Spinors On Singular Spaces And The Topology Of Causal Fermion Systems

Author : Felix Finster,Niky Kamran
Publisher : American Mathematical Soc.
Page : 83 pages
File Size : 43,8 Mb
Release : 2019-06-10
Category : Electronic
ISBN : 9781470436216

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Spinors on Singular Spaces and the Topology of Causal Fermion Systems by Felix Finster,Niky Kamran Pdf

Causal fermion systems and Riemannian fermion systems are proposed as a framework for describing non-smooth geometries. In particular, this framework provides a setting for spinors on singular spaces. The underlying topological structures are introduced and analyzed. The connection to the spin condition in differential topology is worked out. The constructions are illustrated by many simple examples such as the Euclidean plane, the two-dimensional Minkowski space, a conical singularity, a lattice system as well as the curvature singularity of the Schwarzschild space-time. As further examples, it is shown how complex and Kähler structures can be encoded in Riemannian fermion systems.

Author : Felix Finster
Publisher : Springer
Page : 548 pages
File Size : 45,6 Mb
Release : 2016-08-19
Category : Science
ISBN : 9783319420677

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The Continuum Limit of Causal Fermion Systems by Felix Finster Pdf

This monograph introduces the basic concepts of the theory of causal fermion systems, a recent approach to the description of fundamental physics. The theory yields quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for describing and analyzing non-smooth geometries and "quantum geometries". The dynamics is described by a novel variational principle, called the causal action principle. In addition to the basics, the book provides all the necessary mathematical background and explains how the causal action principle gives rise to the interactions of the standard model plus gravity on the level of second-quantized fermionic fields coupled to classical bosonic fields. The focus is on getting a mathematically sound connection between causal fermion systems and physical systems in Minkowski space. The book is intended for graduate students entering the field, and is furthermore a valuable reference work for researchers in quantum field theory and quantum gravity.

Author : Felix Finster,Domenico Giulini,Johannes Kleiner,Jürgen Tolksdorf
Publisher : Springer Nature
Page : 302 pages
File Size : 55,5 Mb
Release : 2020-04-09
Category : Science
ISBN : 9783030389413

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Progress and Visions in Quantum Theory in View of Gravity by Felix Finster,Domenico Giulini,Johannes Kleiner,Jürgen Tolksdorf Pdf

This book focuses on a critical discussion of the status and prospects of current approaches in quantum mechanics and quantum field theory, in particular concerning gravity. It contains a carefully selected cross-section of lectures and discussions at the seventh conference “Progress and Visions in Quantum Theory in View of Gravity” which took place in fall 2018 at the Max Planck Institute for Mathematics in the Sciences in Leipzig. In contrast to usual proceeding volumes, instead of reporting on the most recent technical results, contributors were asked to discuss visions and new ideas in foundational physics, in particular concerning foundations of quantum field theory. A special focus has been put on the question of which physical principles of quantum (field) theory can be considered fundamental in view of gravity. The book is mainly addressed to mathematicians and physicists who are interested in fundamental questions of mathematical physics. It allows the reader to obtain a broad and up-to-date overview of a fascinating active research area.

Author : Felix Finster,Johannes Kleiner,Christian Röken,Jürgen Tolksdorf
Publisher : Birkhäuser
Page : 518 pages
File Size : 41,8 Mb
Release : 2016-02-24
Category : Science
ISBN : 9783319269023

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Quantum Mathematical Physics by Felix Finster,Johannes Kleiner,Christian Röken,Jürgen Tolksdorf Pdf

Quantum physics has been highly successful for more than 90 years. Nevertheless, a rigorous construction of interacting quantum field theory is still missing. Moreover, it is still unclear how to combine quantum physics and general relativity in a unified physical theory. Attacking these challenging problems of contemporary physics requires highly advanced mathematical methods as well as radically new physical concepts. This book presents different physical ideas and mathematical approaches in this direction. It contains a carefully selected cross-section of lectures which took place in autumn 2014 at the sixth conference ``Quantum Mathematical Physics - A Bridge between Mathematics and Physics'' in Regensburg, Germany. In the tradition of the other proceedings covering this series of conferences, a special feature of this book is the exposition of a wide variety of approaches, with the intention to facilitate a comparison. The book is mainly addressed to mathematicians and physicists who are interested in fundamental questions of mathematical physics. It allows the reader to obtain a broad and up-to-date overview of a fascinating active research area.

