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A novel method to calculate single-particle fabric tensors and related geometric characteristics (contact, free and particle area, particle volume and contact centroid) of assemblies of spherical particles has been developed. A standard contact detection stage is used to identify overlapping particles. The shape of each contact region is obtained as the intersection of a circle, obtained in the absence of any contact impingement, and a convex polygon, determined by halfplane intersection based on directed lines obtained from the intersection of contact planes. An underlying power diagram is used to subdivide volume and area between particles, but calculations are performed particle-wise, resulting in an efficient algorithm with a structure ideally suited for parallelisation. Numerical tests performed on loose and dense particle assemblies and comparison (of single-particle free area and particle volume) with a reference method (POWERSASA) indicate that the method is robust and highly accurate. The method will be useful not only for geometric analysis but also as a basis for subsequent developments of strain measures and computational procedures for highly deformed granular materials. Moreover, it constitutes a viable alternative for calculation of solvent accessible surface areas, considering its robustness, efficiency and ease of parallelisation.

A novel method to calculate single-particle fabric tensors and related geometric characteristics (contact, free and particle area, particle volume and contact centroid) of assemblies of spherical particles has been developed. A standard contact detection stage is used to identify overlapping particles. The shape of each contact region is obtained as the intersection of a circle, obtained in the absence of any contact impingement, and a convex polygon, determined by halfplane intersection based on directed lines obtained from the intersection of contact planes. An underlying power diagram is used to subdivide volume and area between particles, but calculations are performed particle-wise, resulting in an efficient algorithm with a structure ideally suited for parallelisation. Numerical tests performed on loose and dense particle assemblies and comparison (of single-particle free area and particle volume) with a reference method (POWERSASA) indicate that the method is robust and highly accurate. The method will be useful not only for geometric analysis but also as a basis for subsequent developments of strain measures and computational procedures for highly deformed granular materials. Moreover, it constitutes a viable alternative for calculation of solvent accessible surface areas, considering its robustness, efficiency and ease of parallelisation.

A novel method to calculate single-particle fabric tensors and related geometric characteristics (contact, free and particle area, particle volume and contact centroid) of assemblies of spherical particles has been developed. A standard contact detection stage is used to identify overlapping particles. The shape of each contact region is obtained as the intersection of a circle, obtained in the absence of any contact impingement, and a convex polygon, determined by halfplane intersection based on directed lines obtained from the intersection of contact planes. An underlying power diagram is used to subdivide volume and area between particles, but calculations are performed particle-wise, resulting in an efficient algorithm with a structure ideally suited for parallelisation. Numerical tests performed on loose and dense particle assemblies and comparison (of single-particle free area and particle volume) with a reference method (POWERSASA) indicate that the method is robust and highly accurate. The method will be useful not only for geometric analysis but also as a basis for subsequent developments of strain measures and computational procedures for highly deformed granular materials. Moreover, it constitutes a viable alternative for calculation of solvent accessible surface areas, considering its robustness, efficiency and ease of parallelisation.

A novel method to calculate single-particle fabric tensors and related geometric characteristics (contact, free and particle area, particle volume and contact centroid) of assemblies of spherical particles has been developed. A standard contact detection stage is used to identify overlapping particles. The shape of each contact region is obtained as the intersection of a circle, obtained in the absence of any contact impingement, and a convex polygon, determined by halfplane intersection based on directed lines obtained from the intersection of contact planes. An underlying power diagram is used to subdivide volume and area between particles, but calculations are performed particle-wise, resulting in an efficient algorithm with a structure ideally suited for parallelisation. Numerical tests performed on loose and dense particle assemblies and comparison (of single-particle free area and particle volume) with a reference method (POWERSASA) indicate that the method is robust and highly accurate. The method will be useful not only for geometric analysis but also as a basis for subsequent developments of strain measures and computational procedures for highly deformed granular materials. Moreover, it constitutes a viable alternative for calculation of solvent accessible surface areas, considering its robustness, efficiency and ease of parallelisation.

A novel method to calculate single-particle fabric tensors and related geometric characteristics (contact, free and particle area, particle volume and contact centroid) of assemblies of spherical particles has been developed. A standard contact detection stage is used to identify overlapping particles. The shape of each contact region is obtained as the intersection of a circle, obtained in the absence of any contact impingement, and a convex polygon, determined by halfplane intersection based on directed lines obtained from the intersection of contact planes. An underlying power diagram is used to subdivide volume and area between particles, but calculations are performed particle-wise, resulting in an efficient algorithm with a structure ideally suited for parallelisation. Numerical tests performed on loose and dense particle assemblies and comparison (of single-particle free area and particle volume) with a reference method (POWERSASA) indicate that the method is robust and highly accurate. The method will be useful not only for geometric analysis but also as a basis for subsequent developments of strain measures and computational procedures for highly deformed granular materials. Moreover, it constitutes a viable alternative for calculation of solvent accessible surface areas, considering its robustness, efficiency and ease of parallelisation.

