The paper shows how Hamiltonian diffeomorphisms and homeomorphisms can be broken down into smaller, manageable pieces.
problem Fragmenting Hamiltonian diffeomorphisms and homeomorphisms on surfaces.
method Develops a C0-fragmentation property for Hamiltonian diffeomorphisms and homeomorphisms on surfaces, proving it with a Lipschitz estimate. result Hamiltonian diffeomorphisms and homeomorphisms can be decomposed into smaller, compactly supported pieces with a Lipschitz estimate on the C0-norm. We generalize the "hamiltonian topology" on hamiltonian isotopies to an intrinsic "symplectic topology" on the space of symplectic isotopies. We use it to define the group SSympeo(M,ω) of strong symplectic homeomorphisms, which generalizes the group Hameo(M,ω) of hamiltonian homeomorphisms introduced by Oh and Mull…
New spectral invariants recover Calabi invariant for surface dynamics.
problem Understanding Hamiltonian homeomorphisms and spectral invariants.
method Defining new spectral invariants for Lagrangian links in surfaces.
result Our invariants recover the Calabi invariant and resolve open questions.
The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.
problem Understanding K-contact manifolds with minimal closed Reeb orbits and their homeomorphism properties.
method Using Boothby-Wang fibration and Hamiltonian torus actions, the study constructs and analyzes K-contact manifolds.
result The existence of K-contact manifolds with minimal closed Reeb orbits that are not homeomorphic to spheres and have unique cohomology rings.
In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for…
This paper shows how pseudo-Anosov flows represent stable Hamiltonian classes and limits the ways 3-manifolds can be obtained from knots.
problem Understanding the canonical representatives of stable Hamiltonian classes and their implications for 3-manifolds.
method Explains the analogy between pseudo-Anosov flows and stable Hamiltonian classes and generalizes an argument to limit the ways 3-manifolds can be obtained from knots.
result There are finitely many pseudo-Anosov flows admitting positive Birkhoff sections on any given rational homology 3-sphere, and any 3-manifold can be obtained in at most finitely many ways as p/q surgery on a fibered hyperbolic knot in S3. The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian G-manifold is a strat…
We investigate polyhedral 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it k-Hamiltonian} if it contains the full k-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so…
The paper studies Hamiltonian flows for pseudo-Anosov mapping classes on surfaces.
problem Understanding the dynamics of pseudo-Anosov mapping classes on Teichmüller spaces.
method Explicit formulae for Hamiltonian flows generated by invariant functions.
result Hamiltonian flows coincide with the action of pseudo-Anosov homeomorphisms at time one.
We define the symplectic displacement energy of a non-empty subset of a compact symplectic manifold as the infimum of the Hofer-like norm [5] of symplectic diffeomorphisms that displace the set. We show that this energy (like the usual displacement energy defined using Hamiltonian diffeomorphisms) is a strictly positiv…
Floer theory connects dynamics on surfaces to their chain-level theory.
problem Connecting dynamics on surfaces to their Floer theory.
method Using capped 1-periodic orbits and ideas from Hofer-Wysocki-Zehnder's theory.
result Definition and computation of novel spectral invariants.
New method finds hyperelliptic 4-manifolds from polytope vector-colorings.
problem Finding hyperelliptic 4-manifolds from polytope vector-colorings.
method Introducing Hamiltonian subcomplexes and their corresponding subgroups.
result For dimensions ≤ 4, there is a bijection between Hamiltonian subcomplexes and hyperelliptic involutions.
We provide new insight into the analysis of N-body problems by studying a compactification MN of R3N that is compatible with the analytic properties of the N-body Hamiltonian HN. We show that our compactification coincides with the compactification introduced by Vasy using blow-ups in order to stu…
The study classifies manifolds realized as orbit spaces of non-free Z2^k actions.
problem Classifying manifolds realized as orbit spaces of non-free Z2^k actions.
method Examining actions of subgroups H on real moment-angle manifolds and analyzing orbit spaces.
result Constructs series of manifolds homeomorphic to S^n and manifolds admitting hyperelliptic involutions.
Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.
problem Extending diffeomorphisms to global mappings of manifolds.
method Elementary argument for diffeomorphisms, deep results for homeomorphisms and bi-Lipschitz mappings.
result Extension of Palais' result to homeomorphisms and bi-Lipschitz mappings.
