Free delivery worldwide. Bestselling Series. Harry Potter. Popular Features. New Releases. Product details Format Hardback pages Dimensions x x Illustrations note 26 Illustrations, black and white; XVI, p. Other books in this series. Add to basket. Mathematical Image Processing Kristian Bredies. Frames and Bases Ole Christensen. Morrey Spaces David Adams. Approximation Theory Ole Christensen. Functions, Spaces, and Expansions Ole Christensen.
Representations, Wavelets, and Frames Palle E. Harmonic Analysis and Applications Christopher Heil. Back cover copy Differential geometry techniques have very useful and important applications in partial differential equations and quantum mechanics. Historically, problems in these areas were approached using the Fourier transform or path integrals, although in some cases e.
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New geometric methods are introduced in the work that have the advantage of providing quantitative or at least qualitative descriptions of operators, many of which cannot be treated by other methods. And, conservation laws of the Euler-Lagrange equations are employed to solve the equations of motion qualitatively when quantitative analysis is not possible. Main topics include: Lagrangian formalism on Riemannian manifolds; energy momentum tensor and conservation laws; Hamiltonian formalism; Hamilton-Jacobi theory; harmonic functions, maps, and geodesics; fundamental solutions for heat operators with potential; and a variational approach to mechanical curves.
The text is enriched with good examples and exercises at the end of every chapter. Geometric Mechanics on Riemannian Manifolds is a fine text for a course or seminar directed at graduate and advanced undergraduate students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics. It is also an ideal resource for pure and applied mathematicians and theoretical physicists working in these areas.
Review Text " This book, which contains some very interesting ideas and results, is primarily oriented towards graduate or advanced undergraduate students in mathematics and theoretical physics with interests in differential geometry, the calculus of variations and the study of PDE's, as well as in classical and quantum mechanics. In addition, for more experienced researchers in these fields, it may be a useful resource, written in a style that makes it easily accessible to a wide audience Thus, the book should be of interest for anyone working in these fields, from advanced undergraduate students to experts.
The book is written in a very pedagogical manner and does not assume many prerequisites, therefore it is quite appropriate to be used for special courses or for self-study. I have to mention that all chapters end with a number of well-chosen exercises that will imporve the understanding of the material and, also, that there are a lot of worked examples that will serve the same purpose.
L, No. The authors make a concerted effort to simplify proofs taken from many sources [so] researchers will readily fin dthe infromations they seek, while students can develop their skills by filling in details of proofs, as well as by using the problem sets that end each chapter. After showing that the unique submaximal model that does not descend to a conformal structure is Cayley-isotrivially flat, we will focus on Cayley structures and explore several geometries arising from them.
Finally we formulate such structures in terms of a dispersionless Lax pair and study the resulting system of PDEs. This work is partly joint with W. In the second part of the talk, I will present some recent results about plurisubharmonic function and the Monge-Ampere equation on almost complex manifolds.
For horizontal curves this curvature coincides with Euclidean curvature of its ortogonal projection onto XY-plane. The idea is based on the following fact: the image of the ortogonal projection into XY-plane of any geodesic in Heisenberg group is an arc of a circle. For any two points in Heisenberg group we define a geodesic radius of curvature which is the radius of the circle arc obtained by a the projection from the unique geodesic connecting those two points. The aim of the talk is to show the similarities between the role played by the intrinsic curvature in Heisenberg group and "normal" curvature, and between the geodesic radius of the curvature and the Menger curvature in Euclidean space.
It is well-known that these two methods give inconsistent results, and some researchers asked the question when one of the above-mentioned dynamics is a subset of another one. We show a simple method of adressing such a question based on the ideas of W. We provide a detailed answer for a relatively big class of non-invariant Chaplygin systems.
The work is based on a joint paper with Witold Respondek to appear in J. It is well-known that their generators must be principal null directions of the Weyl tensor. What is more, their leaf space is endowed with the structure of a CR manifold. In this talk I will give the integrability condition for the existence of SCNGs in dimension greater than four, and show that remarkably, in even dimension, the connection between SCNGs and almost CR structures still subsist under relatively mild curvature conditions on the Weyl tensor.
Finally, one can play a similar game in split signature: under suitable curvature prescriptions, SCNGs induce Lagrange contact structures and projective structures. The key role is played by homogeneous symplectic and Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal GL 1;R -bundle structure on the manifold and not just to a vector field. This allows for working with nontrivial line bundles that drastically simplifies the picture.
Contact manifolds of degree 2 and contact analogs of Courant algebroids are studied as well. Based on a joint work with A. Bruce and K. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. A joint work with Andriy Panasyuk. Gromov, using estimates on the Hopf invariant, the second by L. Guth, using Steenrod squares. I'll discuss results on sub-Riemannian isometries of the structures and present a simple construction of a canonical connection associated to the structures.
