Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
A closed 4-manifold (or, more generally, a finite PD4-space) has a finitely dominated infinite regular covering space if and only if either its universal covering space is finitely dominated or it is finitely covered by the mapping torus of a self homotopy equivalence of a PD3-complex.
Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.
problem Constructing moduli spaces over Riemann surfaces.
method Finite-dimensional construction using holomorphic symplectic reduction.
result Moduli spaces over Riemann surfaces as stratified holomorphic symplectic spaces.
We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse of finite spaces induces a simplicial …
We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition comp…
Survey of recent results on homogeneous finite-dimensional spaces.
problem Understanding properties of homogeneous finite-dimensional spaces.
method Discussion of recent results and unsolved problems.
result Discussion of recent results and unsolved problems.
Smooth distributions on subcartesian spaces can be globally finitely generated.
problem Understanding smooth distributions on subcartesian spaces.
method Embedding in Euclidean space, Whitney Embedding Theorem, and distribution theory.
result Smooth generalized distributions and subbundles on connected subcartesian spaces are globally finitely generated.
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.
Study on minimal submanifolds with finite curvature in Euclidean space.
problem Finite diffeomorphism types of complete immersed minimal submanifolds with finite total curvature.
method Adapted method from Chodosh, Ketover, and Maximo for hypersurfaces to submanifolds of arbitrary codimension.
result Proved finite diffeomorphism types for complete immersed minimal submanifolds with finite total curvature.
The paper shows plentiful non-homotopy finite Poincaré duality spaces.
problem The existence of non-homotopy finite Poincaré duality spaces.
method Constructing a finitely dominated Poincaré space with a non-trivial 2-divisible element in the reduced Grothendieck group.
result The existence of finitely dominated Poincaré spaces that are not homotopy finite.
The kth finite subset space of a topological space X is the space exp_k X of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of configuration spaces of distinct unordered points in X. We show that the finite subset …
Finite spaces can be or not coproducts of subspaces.
problem When finite-dimensional diffeological vector spaces are coproducts of their subspaces.
method Reviewing the question in diffeological vector spaces and comparing with other categories.
result Finite-dimensional spaces can be coproducts, but not always.
Researchers find Frobenius manifold structures on orbits spaces of finite groups.
problem Understanding Frobenius manifold structures on orbits spaces of finite groups.
method Applying Dubrovin's method to various orbits spaces of linear representations of finite groups.
result Discoveries of non-trivial Frobenius manifold structures on orbits spaces.
The study classifies flows of finite curvature in 3D space.
problem Classifying flows of finite curvature in 3D space.
method Partial classification of eternal mean convex flows.
result Topologically nonplanar flows must exit a catenoid.
Proves symmetric spaces for certain quasi-isometric properties.
problem Understanding quasi-isometric spaces and their properties.
method Analyzes geometric and geometrically finite quasi-actions.
result Proves quasi-isometric spaces are symmetric or real line.
Finite groups of RAAGs act on a contractible space without fixed points.
problem Characterizing finite subgroups of RAAG outer automorphisms.
method Analyzing the action of finite subgroups on a contractible space.
result Finite subgroups of RAAG outer automorphisms fix a point in the space.
Convex cores found for group actions on median spaces.
problem Understanding group actions on median spaces without metric or topology.
method Introduced convex cores for actions on finite-rank median algebras.
result Actions on median spaces have nonempty convex cores.
Constructs finite element spaces for (p,q)-forms, excluding one subspace.
problem Constructing finite element spaces for (p,q)-forms. method Piecewise polynomial finite element spaces for all natural subspaces of (p,q)-forms, excluding one subspace. result Recovers known finite element spaces and introduces new ones.
Finite type and finitely generated homotopy groups for manifold automorphisms.
problem Finite type and homotopy group properties of manifold automorphism spaces.
method Analyzing the classifying space of diffeomorphism groups and using simple homotopy theory.
result The classifying space of diffeomorphism groups has finitely generated homotopy groups.
The paper explores the structure of Reeb spaces for smooth functions on manifolds.
problem Understanding the structure of Reeb spaces for smooth functions on manifolds.
method Proving the structure of Reeb spaces and showing that any graph can be realized as a Reeb space.
result The Reeb space of a smooth function on a closed manifold with finitely many critical values has a graph structure.
Characterizes Stein surfaces with finite homotopy rank-sum.
problem Finite homotopy rank-sum in Stein spaces.
method Rational homotopy theory, classification of Stein surfaces.
result Affine Stein surfaces with finite fundamental group are either simply connected or of order 2.
We prove a version of the Tits alternative for groups acting on complete, finite rank median spaces. This shows that group actions on finite rank median spaces are much more restricted than actions on general median spaces. Along the way, we extend to median spaces the Caprace-Sageev machinery and part of Hagen's theor…
Finiteness predicts dualities in quantum gravity.
problem Finiteness of quantum gravity amplitudes in fully compactified theories.
method Relating moduli space compactifiability to duality group representations.
result Finiteness requires compact moduli spaces and semisimple duality group representations.
