118 research outputs found
Computation and Physics in Algebraic Geometry
Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra.
First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case.
Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature.
Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry
Recommended from our members
Highly symmetric embeddings of graphs on surfaces
This thesis considers highly symmetric maps, that is embeddings of graphs in surfaces such that the automorphism group is “large”. This may be when the automorphism group of the map acts regularly on the flag-set of the map, as for the fully regular maps studied in Part I. In contrast, Part II focusses on a class of maps where the automorphism group has (up to) two orbits on the flag-set and may not be edge-transitive.
Part I is dedicated to advancing the understanding of fully regular maps with external symmetries. Chapter 2 proves that for arbitrary valency greater than three, a fully regular map with Trinity symmetry exists, extending the previously-known existence of such a map for every even valency. Chapter 3 addresses a group of operators which acts on fully regular maps whose automorphism group is isomorphic to SL(2, 2^α). The group of operators, which depends on the value of α and is defined more precisely in Chapter 3, includes the dual and Petrie operators as well as the allowable hole operators. One approach is by exploring the orbits of this group as it acts on the space of all maps with automorphism group isomorphic to SL(2, 2^α) for the given α. A detailed investigation is presented for the group of operators acting on the set consisting of all maps with automorphism group A5 which is isomorphic to SL(2, 4).
In Part II, the focus is on edge-biregular maps. These maps can be identified with group presentations which have a particular form, namely they are generated by four involutions which partition into two distinct sets each consisting of a pair of commuting involutions. Edge-biregular maps correspond to the most symmetric examples of maps with bipartite medial graph. By the definition, each edge-biregular map inherits a two-colouring on the edges, and so long as the map is not degenerate in some way, both the valency and the face length are even. In Chapter 4 these maps are introduced, foundations are laid and degeneracies are addressed. Chapter 5 is a partial classification covering edge-biregular maps whose colour-preserving automorphism group is dihedral, and/or whose surface has Euler characteristic which is either non-negative or negative and prime. The context for Chapter 6 is edge-biregular maps whose underlying group is symmetric or alternating. A genuinely edge-biregular map is an edge-biregular map which (when disregarding the colouring of edges) is not a fully regular map. The chapter includes a proof that, with the exception of some small cases, a genuinely edge-biregular map of every feasible type exists such that the colour preserving automorphism group is symmetric or alternating.</br
Efficient finite element methods for solving high-frequency time-harmonic acoustic wave problems in heterogeneous media
This thesis focuses on the efficient numerical solution of frequency-domain wave propagation problems using finite element methods. In the first part of the manuscript, the development of domain decomposition methods is addressed, with the aim of overcoming the limitations of state-of-the art direct and iterative solvers. To this end, a non-overlapping substructured domain decomposition method with high-order absorbing conditions used as transmission conditions (HABC DDM) is first extended to deal with cross-points, where more than two subdomains meet. The handling of cross-points is a well-known issue for non-overlapping HABC DDMs. Our methodology proposes an efficient solution for lattice-type domain partitions, where the domains meet at right angles. The method is based on the introduction of suitable relations and additional transmission variables at the cross-points, and its effectiveness is demonstrated on several test cases. A similar non-overlapping substructured DDM is then proposed with Perfectly Matched Layers instead of HABCs used as transmission conditions (PML DDM). The proposed approach naturally considers cross-points for two-dimensional checkerboard domain partitions through Lagrange multipliers used for the weak coupling between subproblems defined on rectangular subdomains and the surrounding PMLs. Two discretizations for the Lagrange multipliers and several stabilization strategies are proposed and compared. The performance of the HABC and PML DDM is then compared on test cases of increasing complexity, from two-dimensional wave scattering in homogeneous media to three-dimensional wave propagation in highly heterogeneous media. While the theoretical developments are carried out for the scalar Helmholtz equation for acoustic wave propagation, the extension to elastic wave problems is also considered, highlighting the potential for further generalizations to other physical contexts. The second part of the manuscript is devoted to the presentation of the computational tools developed during the thesis and