The talk is based on a joint work with Marek Grochowski. Using Cartan's method, we will solve the local equivalence problem of causal structures and give a geometric interpretation of their fundamental invariants. We will mostly focus on special classes of causal geometries in dimension four, referred to as half-flat and locally isotrivial, and study several twistorial constructions arising from them.
Javier de Lucas Araujo University of Warsaw : Poisson-Hopf algebra deformations of a class of Hamiltonian systems ABSTRACT: This talk is devoted to the use the theory of deformation of Hopf-algebras to construct Hamiltonian systems on a symplectic manifold and to study their constants of the motion, multi-dimensional generalisations, and physical applications.
First, I will survey the theory of deformation of Hopf algebras by introducing co-algebras, bi-algebras, antipode mappings, Hopf and Poisson-Hopf algebras, the dual principle, and the deformation of Hopf algebras. I will detail some classical examples of Hopf algebras: the universal enveloping algebra and their associated quantum groups, or the Konstant-Kirillov-Souriau Poisson algebra and its quantum deformations.
In the second part of the talk, I will use representations of Poisson-Hopf algebras to construct Hamiltonian systems on a symplectic manifold. The representation of a universal enveloping algebra will give rise to a certain Hamiltonian system, a so-called Lie--Hamilton system, whereas its deformation will lead to a one-parametric deformation of the Lie--Hamilton system. The centers of Hopf algebras and their so-called antipodes will give rise to constants of motion of the Lie--Hamilton system and its deformations; the coalgebra structure will lead to multi-dimensional generalisations of the Lie--Hamilton system.
As a final example, I will deform a t-dependent frequency Smorodinsky--Winternitz oscillator to obtain and to analyse a t-dependent frequency oscillator with a mass depending on the position and a Rosochatius-Winternitz potential term. They are special cases of cone structures with conic connections.
We give an overview of the subject, emphasizing the interaction of differential geometric methods and algebraic geometric methods. In this talk I will try to describe how to determine if two given ones are locally nonequivalent. VMRT is a fundamental tool in the program of studying the varieties that are covered by rational curves. The latter may be thought of as the closest analogoues to the notion of a line in the familiar Euclidean geometry, playing a similar role as geodesics in Riemannian geometry.
I will focus on the case when the underlying variety is a complex contact manifold. More precisely, when the contact manifold is homogeneous with respect to a Lie group G.
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The review part of this talk is based on the paper "Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems" by J. Alekseevky, J. Gutt, G. Manno and the speaker, recently accepted by Communications in Contemporary Mathematics.
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The talk will outline how this theorem can be generalised in the context of quasiconformal mappings in metric measure spaces, bringing to the fore the significance of Loewner conditions. There are also more general results by Balogh and Koskela concerning porous sets which I will outline. The range of this map is characterised by Calyey's description of pairs poristic conics inscribed and circumscribed in a triangle. This is an example of a more general twistor construction, when the twistor space fibers holomorphicaly over a projective plane.
The resulting twistor correspondence provides a solution to a system of nonlinear equations for an anti-self-dual conformal structure.
Geometric Mechanics on Riemannian Manifolds : Applications to Partial Differential Equations
Malament stating that the class of continuous timelike curves determines the topology of spacetime. The aim of my talk is to generalize this result to a certain class of sub-Lorentzian manifolds, as well as to some control systems and differential inclusions. We will see that any submaximally symmetric quaternionic manifold arises by the construction and that the standard submaximally symmetric quaternionic model arises from the unique submaximally symmetric c-projective model.
This suggests that the submaximally symmetric quaternionic structure should be also unique. Finally we will discuss the dimension of quaternionic symmetries of the Calabi metric showing that the dimension of the algebra of quaternionic symmetries is not fully determined by the dimension of algebra of c-projective symmetries of the submanifold.
In this talk, the notion of rolling is extended to a pair of Finsler surfaces which gives rise to a rank 2 distribution on a 6-dimensional manifold. Special classes of rolling Finsler surfaces will be discussed. Finally, a geometric formulation of rolling for a pair of Cartan geometries of the same type on manifolds of equal dimensions will be given. To determine if an extremal of a given variational problem is indeed minimal, one needs to study the definiteness of the second variation.
In general this is a difficult problem. However, for one-dimensional problems of mechanical type a clever use of the Sturm-Liouville theory allows to prove or exclude minimality from very simple global geometric properties of the extremal. Those problems are strongly connected with open rank-one conjecture posed by Morrey in , known in the multidimensional calculus of variations. This is still not the case of the second order conditions, where usually very strong assumptions are imposed on optimal controls.