© 2024 The Authors. Published by the Royal Society under the terms of theCreative Commons Attribution Licensehttp://creativecommons.org/licenses/by/4.0, We prove that the correspondence between Reeb and Beltrami vector fields presented in Etnyre & Ghrist (Etnyre, Ghrist 2000 Nonlinearity 13, 441–458 (doi:10.1088/0951-7715/13/2/306)) can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a corollary of this correspondence, we show that energy levels above the maximum of the potential energy of mechanical Hamiltonian systems can be viewed as stationary fluid flows, though the metric is not prescribed. In particular, we showcase the emblematic example of the n-body problem and focus on the Kepler problem. We explicitly construct a compatible Riemannian metric that makes the Kepler problem of celestial mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold, the Kepler–Euler flow., Peer Reviewed, Postprint (author's final draft)

—In this short note, we prove that singular Reeb vector fields associated with generic b-contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) 2N or an infinite number of escape orbits, where N denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of b-Beltrami vector fields that are not b-Reeb. The proof is based on a more detailed analysis of the main result in [19]., Peer Reviewed, Postprint (author's final draft)

The main content of this thesis is two papers. The first paper defines a canonical $\Q$-grading of the embedded contact homology chain complex under certain conditions. ECH is an invariant of 3-manifolds isomorphic to the Seiberg-Witten Floer cohomology as defined by Kronheimer and Mrowka. The underlying chain complex comes with a relative $\Z/p$-grading, or, an absolute $\Z/p$-grading after fixing a reference generator. However, when $p=0$, we show that one can actually specify an absolute grading of the chain complex without having to choose a reference generator. Explicitly, that grading set is a subset of $\Q$ isomorphic to $\Z$. As a consequence, $ECH(Y,\lambda)$ is canonically $\Q$-graded whenever $H_1(Y)$ is torsion. The second paper is an application of ECH capacities to the problem of embedding ellipsoids into irrational ellipsoids. More specifically, we analyze lattice point counting functions to deduce that the existence of an accumulation point of an infinite staircase is impossible for all irrational targets minus a set of irrationals in bijection with $\N$, for which the question has yet to be resolved.

Reeb flows are a rich, ubiquitous class of dynamical systems arising in symplectic geometry, which include billiard systems, many-body orbital systems, geodesic flows and many Hamiltonian flows. Convexity hypotheses play an important, albeit mysterious, role in the study of these flows. In this thesis, we discuss several new results in the study of convexity in symplectic geometry and Reeb dynamics.In Chapter 1, we resolve a longstanding open problem on the intrinsic characterization of Reeb flows arising from Hamiltonian flows on the convex boundaries. Namely, we prove that dynamically convex Reeb flows, introduced by Hofer-Wysocki-Zehnder, are not all convex. Our proof uses a novel relation between Riemannian geometry and Reeb dynamics, and uses constructions of Abbondandolo-Bramham-Hryniewicz-Salomao.In Chapter 2, we describe a powerful new framework for computationally modelling Reeb dynamics on the boundaries of convex polytopes. We apply this framework to provide new evidence and examples relating to the Viterbo conjecture, a major open problem in Reeb dynamics and quantitative symplectic geometry.In Chapter 3, we study convex toric domains and toric surfaces. A longstanding conjecture in toric geometry states that the Gromov width is monotonic under inclusion of moment polytopes of closed toric varieties. We use methods from toric geometry and ECH to prove a generalization of this conjecture in dimension 4.

In this thesis I prove two isomorphisms in the local symplectic homology of a non iterated, isolated Reeb orbit. The isomorphisms are in $S^1$-equivariant and nonequivariant symplectic homology relating the local Floer homology group of the orbit to that of the return map. The $S^1$-equivariant symplectic homology isomorphism can be stated succinctly as the local $S^1$-equivariant symplectic homology of an uniterated isolated Reeb orbit is isomorphic to the local Hamiltonian Floer homology of the return map. An application of this result gives the equivalence of two different definitions of a Reeb orbit being a symplectically degenerate maximum.

This thesis consists of an introduction and two research papers in the fields of symplectic and contact geometry. The focus of the thesis is on Floer theory and symplectic field theory. In Paper I we show that the partially wrapped Floer cohomology of a cotangent fiber stopped by the unit conormal of a submanifold, is equivalent to chains of based loops on the complement of the submanifold in the base. For codimension two knots in the n-sphere we show that there is a relationship between the wrapped Floer cohomology algebra of the fiber and the Alexander invariant of the knot. This allows us to exhibit codimension two knots with infinite cyclic knot group such that the union of the unit conormal of the knot and the boundary of a cotangent fiber is not Legendrian isotopic to the union of the unit conormal of the unknot union the boundary and the same cotangent fiber. In Paper II we study the Chekanov-Eliashberg dg-algebra which is a holomorphic curve invariant associated to a smooth Legendrian submanifold. We extend this definition to singular Legendrians. Using the new definition we formulate and prove a surgery formula relating the wrapped Floer cohomology algebra of the co-core disk of a stop with the Chekanov-Eliashberg dg-algebra of its attaching locus interpreted as the Weinstein neighborhood of a singular Legendrian. A special case of this surgery formula, when the Legendrian is non-singular, establishes a proof of a conjecture by Ekholm-Lekili and independently by Sylvan. We furthermore provide an alternative geometric proof of the pushout diagrams for partially wrapped Floer cohomology and the stop removal formulas of Ganatra-Pardon-Shende.

Reeb flows are a rich, ubiquitous class of dynamical systems arising in symplectic geometry, which include billiard systems, many-body orbital systems, geodesic flows and many Hamiltonian flows. Convexity hypotheses play an important, albeit mysterious, role in the study of these flows. In this thesis, we discuss several new results in the study of convexity in symplectic geometry and Reeb dynamics.In Chapter 1, we resolve a longstanding open problem on the intrinsic characterization of Reeb flows arising from Hamiltonian flows on the convex boundaries. Namely, we prove that dynamically convex Reeb flows, introduced by Hofer-Wysocki-Zehnder, are not all convex. Our proof uses a novel relation between Riemannian geometry and Reeb dynamics, and uses constructions of Abbondandolo-Bramham-Hryniewicz-Salomao.In Chapter 2, we describe a powerful new framework for computationally modelling Reeb dynamics on the boundaries of convex polytopes. We apply this framework to provide new evidence and examples relating to the Viterbo conjecture, a major open problem in Reeb dynamics and quantitative symplectic geometry.In Chapter 3, we study convex toric domains and toric surfaces. A longstanding conjecture in toric geometry states that the Gromov width is monotonic under inclusion of moment polytopes of closed toric varieties. We use methods from toric geometry and ECH to prove a generalization of this conjecture in dimension 4.