Study on homeomorphism groups of manifolds using set theory.
problem Relationship between set theory and homeomorphism groups of manifolds.
method First-order rigidity, type versus conjugacy, axiom of constructibility, projective determinacy.
result Under V=L, homeomorphism groups of manifolds are first-order rigid and conjugacy class is determined by type.
New proof associates partitions to isotopic pseudo-Anosov homeomorphisms.
problem Stable and unstable foliations for pseudo-Anosov homeomorphisms.
method Geometric realization of Fathi's result for isotopic homeomorphisms.
result Associated stable and unstable partitions for isotopic pseudo-Anosov homeomorphisms.
Uniform interpretation of group theory in manifold homeomorphisms.
problem Understanding group properties in manifold homeomorphisms.
method First order theory interpretation of second order group theory.
result Many group theory problems encoded in homeomorphism groups.
Proves existence of sentences to identify homeomorphic manifolds.
problem Identifying homeomorphic manifolds using group properties.
method Defines sentences in group language to match homeomorphic manifolds.
result Existence of sentences to distinguish homeomorphic manifolds.
New homeomorphism found in Klein bottle group.
problem Understanding homeomorphisms of Klein bottle.
method Using recent results on commutator length.
result Existence of homeomorphism with positive stable commutator length.
Study weak conjugacy in surface homeomorphisms.
problem Understanding weak conjugacy in homeomorphisms of surfaces.
method Exploring the group of homeomorphisms isotopic to the identity.
result New insights into weak conjugacy relations.
Paper explains dynamics of homeomorphisms to mapping tori geometry.
problem Understanding dynamics of end-periodic homeomorphisms.
method Illustration-driven overview of recent results.
result Analogue of Brock's theorem for infinite-type surfaces.
This work generalizes Hamiltonian mechanics using closed differential forms.
problem Hidden invariants in classical Hamiltonian mechanics.
method Establishes a novel correspondence between generalized Hamiltonian mechanics and multisymplectic geometry.
result Key theorems linking classical and generalized Hamiltonian systems.
Study investigates Hamiltonian systems in fibered almost-symplectic manifolds.
problem Existence and properties of non-vertical fully-Hamiltonian vector fields in almost symplectic manifolds with Lagrangian fibrations.
method Investigates vector fields in 2n-dimensional almost symplectic manifolds with Lagrangian fibrations, focusing on partially-Hamiltonian and fully-Hamiltonian vector fields.
result Non-vertical fully-Hamiltonian vector fields exist under certain genericity conditions and can be reduced to families of symplectic-Hamiltonian vector fields.
Holographic energy equals Hamiltonian energy.
problem Equating holographic and Hamiltonian energies.
method Relative holographic and Hamiltonian energy comparison.
result Holographic energy is identical to Hamiltonian energy.
Develops Hamiltonian Score Matching and Generative Flows for machine learning.
problem Estimating score functions and designing generative models.
method Introduces Hamiltonian velocity predictors (HVPs) for score matching and generative flows.
result Hamiltonian Generative Flows (HGFs) rival leading generative modeling techniques.
Summing Hamiltonian manifolds with a common submanifold.
problem Combining Hamiltonian manifolds with a shared submanifold.
method Establishing symplectic reduction and comparing Chern classes.
result Symplectic reduction of the sum agrees with the sum of reductions.
Unified framework recovers and improves classical Brouwer homeomorphism results.
problem Classical Brouwer homeomorphism theory and its dynamics.
method Unified foliated framework combining Le Calvez's and Handel's methods.
result Recovery and improvement of classical results in Brouwer homeomorphism theory.
Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.
problem Understanding the structure of homeomorphisms in Euclidean space.
method Proving every orientation-preserving homeomorphism can be written as a commutator of two such homeomorphisms.
result Every orientation-preserving homeomorphism of Euclidean space is a commutator of two homeomorphisms.
The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.
problem Investigating stability properties of Hamiltonian Poisson integrators.
method Examples of Lotka-Volterra dynamics and numerical investigations of a non-integrable system are used.
result The existence of a modified Hamiltonian is crucial for the stability of Hamiltonian Poisson integrators.