Author : Luigi Ambrosio,Andrea Mondino,Giuseppe Savaré
Publisher : American Mathematical Soc.
Page : 121 pages
File Size : 55,8 Mb
Release : 2020-02-13
Category : Education
ISBN : 9781470439132

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Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces by Luigi Ambrosio,Andrea Mondino,Giuseppe Savaré Pdf

The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.

Author : Michael Handel,Lee Mosher
Publisher : American Mathematical Soc.
Page : 276 pages
File Size : 55,5 Mb
Release : 2020-05-13
Category : Education
ISBN : 9781470441135

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Subgroup Decomposition in Out(Fn) by Michael Handel,Lee Mosher Pdf

In this work the authors develop a decomposition theory for subgroups of Out(Fn) which generalizes the decomposition theory for individual elements of Out(Fn) found in the work of Bestvina, Feighn, and Handel, and which is analogous to the decomposition theory for subgroups of mapping class groups found in the work of Ivanov.

Author : Gonzalo Fiz Pontiveros,Simon Griffiths,Robert Morris
Publisher : American Mathematical Soc.
Page : 125 pages
File Size : 43,6 Mb
Release : 2020-04-03
Category : Education
ISBN : 9781470440718

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The Triangle-Free Process and the Ramsey Number R(3,k) by Gonzalo Fiz Pontiveros,Simon Griffiths,Robert Morris Pdf

The areas of Ramsey theory and random graphs have been closely linked ever since Erdős's famous proof in 1947 that the “diagonal” Ramsey numbers R(k) grow exponentially in k. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the “off-diagonal” Ramsey numbers R(3,k). In this model, edges of Kn are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted Gn,△. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kim's celebrated result that R(3,k)=Θ(k2/logk). In this paper the authors improve the results of both Bohman and Kim and follow the triangle-free process all the way to its asymptotic end.

Author : Cristian Gavrus,Sung-Jin Oh
Publisher : American Mathematical Soc.
Page : 94 pages
File Size : 47,6 Mb
Release : 2020-05-13
Category : Education
ISBN : 9781470441111

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Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data by Cristian Gavrus,Sung-Jin Oh Pdf

In this paper, the authors prove global well-posedness of the massless Maxwell–Dirac equation in the Coulomb gauge on R1+d(d≥4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell–Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell–Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru, which says that the most difficult part of Maxwell–Dirac takes essentially the same form as Maxwell-Klein-Gordon.

Author : Jaroslav Nešetřil,Patrice Ossona de Mendez
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 41,9 Mb
Release : 2020-04-03
Category : Education
ISBN : 9781470440657

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A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth by Jaroslav Nešetřil,Patrice Ossona de Mendez Pdf

In this paper the authors introduce a general framework for the study of limits of relational structures and graphs in particular, which is based on a combination of model theory and (functional) analysis. The authors show how the various approaches to graph limits fit to this framework and that the authors naturally appear as “tractable cases” of a general theory. As an outcome of this, the authors provide extensions of known results. The authors believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse structures. First, the authors consider limits of structures with bounded diameter connected components and prove that in this case the convergence can be “almost” studied component-wise. They also propose the structure of limit objects for convergent sequences of sparse structures. Eventually, they consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role as “elementary bricks” these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of modeling the authors introduce here. Their example is also the first “intermediate class” with explicitly defined limit structures where the inverse problem has been solved.

Author : S. V. Ivanov
Publisher : American Mathematical Soc.
Page : 106 pages
File Size : 45,9 Mb
Release : 2020-05-13
Category : Education
ISBN : 9781470441432

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The Bounded and Precise Word Problems for Presentations of Groups by S. V. Ivanov Pdf

The author introduces and studies the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be solved in nondeterministic polynomial time, and the precise word problem is in PSPACE, i.e., it can be solved in polynomial space. The main technical result of the paper states that, for certain finite presentations of groups, which include the Baumslag-Solitar one-relator groups and free products of cyclic groups, the bounded word problem and the precise word problem can be solved in polylogarithmic space. As consequences of developed techniques that can be described as calculus of brackets, the author obtains polylogarithmic space bounds for the computational complexity of the diagram problem for free groups, for the width problem for elements of free groups, and for computation of the area defined by polygonal singular closed curves in the plane. The author also obtains polynomial time bounds for these problems.

