Surface groups found in interval diffeomorphisms without global fix points.
problem Existence of surface groups in interval diffeomorphisms.
method Analyzing the structure of diffeomorphisms of the interval.
result Surface groups with specific properties in interval diffeomorphisms.
We prove a homological stability theorem for moduli spaces of manifolds of dimension 2n, for attaching handles of index at least n, after these manifolds have been stabilised by countably many copies of Sn×Sn. Combined with previous work of the authors, we obtain an analogue of the Madsen--Weiss theorem …
The leaf space of a Killing Riemannian foliation is a diffeological quasifold.
problem Understanding the structure of leaf spaces of Riemannian foliations.
method Analyzing the holonomy groupoid and using diffeological spaces.
result The leaf space of a Killing Riemannian foliation is a diffeological quasifold.
The paper constructs and analyzes self-similar blowup solutions for a wave map equation.
problem Existence and stability of self-similar blowup solutions for a wave map equation.
method Construction of self-similar solutions, detailed nonlinear stability analysis, spectral analysis of linearized operators.
result Sharp semigroup bounds and nonlinear stability of all discretely self-similar profiles in all dimensions.
Kropholler's class of groups is the smallest class of groups which contains all finite groups and is closed under the following operator: whenever G admits a finite-dimensional contractible G-CW-complex in which all stabilizer groups are in the class, then G is itself in the class. Kropholler's class admits a hie…
Countable modular groups found on surfaces with infinite type.
problem Finding modular groups of infinite type surfaces.
method Proving countable modular groups for orientable infinite type surfaces.
result Every orientable infinite type surface has a countable modular group.
Study chaotic behavior in homeomorphism groups of countable products of spaces.
problem Investigate chaotic behavior in homeomorphism groups of countable products of various metrizable topological spaces.
method Construct numerous examples of chaotic groups of homeomorphisms of countable products of spaces.
result New chaotic groups of homeomorphisms of countable products of various metrizable topological spaces are discovered.
Finite groups can be represented as origami automorphisms, extended to countable groups.
problem Representing countable groups as automorphisms of origamis.
method Considering origamis on the Loch Ness monster.
result Every countable group can be represented as origami automorphisms.
We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H whe…
It is well known that a countable group admits a left-invariant total order if and only if it acts faithfully on R by orientation preserving homeomorphisms. Such group actions are special cases of group actions on simply connected 1-manifolds, or equivalently, actions on oriented order trees. We characterize a class of…
Extends inf-convolution to countable risk measures for risk sharing.
problem Limited inf-convolution theory to finite sets of risk measures.
method Extends inf-convolution to countable sets, investigates properties and results.
result Generalizes known properties and results to countable case.
We prove that two countable locally finite-by-abelian groups G,H endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely-generated or both are infinitely generated. On the other hand, we show that each countable group…
Every countable compact subset of sphere is tame.
problem Characterizing compact subsets of spheres.
method Proving homeomorphic complements imply homeomorphic subsets.
result Wild subspaces like Antoine contain Cantor sets.
Finite approximations help reconstruct countable metric and ultrametric spaces.
problem Reconstructing countable metric and ultrametric spaces.
method Topological reconstruction using inverse limits of finite T0 spaces. result Countable metric and ultrametric spaces can be reconstructed as finite approximations.
Let G be a countable group that splits as a free product of groups of the form G=G1∗⋯∗Gk∗FN, where FN is a finitely generated free group. We identify the closure of the outer space PO(G,{G1,…,Gk}) for the axes topology with the space of projective minimal, \emph{very small} …
We show that all non-trivial continuous endomorphisms of the circle group are topologically mixing. We also show that there exists a large infinite class of continuous endomorphisms of any n-dimensional torus group which are topologically mixing. Lastly, we prove that any continuous endomorphism on an abelian polish se…
New findings on group actions and stabilizers of infinite sets.
problem Understanding finiteness properties of stabilizers in group actions.
method Analyzing oligomorphic actions and permutational wreath products.
result Existence of finite subsets with non-FP∞ stabilizers. Compact groups can be represented as dessin automorphisms.
problem Representing all countable groups as automorphisms of dessins.
method Constructive proof using non-compact dessins.
result Any tame action of a countable group is realizable as a dessin automorphism.
Generalizes theorem for topological G-manifolds with linear Lie groups G.
problem Understanding topological G-manifolds with linear Lie groups G. method Using countable CW complexes and Palais-proper actions.
result Topological G-manifolds have G-homotopy type of countable G-CW complexes. Artin groups of hyperbolic type are boundary amenable and have rigid properties.
problem Characterizing rigidity and measure equivalence properties of Artin groups.
method Analyzing boundary amenability, measure equivalence, and fixed set graphs.
result Measure equivalent Artin groups of hyperbolic type have isomorphic fixed set graphs.
We compute the asymptotic dimension of the rationals given with an invariant proper metric. Also, we show that a countable torsion abelian group taken with an invariant proper metric has asymptotic dimension zero.
The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are paramet…
We characterize language generation with stability and breadth, proving impossibility results.
problem Characterizing and proving impossibility results for language generation with stability and breadth.
method Analysis of existing notions of breadth and stability, proving lower bounds.
result Proven impossibility of generating with higher perplexity or lower hallucination rate for stable generators.
For every countable group G we construct a compact path connected subspace K of R^4 whose fundamental group is isomorphic to G. Our construction is much simpler than the one found recently by Virk.
Classifies when homeomorphism groups of stable surfaces have automatic continuity.
problem Determining when homeomorphism groups of stable surfaces are continuous.
method Developed a general framework to prove automatic continuity for homeomorphism groups, applied to stable surfaces and Stone spaces.
result Classification of stable surfaces with respect to automatic continuity of their homeomorphism groups.