which were used to produce all the numerical results: GmshFEM, a new C++ finite element library based on the application programming interface of the open-source finite element mesh generator Gmsh; and GmshDDM, a distributed domain decomposition library based on GmshFEM.Cette thèse porte sur la résolution numérique efficace de problèmes de propagation d'ondes dans le domaine fréquentiel avec la méthode des éléments finis. Dans la première partie du manuscrit, le développement de méthodes de décomposition de domaine est abordé, dans le but de surmonter les limitations des solveurs directs et itératifs de l'état de l'art. À cette fin, une méthode de décomposition de domaine sous-structurée sans recouvrement avec des conditions absorbante d'ordre élevé utilisées comme conditions de transmission (HABC DDM) est d'abord étendue pour traiter les points de jonction, où plus de deux sous-domaines se rencontrent. Le traitement des points de jonction est un problème bien connu pour les HABC DDM sans recouvrement. La méthodologie proposée mène à une solution efficace pour les partitions en damier, où les domaines se rencontrent à angle droit. La méthode est basée sur l'introduction de variables de transmission supplémentaires aux points de jonction, et son efficacité est démontrée sur plusieurs cas-tests. Une DDM sans recouvrement similaire est ensuite proposée avec des couches parfaitement adaptées au lieu des HABC (DDM PML). L'approche proposée prend naturellement en compte les points de jonction des partitions de domaine en damier par le biais de multiplicateurs de Lagrange couplant les sous-domaines et les couches PML adjacentes. Deux discrétisations pour les multiplicateurs de Lagrange et plusieurs stratégies de stabilisation sont proposées et comparées. Les performances des DDM HABC et PML sont ensuite comparées sur des cas-tests de complexité croissante, allant de la diffraction d'ondes dans des milieux homogènes bidimensionnelles à la propagation d'ondes tridimensionnelles dans des milieux hautement hétérogènes. Alors que les développements théoriques sont effectués pour l'équation scalaire de Helmholtz pour la simulation d'ondes acoustiques, l'extension aux problèmes d'ondes élastiques est également considérée, mettant en évidence le potentiel de généralisation des méthodes développées à d'autres contextes physiques. La deuxième partie du manuscrit est consacrée à la présentation des outils de calcul développés au cours de la thèse et qui ont été utilisés pour produire tous les résultats numériques : GmshFEM, une nouvelle bibliothèque d'éléments finis C++ basée sur le générateur de maillage open-source Gmsh ; et GmshDDM, une bibliothèque de décomposition de domaine distribuée basée sur GmshFEM
The Fifteenth Marcel Grossmann Meeting
The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity
On a generalization of median graphs: -median graphs
Median graphs are connected graphs in which for all three vertices there is a
unique vertex that belongs to shortest paths between each pair of these three
vertices. To be more formal, a graph is a median graph if, for all , it holds that where
denotes the set of all vertices that lie on shortest paths connecting
and . In this paper we are interested in a natural generalization of
median graphs, called -median graphs. A graph is a -median graph, if
there are vertices such that, for all , it holds that , . By definition, every median graph with vertices is an -median graph.
We provide several characterizations of -median graphs that, in turn, are
used to provide many novel characterizations of median graphs
Twin-width VIII: delineation and win-wins
We introduce the notion of delineation. A graph class is said
delineated if for every hereditary closure of a subclass of
, it holds that has bounded twin-width if and only if
is monadically dependent. An effective strengthening of
delineation for a class implies that tractable FO model checking
on is perfectly understood: On hereditary closures of
subclasses of , FO model checking is fixed-parameter tractable
(FPT) exactly when has bounded twin-width. Ordered graphs
[BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively
delineated, while subcubic graphs are not. On the one hand, we prove that
interval graphs, and even, rooted directed path graphs are delineated. On the
other hand, we show that segment graphs, directed path graphs, and visibility
graphs of simple polygons are not delineated. In an effort to draw the
delineation frontier between interval graphs (that are delineated) and
axis-parallel two-lengthed segment graphs (that are not), we investigate the
twin-width of restricted segment intersection classes. It was known that
(triangle-free) pure axis-parallel unit segment graphs have unbounded
twin-width [BGKTW, SODA '21]. We show that -free segment graphs, and
axis-parallel -free unit segment graphs have bounded twin-width, where
is the half-graph or ladder of height . In contrast, axis-parallel
-free two-lengthed segment graphs have unbounded twin-width. Our new
results, combined with the known FPT algorithm for FO model checking on graphs
given with -sequences, lead to win-win arguments. For instance, we derive
