Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

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Introduction to the Modern Theory of Dynamical Systems

WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

완효성 비료와 4종복비의 혼용 시용이 팔레놉시스의 생육에 미치는 효과

Let 
$\Bbb L$
 be a convex superlinear Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mane in [M3]. We show that the critical value of the lift of 
$\Bbb L$
 to a covering of N equals the infimum of the values of k such that the energy level k bounds an exact Lagrangian graph in the cotangent bundle of the covering. As a consequence, we show that up to reparametrization, the dynamics of the Euler-Lagrange flow of 
$\Bbb L$
 on an energy level that contains supports of minimizing measures with non-zero rotation vector can be reduced to Finsler metrics. We also show that if the Euler-Lagrange flow of 
$\Bbb L$
 on the energy level k is Anosov, then k must be strictly bigger than the critical value c

 u
 (
 
$\Bbb L$
 ) of the lift of L to the universal covering of N. It follows that given k < c

 u
 (
 
$\Bbb L$
 ), there exists a potential 
$\psi$
 with arbitrarily small C

 2-norm such that the energy level k of 
$\Bbb L + \psi$
 possesses conjugate points. Finally we show the existence of weak KAM solutions for coverings of N and we explain the relationship between Fathi's results in [F1,2] and Mane's critical values and action potentials.

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Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values

We show that any exact magnetic flow on a closed surface has periodic orbits in all energy levels. Moreover, we give homological and homotopical properties of these periodic orbits in terms of the Mane's critical values of the corresponding Lagrangian. We also prove that if M is not the 2-torus, the energy level k is of contact type if and only if k > c 0 , where c 0 is Mane's strict critical value. When M is the 2-torus, we give examples for which the energy level c 0 is of contact type.

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Periodic orbits for exact magnetic flows on surfaces

We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case.

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Tensor tomography on surfaces

The authors survey recent progress in the problem of recovering a tensor field from its integrals along geodesics. Several open problems are also proposed.

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Tensor tomography: Progress and challenges

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S1-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.¶We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.¶We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.¶We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S4, ℂP2, $\overline {\mathbb{C}P} ^2$,S2×S2and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S4,ℂP2,S2×S2,ℂP2$\# \overline {\mathbb{C}P} ^2$ or ℂP2# ℂP2. Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S5,S3×S2, then on trivial S3-bundle over S2 or the Wu-manifold SU(3)/SO(3).

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Gabriel P. Paternain

Papers

Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values

Periodic orbits for exact magnetic flows on surfaces

Tensor tomography on surfaces

Tensor tomography: Progress and challenges

Minimal entropy and collapsing with curvature bounded from below