In this thesis I prove two isomorphisms in the local symplectic homology of a non iterated, isolated Reeb orbit. The isomorphisms are in $S^1$-equivariant and nonequivariant symplectic homology relating the local Floer homology group of the orbit to that of the return map. The $S^1$-equivariant symplectic homology isomorphism can be stated succinctly as the local $S^1$-equivariant symplectic homology of an uniterated isolated Reeb orbit is isomorphic to the local Hamiltonian Floer homology of the return map. An application of this result gives the equivalence of two different definitions of a Reeb orbit being a symplectically degenerate maximum.

This thesis consists of an introduction and two research papers in the fields of symplectic and contact geometry. The focus of the thesis is on Floer theory and symplectic field theory. In Paper I we show that the partially wrapped Floer cohomology of a cotangent fiber stopped by the unit conormal of a submanifold, is equivalent to chains of based loops on the complement of the submanifold in the base. For codimension two knots in the n-sphere we show that there is a relationship between the wrapped Floer cohomology algebra of the fiber and the Alexander invariant of the knot. This allows us to exhibit codimension two knots with infinite cyclic knot group such that the union of the unit conormal of the knot and the boundary of a cotangent fiber is not Legendrian isotopic to the union of the unit conormal of the unknot union the boundary and the same cotangent fiber. In Paper II we study the Chekanov-Eliashberg dg-algebra which is a holomorphic curve invariant associated to a smooth Legendrian submanifold. We extend this definition to singular Legendrians. Using the new definition we formulate and prove a surgery formula relating the wrapped Floer cohomology algebra of the co-core disk of a stop with the Chekanov-Eliashberg dg-algebra of its attaching locus interpreted as the Weinstein neighborhood of a singular Legendrian. A special case of this surgery formula, when the Legendrian is non-singular, establishes a proof of a conjecture by Ekholm-Lekili and independently by Sylvan. We furthermore provide an alternative geometric proof of the pushout diagrams for partially wrapped Floer cohomology and the stop removal formulas of Ganatra-Pardon-Shende.

This thesis consists of an introduction and two research papers in the fields of symplectic and contact geometry. The focus of the thesis is on Floer theory and symplectic field theory. In Paper I we show that the partially wrapped Floer cohomology of a cotangent fiber stopped by the unit conormal of a submanifold, is equivalent to chains of based loops on the complement of the submanifold in the base. For codimension two knots in the n-sphere we show that there is a relationship between the wrapped Floer cohomology algebra of the fiber and the Alexander invariant of the knot. This allows us to exhibit codimension two knots with infinite cyclic knot group such that the union of the unit conormal of the knot and the boundary of a cotangent fiber is not Legendrian isotopic to the union of the unit conormal of the unknot union the boundary and the same cotangent fiber. In Paper II we study the Chekanov-Eliashberg dg-algebra which is a holomorphic curve invariant associated to a smooth Legendrian submanifold. We extend this definition to singular Legendrians. Using the new definition we formulate and prove a surgery formula relating the wrapped Floer cohomology algebra of the co-core disk of a stop with the Chekanov-Eliashberg dg-algebra of its attaching locus interpreted as the Weinstein neighborhood of a singular Legendrian. A special case of this surgery formula, when the Legendrian is non-singular, establishes a proof of a conjecture by Ekholm-Lekili and independently by Sylvan. We furthermore provide an alternative geometric proof of the pushout diagrams for partially wrapped Floer cohomology and the stop removal formulas of Ganatra-Pardon-Shende.

Published under the PNAS license, Can every physical system simulate any Turing machine? This is a classical problem which is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore asked in [15] if hydrodynamics is capable of performing computations. More recently, Tao launched a programme based on the Turing completeness of the Euler equations to address the blow up problem in the Navier-Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem [7] to quantum field theories [11]. To the best of our knowledge, the existence of undecidable particle paths of 3D fluid flows has remained an elusive open problem since Moore's works in the early 1990's. In this article we construct a Turing complete stationary Euler flow on a Riemannian S3 and speculate on its implications concerning Tao's approach to the blow up problem in the Navier-Stokes equations., This work was partially supported by ICMAT–Severo OchoaGrant CEX2019-000904-S., Peer Reviewed, Postprint (author's final draft)

This thesis consists of an introduction and two research papers in the fields of symplectic and contact geometry. The focus of the thesis is on Floer theory and symplectic field theory. In Paper I we show that the partially wrapped Floer cohomology of a cotangent fiber stopped by the unit conormal of a submanifold, is equivalent to chains of based loops on the complement of the submanifold in the base. For codimension two knots in the n-sphere we show that there is a relationship between the wrapped Floer cohomology algebra of the fiber and the Alexander invariant of the knot. This allows us to exhibit codimension two knots with infinite cyclic knot group such that the union of the unit conormal of the knot and the boundary of a cotangent fiber is not Legendrian isotopic to the union of the unit conormal of the unknot union the boundary and the same cotangent fiber. In Paper II we study the Chekanov-Eliashberg dg-algebra which is a holomorphic curve invariant associated to a smooth Legendrian submanifold. We extend this definition to singular Legendrians. Using the new definition we formulate and prove a surgery formula relating the wrapped Floer cohomology algebra of the co-core disk of a stop with the Chekanov-Eliashberg dg-algebra of its attaching locus interpreted as the Weinstein neighborhood of a singular Legendrian. A special case of this surgery formula, when the Legendrian is non-singular, establishes a proof of a conjecture by Ekholm-Lekili and independently by Sylvan. We furthermore provide an alternative geometric proof of the pushout diagrams for partially wrapped Floer cohomology and the stop removal formulas of Ganatra-Pardon-Shende.