FPT algorithms for -Ladder on visibility graphs of 1.5D terrains, and
-Independent Set on visibility graphs of simple polygons.Comment: 51 pages, 19 figure
Advanced Concepts in Particle and Field Theory
Uniting the usually distinct areas of particle physics and quantum field theory, gravity and general relativity, this expansive and comprehensive textbook of fundamental and theoretical physics describes the quest to consolidate the elementary particles that are the basic building blocks of nature. Designed for advanced undergraduates and graduate students and abounding in worked examples and detailed derivations, as well as historical anecdotes and philosophical and methodological perspectives, this textbook provides students with a unified understanding of all matter at the fundamental level. Topics range from gauge principles, particle decay and scattering cross-sections, the Higgs mechanism and mass generation, to spacetime geometries and supersymmetry. By combining historically separate areas of study and presenting them in a logically consistent manner, students will appreciate the underlying similarities and conceptual connections across these fields. This title, first published in 2015, has been reissued as an Open Access publication
Aspects of Quantum Field Theory in Enumerative Graph Theory
While a quantum field theorist has many uses for mathematics of all kinds, the relationship between quantum field theory and mathematics is far too fluid in the world of modern research to be described as the simple provision of mathematical tools to physicists, as Feynman often framed it. Problems large and small of a seemingly purely mathematical nature often arise directly from a physical setting. In this thesis we focus on two combinatorial problems with deep physical motivations.
The first of these is the Quadrangulation Conjecture of Jackson and Visentin, which asks for a bijective proof of an identity relating numbers of maps to numbers of maps which are quadrangulations. We provide a set of auxiliary bijections culminating in a bijection between maps with marked spanning trees and chord diagrams with partitions of the chords into a non-crossing part and a ‘genus-g’ part, and a bijection between these partitioned chord diagrams and four-regular maps with marked Euler tours.
The second problem comes from the CHY integral formulation of tree-level Feynman integrals in supersymmetric Yang-Mills theory, but amounts to the enumeration of ways to decompose 4-regular graphs into pairs of edge-disjoint Hamiltonian cycles. We show that for any graph which is the edge-disjoint union of an arbitrary 2-regular graph and a cycle, there are at least (n−2)!/4 ways to decompose the result into two full cycles. Moreover, if the chosen 2-regular graph consists of only even cycles this bound improves to (n − 2)!/2. Further, if the graph consists only of 2-cycles, we obtain the exact number of decompositions, which is (1/2) (n−2)!!S_H^±(n/2−1,1), where S_H^±(a,b) is the so-called signed Hultman number. Interestingly, this combinatorial problem turns out to have further connections to the study of genomic rearrangements in bioinformatics
Wave Propagation in Materials for Modern Applications
In the recent decades, there has been a growing interest in micro- and nanotechnology. The advances in nanotechnology give rise to new applications and new types of materials with unique electromagnetic and mechanical properties. This book is devoted to the modern methods in electrodynamics and acoustics, which have been developed to describe wave propagation in these modern materials and nanodevices. The book consists of original works of leading scientists in the field of wave propagation who produced new theoretical and experimental methods in the research field and obtained new and important results. The first part of the book consists of chapters with general mathematical methods and approaches to the problem of wave propagation. A special attention is attracted to the advanced numerical methods fruitfully applied in the field of wave propagation. The second part of the book is devoted to the problems of wave propagation in newly developed metamaterials, micro- and nanostructures and porous media. In this part the interested reader will find important and fundamental results on electromagnetic wave propagation in media with negative refraction index and electromagnetic imaging in devices based on the materials. The third part of the book is devoted to the problems of wave propagation in elastic and piezoelectric media. In the fourth part, the works on the problems of wave propagation in plasma are collected. The fifth, sixth and seventh parts are devoted to the problems of wave propagation in media with chemical reactions, in nonlinear and disperse media, respectively. And finally, in the eighth part of the book some experimental methods in wave propagations are considered. It is necessary to emphasize that this book is not a textbook. It is important that the results combined in it are taken “from the desks of researchers“. Therefore, I am sure that in this book the interested and actively working readers (scientists, engineers and students) will find many interesting results and new ideas
- …