Study shows infinite-dimensional third bounded cohomology for non-orientable surfaces.
problem Understanding the third bounded cohomology of non-orientable surfaces.
method Analyzing measure-preserving homeomorphisms of non-orientable surfaces.
result Third bounded cohomology is infinite-dimensional.
In this paper we introduce a general notion of weak extension property for embeddings induced by a group actions. As an example, for the group H(M, m) of measure-preserving homeomorphisms of a noncompact manifold M, we deduce weak type extension theorems, and as an application we exhibit the local contractibility of th…
We prove that manifolds with complicated enough fundamental group admit measure-preserving homeomorphisms which have positive stable fragmentation norm with respect to balls of bounded measure.
Suppose M is a noncompact connected 2-manifold and m is a good Radon measure of M with m(partial M) = 0. Let H(M)_0 denote the identity component of the group of homeomorphisms of M equipped with the compact-open topology and let H(M; m)_0 denote the identity component of the subgroup consisting of m-preserving homeomo…
Let G be a group acting on the plane by orientation-preserving homeomorphisms. We show that if for some k>0 there is a ball of radius r > k/\sqrt{3} such that each point x in the ball satisfies |gx -hx| < k for all g, h in G, and the action of G satisfies a nonwandering hypothesis, then the action has a global fixed po…
The paper extends a measure preserving property to bi-Lipschitz maps between Moran sets.
problem Extending the measure preserving property to Moran sets.
method Analyzing bi-Lipschitz maps between Moran sets.
result Bi-Lipschitz maps between Moran sets preserve measure locally.
The study shows finite measure-preserving isometry groups for certain metric measure spaces.
problem Understanding the structure of isometry groups in metric measure spaces.
method Analyzing synthetic negative Ricci curvature and Bakry-Émery Ricci curvature.
result The measure-preserving isometry group is finite for compact metric measure spaces with specific curvature conditions.
Suppose M is a noncompact connected n-manifold and m is a good Radon measure of M with m(bdry M) = 0. Let H(M; m) denote the group of m-preserving homeomorphisms of M equipped with the compact-open topology and H_E(M; m) denote the subgroup consisting of all h in H(M; m) which fix the ends of M. Each h in H_E(M; m) mov…
Unified theory of measure-preserving diffusions on manifolds.
problem Deriving a complete recipe for measure-preserving diffusions on manifolds.
method Developed a geometric theory that unifies and generalizes previous constructions, relying on intrinsic geometry of the target measure.
result The completeness result is a direct consequence of manifold topology and target measure geometry.
The paper defines and proves non-triviality of volume and Euler classes in bounded cohomology of transformation groups.
problem Defining and proving non-triviality of volume and Euler classes in bounded cohomology.
method Definition and proof of non-triviality of volume and Euler classes in bounded cohomology of transformation groups.
result Non-triviality of volume and Euler classes in bounded cohomology of transformation groups.
Study curvatures of diffeomorphisms on non-orientable surfaces.
problem Computing curvatures of measure-preserving diffeomorphisms on non-orientable surfaces.
method Extending Arnold and Lukatskii's approach, computing curvatures and asymptotics.
result Computed curvatures and asymptotics for the Klein bottle and real projective plane.
We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated s…
Characterizes measures preserving compound mixed renewal process properties.
problem Preserving compound mixed renewal process properties under different probability measures.
method Characterization of progressively equivalent probability measures.
result Any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures.
Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.
problem Existence of non-trivial holomorphic vector fields on compact complex manifolds.
method Vanishing result for measure preserving holomorphic vector fields, Gibbs stability, and log terminal singularities.
result No non-trivial holomorphic vector fields on compact complex manifolds with big anti-canonical line bundle.
We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping cl…
The paper explores arbitrage opportunities in derivative markets under specific conditions.
problem Arbitrage opportunities in derivative markets under different conditions.
method Analyzes the relationship between pricing kernel monotonicity and stochastic arbitrage opportunities.
result Pricing kernel nonmonotonicity is equivalent to stochastic arbitrage opportunities under adequacy.
For a commodity spot price dynamics given by an Ornstein-Uhlenbeck process with Barndorff-Nielsen and Shephard stochastic volatility, we price forwards using a class of pricing measures that simultaneously allow for change of level and speed in the mean reversion of both the price and the volatility. The risk premium i…
We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space …
We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can pe…
Compact embeddings for invariant functions in metric-measure spaces.
problem Embedding functions with symmetry in metric-measure spaces.
method Analyzing H-invariant functions in compact metric-measure spaces, extending to Riemannian manifolds. result Obtained compact Sobolev embeddings for critical exponents.
New expanders found using origami surfaces with spectral gap.
problem Constructing expanders with spectral gap on surfaces of arbitrary genus.
method Affine actions on origami surfaces to achieve spectral gap.
result New expanders distinct from classical ones.
We derive integral tests for the existence and absence of arbitrage in a financial market with one risky asset which is either modeled as stochastic exponential of an Ito process or a positive diffusion with Markov switching. In particular, we derive conditions for the existence of the minimal martingale measure. We al…
New copula models capture volatility and directionality in financial time series.
problem Modeling financial return series with volatility and serial correlation.
method Stationary d-vine copula processes with v-transforms for stochastic volatility and directionality.
result Models can rival and sometimes outperform GARCH family models.
New framework uses simplicial and categorical methods to detect market inconsistencies.
problem Detecting inconsistencies in financial markets using non-measure-preserving transitions.
method Simplicial and categorical formulation of AB type arbitrage in filtered market systems.
result Holonomy along loops reveals global inconsistencies invisible at local levels.
Handlebody groups are rigid under measure equivalence.
problem Proving handlebody groups are rigid under measure equivalence.
method Proving superrigidity for measure equivalence of handlebody groups.
result Every countable group measure equivalent to handlebody groups is virtually isomorphic to them.
We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requ…
For a positive integer n≥3, the collection of n-sided polygons embedded in 3-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded n-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components …
Study graph products of groups, classifying them up to measure equivalence and rigidity.
problem Classifying graph products of groups up to measure equivalence and rigidity.
method Measure-theoretic and structural properties of von Neumann algebras, rigidity theorems.
result Quantified measure equivalence classification and rigidity theorems for graph products.
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.
We introduce an universum of the Polish (=complete separable metric) space - the convex cone of distance matrices and study its geometry. It happened that the generic Polish spaces in this sense of this universum is so called Urysohn spaces defined by P.S.Urysohn in 20-th, and generic metric triple (= metric space with…
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.
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 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.
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.
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.
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.
Positive factorization for pseudoperiodic homeomorphisms on surfaces.
problem Factorization of pseudoperiodic homeomorphisms on surfaces.
method Generalization of classical results on smooth germs of surfaces, topological characterization of monodromies, and use of positive factorization criteria.
result Pseudoperiodic homeomorphisms on surfaces with positive fractional Dehn twist coefficients and screw numbers admit a positive factorization.
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.