This thesis consists of an introduction and two research papers in the fields of symplectic and contact geometry. The focus of the thesis is on Floer theory and symplectic field theory. In Paper I we show that the partially wrapped Floer cohomology of a cotangent fiber stopped by the unit conormal of a submanifold, is equivalent to chains of based loops on the complement of the submanifold in the base. For codimension two knots in the n-sphere we show that there is a relationship between the wrapped Floer cohomology algebra of the fiber and the Alexander invariant of the knot. This allows us to exhibit codimension two knots with infinite cyclic knot group such that the union of the unit conormal of the knot and the boundary of a cotangent fiber is not Legendrian isotopic to the union of the unit conormal of the unknot union the boundary and the same cotangent fiber. In Paper II we study the Chekanov-Eliashberg dg-algebra which is a holomorphic curve invariant associated to a smooth Legendrian submanifold. We extend this definition to singular Legendrians. Using the new definition we formulate and prove a surgery formula relating the wrapped Floer cohomology algebra of the co-core disk of a stop with the Chekanov-Eliashberg dg-algebra of its attaching locus interpreted as the Weinstein neighborhood of a singular Legendrian. A special case of this surgery formula, when the Legendrian is non-singular, establishes a proof of a conjecture by Ekholm-Lekili and independently by Sylvan. We furthermore provide an alternative geometric proof of the pushout diagrams for partially wrapped Floer cohomology and the stop removal formulas of Ganatra-Pardon-Shende.

In this article, we investigate Reeb dynamics on bm-contact manifolds, previously introduced in [37], which are contact away from a hypersurface Zbut satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact bm-contact manifolds without periodic Reeb orbits outside Zare provided. Furthermore, we prove that in dimension 3, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the bm-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex, Postprint (published version)

In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-contact structures, which are tangent to a given smooth hypersurface $Z$ and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called $b^m$-contact forms, having an associated critical hypersurface $Z$. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of those are related to smooth contact structures through a desingularization technique. The problem of existence of $b^m$-contact structures on a given manifold is also tackled in this paper. We prove that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a $b^m$-contact structure., In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-contact structures, which are tangent to a given smooth hypersurface $Z$ and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called $b^m$-contact forms, having an associated critical hypersurface $Z$. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of those are related to smooth contact structures through a desingularization technique. The problem of existence of $b^m$-contact structures on a given manifold is also tackled in this paper. We prove that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a $b^m$-contact structure. In particular, in the $3$-dimensional case, this construction yields the existence of a generic set of surfaces $Z$ such that the pair $(M,Z)$ is a $b^{2k}$-contact manifold and $Z$ is its critical hypersurface. {As a consequence of the desingularization techniques in \cite{gmw1}, we prove the existence of folded contact forms on any almost contact manifold.}, Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA AcademiaPrize 2016. C ́edric Oms is supported by an AFR-Ph.D. grant ofFNR - Luxembourg National Research Fund. Eva Mi-randa and C ́edric Oms are partially supported by the grants reference number MTM2015-69135-P (MINECO/FEDER)and reference number 2017SGR932 (AGAUR). Eva Miranda was supported by aChaire d’Excellenceof theFondationSciences Math ́ematiques de Pariswhen this project started and this work has been supported bya public grant overseenby the French National Research Agency (ANR) as part of the“Investissements d’Avenir”program (reference: ANR-10-LABX-0098). This material is based upon work supported by the National Science Foundation under Grant No.DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, Califor-nia, during the Fall 2018 semester, Peer Reviewed, Preprint

In this article, we investigate Reeb dynamics onbm-contact manifolds, previously introduced in [MO], which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$., In this article, we investigate Reeb dynamics on $b^m$-contact manifolds, previously introduced in \cite{MO}, which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact $b^m$-contact manifolds without periodic Reeb orbits outside $Z$ are provided. Furthermore, we prove that in dimension $3$, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the $b^m$-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are $\R^+$-invariant in the open ends, obtaining as a corollary the existence of periodic $b^m$-Reeb orbits away from the critical set. The study of $b^m$-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three body problem are described by the Reeb vector field of a $b^3$-contact form that admits an infinite number of periodic orbits at the critical set., Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA AcademiaPrize 2016. C ́edric Oms is supported by an AFR-Ph.D. grant of FNR - Luxembourg National Research Fund. Eva Mi-randa and C ́edric Oms are partially supported by the grants reference number MTM2015-69135-P (MINECO/FEDER)and reference number 2017SGR932 (AGAUR). Eva Miranda was supported by aChaire d’Excellenceof theFondationSciences Math ́ematiques de Pariswhen this project started and this work has been supported by a public grant overseenby the French National Research Agency (ANR) as part of the“Investissements d’Avenir”program (reference: ANR-10-LABX-0098). This material is based upon work supported by the National Science Foundation under Grant No.DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, Califor-nia, during the Fall 2018 semester., Preprint

An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action over all periodic orbits of the diffeomorphism is less than or equal to its Calabi invariant.

An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action over all periodic orbits of the diffeomorphism is less than or equal to its Calabi invariant.

[EN] A Toda¿chain symmetry is shown to underlie the van derWaals gas and its close cousin, the ideal gas. Links to contact geometry are explored.

[EN] A Toda¿chain symmetry is shown to underlie the van derWaals gas and its close cousin, the ideal gas. Links to contact geometry are explored.

We consider the problem of feedback control design for thermodynamic systems. Following past contributions, we consider the expression of closed-loop control in the Thermodynamic Phase Space (TPS) using contact geometry. Some of these past contributions identified restricted classes of admissible controllers by restricting the dynamics to a Legendre submanifold in the TPS. In this paper, we consider the problem of controlling a system through damping feedback control. By allowing this particular class of feedback design technique, we characterize the impact of control on the thermodynamic structure of the closed-loop system in the TPS.An example is considered to illustrate the proposed construction.