If K is a compact Lie group and g≥2 an integer, the space K2g is endowed with the structure of a Hamiltonian space with a Lie group valued moment map Φ. Let β be in the centre of K. The reduction Φ−1(β)/K is homeomorphic to a moduli space of flat connections. When K is simply connected, a dire…
The study of periodic subgroups in homeomorphism groups of manifolds.
problem Burnside problem for homeomorphism groups of manifolds.
method Analyzing surface and circle homeomorphism groups, extending Tits alternative.
result Every finitely generated periodic subgroup is finite for most manifolds.
Adapts pivoting technique to circle homeomorphisms for proofs.
problem Probabilistic Tits alternative and exponential synchronization.
method Adapts Gou{ë}zel's pivoting technique.
result Different proofs of probabilistic Tits alternative and exponential synchronization.
No algorithm exists to decide 4-manifold homeomorphism.
problem Deciding homeomorphism for 4-manifolds.
method Demonstrated through a specific example of a connected sum of 12 copies of S^2 × S^2.
result No algorithm exists for deciding homeomorphism of 4-manifolds.
We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2) with an irrational symplectic structure). The underlying smooth manifolds of our…
Proof shows homeomorphism problem is unsolvable.
problem Unsolvable homeomorphism problem in topology.
method Detailed proof of Markov's theorem.
result Existence of unrecognizable manifolds in higher dimensions.
Study covers of sphere with homeomorphisms lifting property.
problem Finite abelian covers of sphere with lifting homeomorphisms.
method Completely determined covers with specific lifting property.
result Properties of finite abelian covers with lifting homeomorphisms.
New algorithms improve MCMC efficiency for complex distributions.
problem High variance and low effective sample size in MCMC samplers.
method Antithetic Riemannian Manifold and Quantum-Inspired Hamiltonian Monte Carlo.
result Improved effective sample size and variance reduction.
Goldman bracket distinguishes surface homeomorphisms.
problem Characterizing homeomorphisms between non-compact surfaces.
method Using the Goldman bracket to distinguish homeomorphisms.
result A homotopy equivalence is a homeomorphism if it preserves the Goldman bracket.
The paper explores deformations of quasi-Hamiltonian spaces to Hamiltonian spaces.
problem Deforming quasi-Hamiltonian spaces to Hamiltonian spaces.
method Introducing and proving examples of deformations, including Lie groups and conjugacy classes.
result Moduli space of flat G-connections deforms to T*G^r+g.
Study shows similar result to Margulis for Cantor set homeomorphisms.
problem Understanding groups of homeomorphisms of Cantor sets.
method Analogous to Margulis's proof for linear groups.
result Groups of homeomorphisms either preserve a measure or contain a free subgroup.
This paper studies the geometry of the group of all co-Hamiltonian diffeomorphisms of a compact cosymplectic manifold (M,ω,η). The fix-point theory for co-Hamiltonian diffeomorphisms is studied, and we use Arnold's conjecture to predict the exact minimum number of fix point that such a diffeomorphism must have (thi…
Let (M,w) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with w. For instance, g could be Kahler, with Kahler form w. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional under Hamiltonian defo…
New integrators preserve geometric structure in Hamiltonian systems.
problem Preserving geometric structure in Hamiltonian systems on Jacobi manifolds.
method Combining Poissonization and symplectic bi-realizations to construct structure-preserving integrators.
result Explicit construction and application of Jacobi Hamiltonian integrators.
The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.
problem Understanding the structure and properties of homeomorphism groups of self-similar 2-manifolds.
method Survey of recent results, exposition of classical results, treatment of stable sets, and proof of new theorems.
result Characterization of homeomorphisms of perfectly self-similar 2-manifolds and extensions of existing results.
The paper develops a theory linking Hamiltonian and quasi-Hamiltonian manifolds.
problem Understanding the deformation of Hamiltonian quasi-Poisson manifolds to Hamiltonian Poisson manifolds.
method Introduces a generalized Hamiltonian deformation theory and constructs a topological quantum field theory.
result Shows that the imploded cross section of the double $D(G)_\imp$ deforms to the implosion of the cotangent bundle $T^*G_\imp$.
In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction on M. We then discuss conditions which guarantee that su…
Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving degree one cohomology class a. We investigate the distortion in Homeo(X,a). Let g be an element of Homeo(X,a). We define a Nielsen-type equivalence relation on the space of g-invari…