In this talk I will first discuss the second-order optimality conditions in the integral form. In the difference with the main approaches of the existing literature, the second order tangents and the second order linearization of control systems will be used to derive the second-order necessary conditions.
This framework allows to replace the control system at hand by a differential inclusion with convex right-hand side. Such a relaxation of control system, and the differential calculus of derivatives of convex set-valued maps lead to fairly general statements. When the end point constraints are absent, the pointwise second order conditions will be stated : the second order maximum principle, the Goh and the Jacobson type necessary optimality conditions for general control systems similar results in the presence of end point constraints are still under investigation.
The talk is intended to be introductory and elements of calculus of set-valued maps will be discussed at the very beginning. The local problem is related to exactness of a Koszul complex. The global version uses H. Cartan Theorems A and B. Another question related to the above is a global version of E.
Cartan Lemma, where the differential forms have singularities. We will show that it can be solved using an algebrac Saito's theorem. The class of distributions naturally arise in the context of special bi-Hamiltonian systems and in the context of certain second order systems of PDEs. I'll also show how the distributions are connected to the contact geometry in dimension 5. An almost quaternionic structure is called submaximally symmetric if it has maximal symmetry dimension amongst those with lesser symmetry dimension than the maximal case.
This is realized both by a quaternionic structure torsion free and by an almost quaternionic structure with vanishing Weyl curvature. Joint work with Boris Kruglikov and Lenka Zalabova. Omid Makhmali McGill University : Local aspects of causal structures and related geometries ABSTRACT: In this talk the study of causal structures will be motivated, which are defined as a field of tangentially non-degenerate projective hypersurfaces in the projectivized tangent bundle of a manifold. They can be realized as a generalization of conformal pseudo-Riemannian structures.
The solution of the local equivalence of causal structures on manifolds of dimension at least four reveals that these geometries are parabolic and the harmonic curvature which is torsion is given by the Fubini cubic forms of the null cones and a generalization of the sectional Weyl curvature. Examples of such geometries will be presented. In dimension four the notion of self-duality for indefinite conformal structures will be extended to causal structures via the existence of a 3-parameter family of surfaces whose tangent planes at each point rule the null cone. We introduce the concept of automorphisms with natural tangent action.
A Generalized Feix--Kaledin construction provides a way to invert this in a special case, i. In this talk we will overview the construction and show how it is related to c-pojective and quaternionic projective cones constructions by S. Armstrong note that in quaternionic case the cone is called Swann bundle. Finally we will discuss the role of the line bundle and investigate its relation with Haydys--Hitchin quaternion-Kahler - hyperkahler correspondence. Arman Taghavi-Chabert University of Turin : Twistor geometry of null foliations ABSTRACT: We give a description of local null foliations on an odd-dimensional complex quadric Q in terms of complex submanifolds of its twistor space defined to be the space of all linear subspaces of Q of maximal dimension.
The resulting descriptions can reveal new relationships among the involved types of structure. Such a reduction determines a partition of the original manifold into three "curved orbits": Two are open submanifolds, each equipped with a Einstein metric, which is asymptotically equivalent to hyperbolic space in a way that can be made precise.
The solution turns out to be equivalent to the classical Fefferman-Graham ambient construction. Applications of these ideas include new results in projective geometry, special Riemannian geometries, and exceptional pseudo-Riemannian holonomy. Christoph Harrach University of Vienna : Poisson transforms for differential forms adapted to homogeneous parabolic geometries ABSTRACT: We present a construction of Poisson transforms between differential forms on homogeneous parabolic geometries and differential forms on Riemannian symmetric spaces tailored to the exterior calculus.
Moreover, we show how their existence and compatibility with natural differential operators can be reduced to invariant computations in finite dimensional representations of reductive Lie groups. To generalize these rigidity results to quasihomogeneous complex manifolds, we study a smooth projective horospherical variety of Picard number one and their geometric structures. Using Cartan geometry, we prove that a geometric structure modeled on a smooth projective horospherical variety of Picard number one is locally equivalent to the standard geometric structure when the geometric structure is defined on a Fano manifold of Picard number one.
In this seminar, we also briefly introduce the origin of this specific problem and horospherical varieties which are completely different from horospheres. Khesin and G. This construction consists in the argument shift method on the Virasoro Lie algebra, in frames of which the three equations are distinguished by the choice of the shift point. If time permits, I will discuss also perspectives of generalizing to this case of some results from the finite-dimensional argument shift method.
It results from the converse theorem by Koebe, that functions having the MVP at every point with all admissable radii are harmonic. Further results by Volterra and Kelogg, known as One-Radius Theorem, and by Hansen and Nadirashvili showed that in case of a bounded domain it is enough to assume MVP with one radius at each point to assert harmonicity.