In dieser Dissertation beschäftigen wir uns mit angepassten Zusammenhängen und ihren (horizontalen) Dirac-Operatoren auf strikt pseudokonvexen CR-Mannigfaltigkeiten. Einen Zusammenhang nennen wir dann angepasst, wenn er die relevanten Daten parallelisiert. Wir beschreiben den Raum der angepassten Zusammenhänge, indem wir ihre Torsionstensoren studieren, von denen gewisse Teile durch die Geometrie der Mannigfaltigkeit festgelegt sind, während andere frei wählbar sind. Als Anwendung betrachten wir die Eigenschaften der Dirac-Operatoren, die zu diesen Zusammenhängen gehören. Weiter betrachten wir horizontale Dirac-Operatoren, die nur in Richtung des horizontalen Bündels H ableiten. Diese Operatoren sind besser an die Sub-Riemannsche Struktur einer CR-Mannigfaltigkeit angepasst als die vollen Dirac-Operatoren. Wir diskutieren, wann diese Operatoren formal selbstadjungiert sind und beweisen eine Weitzenböck-Typ-Formel. Wir konzentrieren uns dann auf den horizontalen Dirac-Operator zum Tanaka-Webster-Zusammenhang. Dieser ändert sich konform kovariant, wenn wir die Kontaktform konform ändern. Für diesen Operator betrachten wir weiterhin zwei Beispiele: Wir betrachten S^1-Bündel über Kähler-Mannigfaltigkeiten, insbesondere berechnen wir für Sphären einen Teil des Spektrums. Außerdem betrachten wir kompakte Quotienten der Heisenberggruppe und berechnen hier in den Dimensionen 3 und 5 das volle Spektrum. Die horizontalen Dirac-Operatoren sind nicht mehr elliptisch, sondern „elliptisch in Richtung von H“. Mithilfe des Heisenbergkalküls stellen wir fest, dass die horizontalen Dirac-Operatoren nicht hypoelliptisch sind. Im Fall des Tanaka-Webster-Zusammenhangs lässt sich aber zeigen, dass der zugehörige Operator auf gewissen Teilen des Spinorbündels hypoelliptisch ist. Dies genügt, um zu beweisen, dass er (nun auf dem gesamten Spinorbündel) ein reines Punktspektrum hat und die Eigenräume, bis auf den Kern, endlich-dimensional sind und aus glatten Eigenspinoren bestehen., In the present thesis, we study adapted connections and their (horizontal) Dirac operators on strictly pseudoconvex CR manifolds. An adapted connection is one that parallelises the relevant data. We describe the space of adapted connections through their torsion tensors, certain parts of which are determined by the geometry of the manifold, while others may be freely chosen. As an application, we study the properties of the Dirac operators induced by these connections. We further consider horizontal Dirac operators, which only derive in the direction of the horizontal bundle H. These operators are more adapted to the essentially sub-Riemannian structure of a CR manifold than the full Dirac operators. We discuss the question of their self-adjointness and prove a Weitzenböck type formula for these operators. Focusing on the horizontal Dirac operator associated with the Tanaka-Webster connection, we show that this operator changes in a covariant way if we change the contact form conformally. Moreover, for this operator we discuss two examples: On S^1-bundles over Kähler manifolds, we can compute part of the spectrum and for compact quotients of the Heisenberg group, we determine the whole spectrum in dimensions three and five. The horizontal Dirac operators are not elliptic, but rather "elliptic in some directions". We review the Heisenberg Calculus for such operators and find that in general, the horizontal Dirac operators are not hypoelliptic. However, in the case of the Tanaka-Webster connection, the associated horizontal Dirac operator is hypoelliptic on certain parts of the spinor bundle and this is enough to prove that its spectrum consists only of eigenvalues and except for the kernel, the corresponding eigenspaces are finite-dimensional spaces of smooth sections.

Práce se zabývá geometrickými vztahy dvojkolí - kolej v podmínkách pražského metra. Je zde popsán vliv kontaktní geometrie na vývoj opotřebení tvarů jízdních obrysů kol a profilů kolejnic. Dále je zde vyhodnoceno měření, které bylo v rámci této diplomové práce provedeno a porovnány měřící systémy. Na závěr je provedeno shrnutí konkrétních faktorů ovlivňujících opotřebení., This thesis is suggesting a geometry relationship wheel set - track in conditions on Prague Metro. There is described contact geometry influence on damage wheel profiles and rail profiles. Then there is the evaluation of the measurement that is made as a part of this thesis and compared measure systems. At the end there is a summary of factors affecting wear., Dopravní fakulta Jana Pernera, Student přednesl obhajobu své diplomové práce, zodpověděl dotaz vedoucího práce a reagoval i na doplňující otázky členů komise.

In this note, we study thermodynamic systems described in the Thermodynamic Phase Space (TPS), commonly referred as the contact geometry approach to thermodynamics. Following classical and recent contributions in the field, we try to fill a gap between the contact geometry endowed with a metric and stability problems considered in the literature. This examination leads to new interpretations of previously obtained results, and highlights new problems and avenue for research.

In this project we present some applications of symplectic and contact geometry on 3-manifolds. In the first section we introduce the notion of symplectic manifolds and various geometrical objects associated with them, such as the moment map and the Delzant polytope. We also investigate symplectic cuts which are used to create new manifolds from given symplectic manifolds. In the second section we review all the basic definitions that arise from contact geometry such as the contact structure, the contact form and the Reeb vector. Several examples are also used in order to clarify these definitions. In the final section we apply the aforementioned notions to the specific case of Lens spaces, which are 3-manifolds. We also propose a possible application on 5-manifolds which would further expand what was discussed here.