Author : Victor Beresnevich,Alan Haynes,Sanju Velani
Publisher : American Mathematical Soc.
Page : 77 pages
File Size : 51,6 Mb
Release : 2020-04-03
Category : Education
ISBN : 9781470440954

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Sums of Reciprocals of Fractional Parts and Multiplicative Diophantine Approximation by Victor Beresnevich,Alan Haynes,Sanju Velani Pdf

Author : Antonio Alarcón,Franc Forstnerič,Francisco J. López
Publisher : American Mathematical Soc.
Page : 77 pages
File Size : 55,5 Mb
Release : 2020-05-13
Category : Education
ISBN : 9781470441616

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New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Rn by Antonio Alarcón,Franc Forstnerič,Francisco J. López Pdf

All the new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables the authors to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, they construct proper non-orientable conformal minimal surfaces in Rn with any given conformal structure, complete non-orientable minimal surfaces in Rn with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits n hyperplanes of CPn−1 in general position, complete non-orientable minimal surfaces bounded by Jordan curves, and complete proper non-orientable minimal surfaces normalized by bordered surfaces in p-convex domains of Rn.

Author : Jean-François Coulombel,Mark Williams
Publisher : American Mathematical Soc.
Page : 143 pages
File Size : 48,5 Mb
Release : 2020-04-03
Category : Education
ISBN : 9781470440374

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Geometric Optics for Surface Waves in Nonlinear Elasticity by Jean-François Coulombel,Mark Williams Pdf

This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. The authors consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an approximate Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which is referred to as “the amplitude equation”, is an integrodifferential equation of nonlocal Burgers type. The authors begin by reviewing and providing some extensions of the theory of the amplitude equation. The remainder of the paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions uε to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength ε, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to uε on a time interval independent of ε. This paper focuses mainly on the case of Rayleigh waves that are pulses, which have profiles with continuous Fourier spectrum, but the authors' method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.

Author : Laurent Berger,Peter Schneider,Bingyong Xie
Publisher : American Mathematical Soc.
Page : 75 pages
File Size : 45,7 Mb
Release : 2020-04-03
Category : Education
ISBN : 9781470440732

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Rigid Character Groups, Lubin-Tate Theory, and (φ,Γ)-Modules by Laurent Berger,Peter Schneider,Bingyong Xie Pdf

The construction of the p-adic local Langlands correspondence for GL2(Qp) uses in an essential way Fontaine's theory of cyclotomic (φ,Γ)-modules. Here cyclotomic means that Γ=Gal(Qp(μp∞)/Qp) is the Galois group of the cyclotomic extension of Qp. In order to generalize the p-adic local Langlands correspondence to GL2(L), where L is a finite extension of Qp, it seems necessary to have at our disposal a theory of Lubin-Tate (φ,Γ)-modules. Such a generalization has been carried out, to some extent, by working over the p-adic open unit disk, endowed with the action of the endomorphisms of a Lubin-Tate group. The main idea of this article is to carry out a Lubin-Tate generalization of the theory of cyclotomic (φ,Γ)-modules in a different fashion. Instead of the p-adic open unit disk, the authors work over a character variety that parameterizes the locally L-analytic characters on oL. They study (φ,Γ)-modules in this setting and relate some of them to what was known previously.

Author : Peter Poláčik
Publisher : American Mathematical Soc.
Page : 87 pages
File Size : 50,5 Mb
Release : 2020-05-13
Category : Education
ISBN : 9781470441128

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Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on R by Peter Poláčik Pdf

The author considers semilinear parabolic equations of the form ut=uxx+f(u),x∈R,t>0, where f a C1 function. Assuming that 0 and γ>0 are constant steady states, the author investigates the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x,0) are near γ for x≈−∞ and near 0 for x≈∞. If the steady states 0 and γ are both stable, the main theorem shows that at large times, the graph of u(⋅,t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). The author proves this result without requiring monotonicity of u(⋅,0) or the nondegeneracy of zeros of f. The case when one or both of the steady states 0, γ is unstable is considered as well. As a corollary to the author's theorems, he shows that all front-like solutions are quasiconvergent: their ω-limit sets with respect to the locally uniform convergence consist of steady states. In the author's proofs he employs phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories {(u(x,t),ux(x,t)):x∈R}, t>0, of the solutions in question.