, 1977, 1983) 3 Geodesics and Geodesic Invariance 3.1 Totally Geodesic Distributions 4 Other Results and Commentaries References What If It Contains a Linear Subspace? 1 Introduction 2 Unirational Hypersurfaces and Complete Intersections 3 Numerical Conditions for K-Linear Spaces in Complete Intersections References Reeb Embeddings and Universality of Euler Flows 1 Introduction 1.1 Hydrodynamics 1.2 Contact Geometry 2 Reeb Embeddings 3 Applications References The Continuous Rank Function for Varieties of Maximal Albanese Dimension and Its Applications 1 Statement of the Results 2 Continuous Rank Function 2.1 Set Up: Pardini's Covering Trick 2.2 Continuous Rank 3 Applications 3.1 BCPS Inequalities 3.2 Classification of the Limit Cases References Developable Surfaces with Prescribed Boundary 1 Introduction 2 The Space of Developable Surfaces 3 The Boundary of a Developable Surface 4 Future Continuation References Configuration Space of a Textile Rectangle 1 Introduction 2 Configuration Space of a Textile Rectangle Using 4 Points 3 Configuration Space of an N-Points Textile Rectangle References The 4/3 Problem for Germs of Isolated Plane Curve Singularities 1 Introduction 2 Deformation Theory 3 Solutions to Dimca and Greuel Question References When Is a Complete Ideal in a Rational Surface Singularity a Multiplier Ideal? 1 Introduction 2 A Reformulation of the Problem via Antinef Closures 3 Working in a Fixed Log-Resolution 4 Comparing Log-Resolutions References "This book comprises an overview of twelve months of intense activity of the research group Geometry, Topology, Algebra, and Applications (GEOMVAP) at the Universitat Politècnica de Catalunya (UPC). Namely, it contains extended abstracts of the group meeting in Cardona and of the international Workshop of Women in Geometry and Topology aligned with a series of workshops in the topic. As such, it includes a panoramic view of the main research interests of the group which focus on varieties and manifolds from the algebraic, topological and differential perspective with a view towards applications. The GEOMVAP group has a long tradition working on various interfaces of algebra, geometry and topology. In the last decade, the group has become active contributor in interdisciplinary science and it is now focused on both a theoretical point of view and the transversal applications to several disciplines including Robotics, Machine Learning, Phylogenetics, Physics and Celestial Mechanics. The increasing interdisciplinarity of modern research and the fact that the boundaries between different areas of mathematics are vanishing, with a constant transfer of problems and techniques between them, makes it difficult to progress without a multidisciplinary approach. GEOMVAP gathers together experts in Algebraic, Symplectic and Arithmetic Geometry to stimulate the interaction between them and to allow the study of each object from different points of view. The book aims at established researchers, as well as at PhD and postdoctoral students who want to learn more about the latest advances in pure and applied Geometry and Topology" -- Pàgina 4 de la coberta