In dieser Dissertation werden Kontaktstrukturen auf beliebigen differenzierbaren Mannigfaltigkeiten ungerader Dimension untersucht. Dies geschiet vermöge der Rabinowitz-Floer-Homologie (RFH), welche 2009 von Cieliebak und Frauenfelder eingeführt wurde. Ein großer Teil der Arbeit widmet sich den technischen Problemen bei der Definition von RFH. Insbesondere wird die Transversalität für die benötigten Modulräume gezeigt. In einem weiteren Abschnitt wird bewiesen, dass RFH im wesentlichen invariant unter subkrittischer Henkelanklebung ist. Schließlich enthält die Arbeit die Berechnung von RFH für einige Brieskorn-Mannigfaltigkeiten. Die dabei gewonnenen Resultate werden dazu verwendet zu zeigen, dass es auf jeder Mannigfaltigkeit, welche füllbare Kontaktstukturen zulässt, entweder unendlich viele verschiedene füllbare Kontaktstrukturen gibt, oder eine Kontaktstruktur mit unendlich vielen verschiedenen Füllungen oder das für alle füllbaren Kontaktstrukturen die RFH von unendlicher Dimension ist für alle Grade., This thesis considers fillable contact structures on odd-dimensional manifolds. For that purpose, Rabinowitz-Floer homology (RFH) is used which was introduced by Cieliebak and Frauenfelder in 2009. A major part of the thesis is devoted to technical problems in the definition of RFH. In particular, it is shown that the moduli spaces involved are cut out transversally. Moreover, it is proved that RFH is essentially invariant under subcritical handle attachment. Finally, RFH is calculated for some Brieskorn manifolds. The obtained results are then used to show for every manifold, which supports fillable contact structures, that there exist either infinitely many different fillable contact structures, or one contact structure with infinitely many different fillings or for every fillable contact structure holds that RFH is infinite dimensional in every degree.

In dieser Dissertation werden Kontaktstrukturen auf beliebigen differenzierbaren Mannigfaltigkeiten ungerader Dimension untersucht. Dies geschiet vermöge der Rabinowitz-Floer-Homologie (RFH), welche 2009 von Cieliebak und Frauenfelder eingeführt wurde. Ein großer Teil der Arbeit widmet sich den technischen Problemen bei der Definition von RFH. Insbesondere wird die Transversalität für die benötigten Modulräume gezeigt. In einem weiteren Abschnitt wird bewiesen, dass RFH im wesentlichen invariant unter subkrittischer Henkelanklebung ist. Schließlich enthält die Arbeit die Berechnung von RFH für einige Brieskorn-Mannigfaltigkeiten. Die dabei gewonnenen Resultate werden dazu verwendet zu zeigen, dass es auf jeder Mannigfaltigkeit, welche füllbare Kontaktstukturen zulässt, entweder unendlich viele verschiedene füllbare Kontaktstrukturen gibt, oder eine Kontaktstruktur mit unendlich vielen verschiedenen Füllungen oder das für alle füllbaren Kontaktstrukturen die RFH von unendlicher Dimension ist für alle Grade., This thesis considers fillable contact structures on odd-dimensional manifolds. For that purpose, Rabinowitz-Floer homology (RFH) is used which was introduced by Cieliebak and Frauenfelder in 2009. A major part of the thesis is devoted to technical problems in the definition of RFH. In particular, it is shown that the moduli spaces involved are cut out transversally. Moreover, it is proved that RFH is essentially invariant under subcritical handle attachment. Finally, RFH is calculated for some Brieskorn manifolds. The obtained results are then used to show for every manifold, which supports fillable contact structures, that there exist either infinitely many different fillable contact structures, or one contact structure with infinitely many different fillings or for every fillable contact structure holds that RFH is infinite dimensional in every degree.

Le sujet de cette thèse est la question du nombre minimal d'orbites de Reeb distinctes sur une variété de contact qui est le bord d'une variété symplectique compacte.L'homologie symplectique $S^1$-équivariante positive est un des outils principaux de cette thèse; elle est construite à partir d'orbites périodiques de champs de vecteurs hamiltoniens sur une variété symplectiquedont le bord est la variété de contact considérée.Nous analysons la relation entre les différentes variantes d'homologie symplectique d'une variété symplectique exacte compacte (domaine de Liouville) et les orbites de Reeb de son bord.Nous démontrons certaines propriétés de ces homologies.Pour un domaine de Liouville plongé dans un autre, nous construisons un morphisme entre leurs homologies.Nous étudions ensuite l'invariance de ces homologies par rapport au choix de la forme de contact sur le bord.Nous utilisons l'homologie symplectique $S^1$-équivariante positive pour donner une nouvelle preuve d'un théorème de Ekeland et Lasrysur le nombre minimal d'orbites de Reeb distinctes sur certaines hypersurfaces dans $R^{2n}$.Nous indiquons comment étendre au cas de certaines hypersurfaces dans certains fibrés en droites complexes négatifs.Nous donnons une caractérisation et une nouvelle façon de calculer l'indice de Conley-Zehnder généralisé, défini par Robbin et Salamon pour tout chemin de matrices symplectiques.Ceci nous a mené à développer de nouvelles formes normales de matrices symplectiques./This thesis deals with the question of the minimal number of distinct periodic Reeb orbits on a contact manifold which is the boundary of a compact symplectic manifold.The positive $S^1$-equivariant symplectic homology is one of the main tools considered in this thesis.It is built from periodic orbits of Hamiltonian vector fields in a symplectic manifold whose boundary is the given contact manifold.Our first result describes the relation between the symplectic, Doctorat en Sciences, info:eu-repo/semantics/nonPublished

Embedded contact homology (ECH) capacities were defined by Hutchings and provide a family of obstructions to embeddings of four-dimensional symplectic manifolds. In Part I of this thesis, we prove that for a four-dimensional Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. This was joint work with Daniel Cristofaro-Gardiner and Michael Hutchings. In Part II of this thesis, we construct topological absolute gradings in Heegaard Floer homology and bordered Floer homology that satify all of the expected properties. This was joint work with Yang Huang. We also show that the isomorphism between Heegaard Floer homology and ECH preserves the absolute gradings.