We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental group of M are isomorphic to G.
New algorithm for countable bandits with optimal regret.
problem Stochastic bandit problem with countably many arms.
method Fully adaptive online learning algorithm with O(log n) expected cumulative regret.
result Achieves optimal regret of O(log n) after any number of plays n.
Study shows non-uniqueness of Brakke flow near flat singular points.
problem Exploring instability of minimal surfaces at flat singular points.
method Analyzes the behavior of stationary varifolds and their blow-ups.
result Proves existence of non-constant Brakke flow near flat singular points.
Study of infinite-string braids as a limit of finite ones.
problem Properties of infinite-string braids.
method Investigate via direct limit of finite n-braid groups.
result Basic properties of countable-string braids.
We show that C^1 hypersurfaces in the Heisenberg group are countably N-rectifiable. As a corollary, this shows that all C^1_H graphs over the xy-plane are countable N-rectifiable, showing the equivalence of this notion of rectifiability with that of Franchi, Serra Cassano and Serapioni for such surfaces.
Study on eigenvalues of Laplacian on anti-de Sitter manifolds.
problem Distribution and stability of eigenvalues in pseudo-Riemannian geometry.
method Analysis of L2-eigenvalues on anti-de Sitter manifolds. result Countably many L2-eigenvalues are stable under small deformations. Quantum isometry groups extend to all countable metric spaces, and loose embeddings help understand metric space relationships.
problem Understanding the quantum isometry groups of all countable metric spaces.
method Defining and studying loose embeddability, showing that 0-dimensional compact metric spaces are generically loosely embeddable into the real line.
result 0-dimensional compact metric spaces are generically loosely embeddable into the real line.
Generalizes manifold results for Lie groups, proving equivariant homotopy type.
problem Hilbert-Smith conjecture for topological G-manifolds. method Generalization of previous results, verification of n-classifying spaces.
result Any Palais-proper topological G-manifold has the equivariant homotopy type of a countable proper G-CW complex. New criterion for smoothing actions with singularities.
problem Smoothability of actions with singular diffeomorphisms.
method Criterion for conjugacy of actions to smooth ones.
result Aperiodic actions by diffeomorphisms with countably many singularities are conjugate to smooth actions.
New model for insurance states using Markov jump processes with non-countable state space.
problem Modeling insurance states with non-countable state spaces.
method Developed a new Thiele's differential equation for continuous time rehabilitation rates.
result Allows for consistent calculation of reserves in disability insurance.
We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group Γ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit e…
We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of GL+(2,R) avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surf…
Causal Set Theory's Hauptvermutung is resolved in two ways, one of which is true.
problem Formulating and resolving the Hauptvermutung in Causal Set Theory.
method Two mathematically well-defined formulations of the Hauptvermutung, one of which is true.
result The Hauptvermutung is true when finite sets are replaced by countable sets.
This note proves equivariant de Rham cohomology for quotient spaces.
problem Computing de Rham cohomology of quotient spaces under group actions.
method Equivariant identification of de Rham complexes using foliation theory.
result Canonical isomorphism of de Rham complexes for quotient spaces.
New findings on asymptotic property C in infinite dimensional spaces.
problem Understanding infinite dimensional spaces with infinite asymptotic dimension.
method Showed preservation of asymptotic property C in infinite products and introduced hyperbolic property C.
result Infinite products and restricted direct products of countable groups with finite asymptotic dimension have asymptotic property C.
The paper proves no exotic actions of diffeomorphism groups on 1-manifolds.
problem The study tackles the exotic actions of diffeomorphism groups on 1-dimensional manifolds.
method The approach involves showing any nontrivial homomorphism has a standard form, with countably many embeddings and conjugate actions.
result The groups Diff^r_c(M) have no countable index subgroups, solving a conjecture of Matsumoto.
The paper proves properties of surfaces with holes and ends.
problem Properties of Riemann surfaces with holes and ends.
method Analyzes Riemann surfaces with countably many ends and discs removed.
result Completes the complex structure of a minimal surface in 3D space.
We study Farrell Nil-groups associated to a finite order automorphism of a ring R. We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite mu…
Compact Lie groups G allow topological G-manifolds to be approximated by finite G-CW complexes.
problem Understanding the homotopy type of topological G-manifolds for compact Lie groups G. method Using countable G-CW complexes to approximate the G-homotopy type of topological G-manifolds. result Topological G-manifolds have the G-homotopy type of finite-dimensional countable G-CW complexes. The purpose of this paper is to present, for all n≥3, very simple examples of continuous maps f:Mn−1→Mn from closed (n−1)-manifolds Mn−1 into closed n-manifold Mn such that even though the singular set S(f) of f is countable and dense, the map f can nevertheless be approximated by an …
The fine curve graph is hyperbolic and contains all countable graphs as induced subgraphs.
problem Characterizing the structure and properties of fine curve graphs.
method Analyzing the hyperbolicity and induced subgraph properties of fine curve graphs and their direct limits.
result The finitary curve graph has diameter 2, contains every countable graph as an induced subgraph, and has the homeomorphism group of the surface as its automorphism group.
We investigate the fundamental group of Griffiths' space, and the first singular homology group of this space and of the Hawaiian Earring by using (countable) reduced tame words. We prove that two such words represent the same element in the corresponding group if and only if they can be carried to the same tame word b…
Characterizes continuity of monotone functionals in mixed topology.
problem Continuity of monotone functionals in mixed topology.
method Characterization through lower semicontinuity and dual representations.
result Continuity in mixed topology is equivalent to dual representation in terms of countably additive measures.