I will present examples showing that none of the assumptions of these theorems can be dropped. The Delsarte conjectured and proved in dimension 3 that in fact this relation is always true, so that MVP on any pair of distinct radii is sufficient. I will present a recent proof of the Delsarte Conjecture in all dimensions and present a counterpart of the conjecture on harmonic manifolds.
The talk is based on joint work with T. We exploit a Riemannian metric that is associated to the system and construct dissipative prolongations of multipeakons near the singular points of the underlying Hamiltonian system. Hence it is worthwhile to study the natural geometric structures they carry, and the associated invariants.
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This approach had been used by A. Smith to classify non-degenerate hydrodynamically integrable hyperbolic Hirota-type PDE in 3 independent variables. I will present some early results of a joint work in progress with G. Manno, G. Moreno and A. Smith, extending the underlying geometry to higher dimensions. In this setting, the Sard conjecture states that S x should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero.
In this seminar, I will present a recent work in collaboration with Ludovic Rifford where we show that the conjecture holds whenever the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces and show that the result holds true under an assumption of non-transversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and some techniques of resolution of singularities.
Adopting this viewpoint allows one to formulate existence of minimizers in much more generality, even when a PDE approach is not available. I will focus on minimizing a p-energy among homotopy classes of maps between certain metric types of spaces. In particular I will discuss the notion of homotopy in the possibly discontinuous case of Sobolev maps, and the proof of existence of a minimizer in this generality. In this talk we present the relations between Lie and Jordan algebras and give the conditions, for which such an equation is reduced to a problem in the corresponding Jordan algebra.
The example model we study is an effective model of a three-level atom interacting with an electric field. They correspond to solitons, solutions of the Korteweg-de Vries equations. Multipeakons obey the system of Hamiltonian ODEs. However, derivative of the Hamiltonian posseses discontinuity.
Geometry, the calculus of variations and geometric analysis
I will discuss several aspects related to the dynamics of multipeakons. Our problem is motivated by nonlinear optics and Bose-Einstein condensates. For instance, in nonlinear optics, a nonlinearity is responsible for nonlinear polarization in a medium and by means of the slowly varying envelope approximation we can study the approximated propagation of the electromagnetic field in the medium. Moreover we discuss how to find the exact propagation of electromagnetic fields in nonlinear media.
A PDE fulfills this property if it possesses "sufficiently many" hydrodynamic reductions. Hydrodynamic reductions are special solutions which can be obtained in a formally analogous way as B. Since that pioneering work, there has been a plethora of spin-offs, where the method of hydrodynamic reductions has been studied, generalized and successfully used in many applications.
Applications to Partial Differential Equations
However, the geometry behind hydrodynamic integrability has been a mystery until , when there appeared the back-to-back papers "Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian" by E. Ferapontov et al. Smith Comm. In this seminar I will review the milestones set by the aforementioned papers, framing them against an appropriate geometric background. Even though I will not announce any new result, I will duly stress a conjecture formulated in Ferapontov paper, which is currently under study by A.
Smith, G. Manno, J. Gutt and myself. More details about the progress of our work will be given by J. Gutt in a forthcoming seminar within this series. Feix and D. Moreover, they generalized this construction for hypercomplex manifolds, where the hypercomplex structure is constructed on a neighbourhood of the zero section of the tangent bundle of a complex manifold with a real analytic connection with curvature of type 1,1. In this talk we will discuss a generalization of this construction to quaternionic geometry.
Finally, we will mention some further directions concerning twisted Armstrong cones and Swann bundles. The presented results are a joint work with D. Calderbank University of Bath. Maciej Nieszporski University of Warsaw : Integrable discretization of Bianchi surfaces ABSTRACT: I will focus on example of Bianchi surfaces to explain what we understand by integrable discretization of class of surfaces, the class which is described by nonlinear integrable system of differential equations.
We also study the singular symplectic forms with singular Martinet hypersurfaces. We prove that the equivalence class of such singular symplectic form-germ is determined by the Martinet hypersurface, the canonical orientation of its regular part and the restriction of the singular symplectic form to its regular part if the Martinet hypersurface is a quasi-homogeneous hypersurface with an isolated singularity. Then I will discuss a recent article about a infinite dimensional Lie structure of a character group of a graded connected Hopf algebra.
Finally, I will show how the reduced path group is embeded in the character Lie group of the shuffle Hopf algebra and discuss some of its propoerties. We give a geometric characterization of systems that become static feedback linearizable after an invertible one-fold prolongation of a suitably chosen control.
They form a particular class of flat systems. Using the notion of Ellie Cartan, they are absolutely equivalent to a trivial system under 1-dimensional prolongation. We propose conditions verifiable by differentiation and algebraic operations describing that class and provide a system of PDE's giving all minimal flat outputs.