Embedded contact homology (ECH) capacities were defined by Hutchings and provide a family of obstructions to embeddings of four-dimensional symplectic manifolds. In Part I of this thesis, we prove that for a four-dimensional Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. This was joint work with Daniel Cristofaro-Gardiner and Michael Hutchings. In Part II of this thesis, we construct topological absolute gradings in Heegaard Floer homology and bordered Floer homology that satify all of the expected properties. This was joint work with Yang Huang. We also show that the isomorphism between Heegaard Floer homology and ECH preserves the absolute gradings.

We continue our study on the partition function for 5D supersymmetric Yang-Mills theory on toric Sasaki-Einstein Yp,q manifolds. Previously, using the localisation technique we have computed the perturbative part of the partition function. In this work we show how the perturbative part factorises into four pieces, each corresponding to the perturbative answer of the same theory on R4×S1. This allows us to identify the equivariant parameters and to conjecture the full partition functions (including the instanton contributions) for Yp,q spaces. The conjectured partition function receives contributions only from singular contact instantons supported along the closed Reeb orbits. At the moment we are not able to prove this fact from the first principles.

We continue our study on the partition function for 5D supersymmetric Yang-Mills theory on toric Sasaki-Einstein Yp,q manifolds. Previously, using the localisation technique we have computed the perturbative part of the partition function. In this work we show how the perturbative part factorises into four pieces, each corresponding to the perturbative answer of the same theory on R4×S1. This allows us to identify the equivariant parameters and to conjecture the full partition functions (including the instanton contributions) for Yp,q spaces. The conjectured partition function receives contributions only from singular contact instantons supported along the closed Reeb orbits. At the moment we are not able to prove this fact from the first principles.

On any simply connected Sasaki-Einstein five dimensional manifold one can construct a super Yang-Mills theory which preserves at least two supersymmetries. We study the special case of toric Sasaki-Einstein manifolds known as Y p,q manifolds. We use the localisation technique to compute the full perturbative part of the partition function. The full equivariant result is expressed in terms of certain special function which appears to be a curious generalisation of the triple sine function. As an application of our general result we study the large N behaviour for the case of single hypermultiplet in adjoint representation and we derive the N 3-behaviour in this case.

We continue our study on the partition function for 5D supersymmetric Yang-Mills theory on toric Sasaki-Einstein Yp,q manifolds. Previously, using the localisation technique we have computed the perturbative part of the partition function. In this work we show how the perturbative part factorises into four pieces, each corresponding to the perturbative answer of the same theory on R4×S1. This allows us to identify the equivariant parameters and to conjecture the full partition functions (including the instanton contributions) for Yp,q spaces. The conjectured partition function receives contributions only from singular contact instantons supported along the closed Reeb orbits. At the moment we are not able to prove this fact from the first principles.

On any simply connected Sasaki-Einstein five dimensional manifold one can construct a super Yang-Mills theory which preserves at least two supersymmetries. We study the special case of toric Sasaki-Einstein manifolds known as Y p,q manifolds. We use the localisation technique to compute the full perturbative part of the partition function. The full equivariant result is expressed in terms of certain special function which appears to be a curious generalisation of the triple sine function. As an application of our general result we study the large N behaviour for the case of single hypermultiplet in adjoint representation and we derive the N 3-behaviour in this case.

To date, bioceramics have not been applied successfully in total knee joint endoprostheses. Sintered bioceramics can be machined only by grinding and polishing processes. Due to high quality requirements, there are significant challenges with regard to these machining technologies. An automated precise economical process chain for the manufacturing of a new all-ceramic knee implant design was developed. It was assumed the geometrical accuracy and the shape of implant contact geometry specified during the manufacturing process has a substantial influence on the wear behavior of the prosthesis. The importance of the surface quality of the ceramic implant surface remains unclear and warrants future examination.

To date, bioceramics have not been applied successfully in total knee joint endoprostheses. Sintered bioceramics can be machined only by grinding and polishing processes. Due to high quality requirements, there are significant challenges with regard to these machining technologies. An automated precise economical process chain for the manufacturing of a new all-ceramic knee implant design was developed. It was assumed the geometrical accuracy and the shape of implant contact geometry specified during the manufacturing process has a substantial influence on the wear behavior of the prosthesis. The importance of the surface quality of the ceramic implant surface remains unclear and warrants future examination.

L'homologie de contact est un invariant homologique pour variétés de contact dont la définition est basée sur l'utilisation de courbes holomorphes. Ce travail de thèse concerne l'étude de cet invariant dans le cas des produits de contact., Doctorat en Sciences, info:eu-repo/semantics/nonPublished

This thesis consists of two parts, which can be read separately. In the first part we study aspects of topological field theories. We show how to topologically twist three-dimensional N=2 supersymmetric Chern-Simons theory using a contact structure on the underlying manifold. This gives us a formulation of Chern-Simons theory together with a set of auxiliary fields and an odd symmetry. For Seifert manifolds, we show how to use this odd symmetry to localize the path integral of Chern-Simons theory. The formulation of three-dimensional Chern-Simons theory using a contact structure admits natural generalizations to higher dimensions. We introduce and study these theories. The focus is on the five-dimensional theory, which can be understood as a topologically twisted version of N=1 supersymmetric Yang-Mills theory. When formulated on contact manifolds that are circle fibrations over a symplectic manifold, it localizes to contact instantons. For the theory on the five-sphere, we show that the perturbative part of the partition function is given by a matrix model. In the second part of the thesis, we study supersymmetric sigma models in the Hamiltonian formalism, both in a classical and in a quantum mechanical setup. We argue that the so called Chiral de Rham complex, which is a sheaf of vertex algebras, is a natural framework to understand quantum aspects of supersymmetric sigma models in the Hamiltonian formalism. We show how a class of currents which generate symmetry algebras for the classical sigma model can be defined within the Chiral de Rham complex framework, and for a six-dimensional Calabi-Yau manifold we calculate the equal-time commutators between the currents and show that they generate the Odake algebra.

This thesis consists of two parts, which can be read separately. In the first part we study aspects of topological field theories. We show how to topologically twist three-dimensional N=2 supersymmetric Chern-Simons theory using a contact structure on the underlying manifold. This gives us a formulation of Chern-Simons theory together with a set of auxiliary fields and an odd symmetry. For Seifert manifolds, we show how to use this odd symmetry to localize the path integral of Chern-Simons theory. The formulation of three-dimensional Chern-Simons theory using a contact structure admits natural generalizations to higher dimensions. We introduce and study these theories. The focus is on the five-dimensional theory, which can be understood as a topologically twisted version of N=1 supersymmetric Yang-Mills theory. When formulated on contact manifolds that are circle fibrations over a symplectic manifold, it localizes to contact instantons. For the theory on the five-sphere, we show that the perturbative part of the partition function is given by a matrix model. In the second part of the thesis, we study supersymmetric sigma models in the Hamiltonian formalism, both in a classical and in a quantum mechanical setup. We argue that the so called Chiral de Rham complex, which is a sheaf of vertex algebras, is a natural framework to understand quantum aspects of supersymmetric sigma models in the Hamiltonian formalism. We show how a class of currents which generate symmetry algebras for the classical sigma model can be defined within the Chiral de Rham complex framework, and for a six-dimensional Calabi-Yau manifold we calculate the equal-time commutators between the currents and show that they generate the Odake algebra.

This thesis consists of two parts, which can be read separately. In the first part we study aspects of topological field theories. We show how to topologically twist three-dimensional N=2 supersymmetric Chern-Simons theory using a contact structure on the underlying manifold. This gives us a formulation of Chern-Simons theory together with a set of auxiliary fields and an odd symmetry. For Seifert manifolds, we show how to use this odd symmetry to localize the path integral of Chern-Simons theory. The formulation of three-dimensional Chern-Simons theory using a contact structure admits natural generalizations to higher dimensions. We introduce and study these theories. The focus is on the five-dimensional theory, which can be understood as a topologically twisted version of N=1 supersymmetric Yang-Mills theory. When formulated on contact manifolds that are circle fibrations over a symplectic manifold, it localizes to contact instantons. For the theory on the five-sphere, we show that the perturbative part of the partition function is given by a matrix model. In the second part of the thesis, we study supersymmetric sigma models in the Hamiltonian formalism, both in a classical and in a quantum mechanical setup. We argue that the so called Chiral de Rham complex, which is a sheaf of vertex algebras, is a natural framework to understand quantum aspects of supersymmetric sigma models in the Hamiltonian formalism. We show how a class of currents which generate symmetry algebras for the classical sigma model can be defined within the Chiral de Rham complex framework, and for a six-dimensional Calabi-Yau manifold we calculate the equal-time commutators between the currents and show that they generate the Odake algebra.

This thesis consists of two parts, which can be read separately. In the first part we study aspects of topological field theories. We show how to topologically twist three-dimensional N=2 supersymmetric Chern-Simons theory using a contact structure on the underlying manifold. This gives us a formulation of Chern-Simons theory together with a set of auxiliary fields and an odd symmetry. For Seifert manifolds, we show how to use this odd symmetry to localize the path integral of Chern-Simons theory. The formulation of three-dimensional Chern-Simons theory using a contact structure admits natural generalizations to higher dimensions. We introduce and study these theories. The focus is on the five-dimensional theory, which can be understood as a topologically twisted version of N=1 supersymmetric Yang-Mills theory. When formulated on contact manifolds that are circle fibrations over a symplectic manifold, it localizes to contact instantons. For the theory on the five-sphere, we show that the perturbative part of the partition function is given by a matrix model. In the second part of the thesis, we study supersymmetric sigma models in the Hamiltonian formalism, both in a classical and in a quantum mechanical setup. We argue that the so called Chiral de Rham complex, which is a sheaf of vertex algebras, is a natural framework to understand quantum aspects of supersymmetric sigma models in the Hamiltonian formalism. We show how a class of currents which generate symmetry algebras for the classical sigma model can be defined within the Chiral de Rham complex framework, and for a six-dimensional Calabi-Yau manifold we calculate the equal-time commutators between the currents and show that they generate the Odake algebra.

A previous simplification of the Dang Van equivalent stress measure for assessment of subsurface initiated rolling contact fatigue (RCF) related to wheel-rail contact is modified. The new criterion is intended for real-time assessment of subsurface RCF from measured wheel-rail contact forces. The only needed parameters in the new expression for the equivalent stress are the vertical force and the wheel and rail radii. Comparisons between the new and the original criterion are carried out and show good agreement for the studied cases of tangent track operations. By employing principles of vehicle dynamics the criterion is further extended to the case of operations in curves. Reasonable consistency was found for curve radii down to approximately 2000 m., QC 20110706 8th Conference on Contact Mechanics and Wear of Rail/Wheel Systems (CM2009), Florence, ITALY, SEP, 2009

In this paper an analysis of the uniformity of temperature distribution on the surface of a semi-infinite friction body, mutual heat interference of discrete contact areas, and the accuracy of the concentrated model of thermal and mechanical contact of rubbing solids in response to the degree of contact discreteness is conducted. The conducted research yields a criterion of selection for the contact geometry model. A classification of heat friction problems that allows the selection of an appropriate model of contact geometry for simulation of heat friction processes is proposed., QC 20160921