A countable CW complex K is quasi-finite (as defined by A.Karasev) if for every finite subcomplex M of K there is a finite subcomplex e(M) such that any map f:A→M, where A is closed in a separable metric space X satisfying XτK, has an extension g:X→e(M). Levin's results imply that none of the Ei…
Outer space for RAAGs is a contractible finite-dimensional space for automorphisms.
problem Constructing a finite-dimensional space for outer automorphisms of RAAGs.
method Constructing a finite-dimensional space OΓ blending features of symmetric spaces and Outer space for free groups. result The space OΓ is contractible, making the quotient a rational classifying space for extOut(AΓ). The simplest condition characterizing quasi-finite CW complexes K is the implication XτhK⟹β(X)τK for all paracompact spaces X. Here are the main results of the paper: Theorem: If {Ks}s∈S is a family of pointed quasi-finite complexes, then their wedge s∈S⋁Ks is quasi-fini…
Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.
problem Understanding geodesic simplices in pseudo-hyperbolic space.
method Cohomological interpretation and necessary/sufficient condition formulation.
result Every ideal geodesic polytope in (2,2) pseudo-hyperbolic space has finite volume. Submanifolds of coordinate finite-type were introduced in HV1. A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of Δ. In the present study we consider coordinate finite-type surfaces in E^4. We give necessary and sufficient conditions for g…
Finite-dimensional spaces of biharmonic functions on manifolds are explored.
problem Characterizing biharmonic functions on open manifolds with nonnegative Ricci curvature.
method Analyzing bounded and polynomial growth biharmonic functions, deriving Weyl bounds, and studying fourth-order operators.
result Finite-dimensional spaces of biharmonic functions with polynomial growth are established.
Extensions of Veech groups using hierarchical hyperbolic spaces.
problem Generalizing Veech groups to non-lattice cases.
method Using a hyperbolic space E^ and a nice action of Γ on it. result Finitely generated Veech groups are hierarchically hyperbolic.
Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proo…
Estimates barycenter in geodesic spaces with finite sample bounds.
problem Estimating the barycenter of a distribution in geodesic spaces.
method Finite sample error bounds, Hoeffding- and Bernstein-type concentration inequalities, efficient algorithms.
result Statistical guarantees for efficient barycenter computation.
The kth finite subset space of a topological space X is the space exp_k X of non-empty subsets of X of size at most k, topologised as a quotient of X^k. Using results from our earlier paper (math.GT/0210315) on the finite subset spaces of connected graphs we show that the kth finite subset space of a connected cell com…
Study shows how certain spaces can be mapped to R^n with specific properties.
problem Characterizing and mapping locally compact median spaces of finite rank.
method Analyzing the combinatorics of halfspaces and properties of isometric actions.
result Spaces with certain actions are isometric to R^n, and orbits are discrete.
Holomorphic analogs of Feynman integrals are shown to be finite.
problem Finite evaluation of holomorphic Feynman integrals.
method Compactification of graph moduli space with metrics.
result Holomorphic Feynman integrals are ultraviolet finite.
The study shows a finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.
problem Finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.
method Analyzing torsion-free groups acting by isometries on hyperbolic metric spaces with bounded entropy and compact quotient.
result The set of such groups is finite and can be estimated based on hyperbolicity constant, entropy, and quotient diameter.
The C-spectral sequence was introduced by Vinogradov in the late Seventies as a fundamental tool for the study of algebro-geometric properties of jet spaces and differential equations. A spectral sequence arise from the contact filtration of the modules of forms on jet spaces of a fibring (or on a differential equation…
Paper proves no specific CMC hypersurfaces in hyperbolic space.
problem Existence of complete noncompact CMC hypersurfaces in hyperbolic space.
method Uses μ-bubbles developed by Gromov and Chodosh-Li-Stryker.
result No complete noncompact CMC hypersurfaces with mean curvature > 1 in hyperbolic space.
Finite presentation for a specific group in 3D handlebody topology.
problem Finding a finite presentation for a specific group in 3D handlebody topology.
method Constructed a finite presentation for the liftable Hilden group to derive the presentation for the balanced superelliptic handlebody group.
result A finite presentation was given for the balanced superelliptic handlebody group.
Finite type for 3-manifolds with boundary.
problem Finite type for 3-manifolds with boundary.
method Contractible space of reducing spheres, homotopy type of finite CW complex.
result B Diff(M rel boundary) has the homotopy type of a finite CW complex.
Hausdorff reflection keeps space shape intact.
problem Preserving shape type in spaces.
method Hausdorff reflection method.
result Hausdorff reflection preserves shape type.
The study of random walks on hyperbolic spaces and Teichmüller spaces, proving central limit theorems and geodesic tracking.
problem Analyzing random walks on hyperbolic and Teichmüller spaces.
method Proving central limit theorems and geodesic tracking using finite moments and logarithmic moments.
result Translation lengths of random isometries satisfy a central limit theorem if and only if the random walk has finite second moment.
We characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.
Artin groups of finite type are not as well understood as braid groups. This is due to the additional geometric properties of braid groups coming from their close connection to mapping class groups. For each Artin group of finite type, we construct a space (simplicial complex) analogous to Teichmueller space that satis…
Study equidistribution for flows on geometrically finite convergence group actions.
problem Counting, mixing and equidistribution for flows on geometrically finite convergence group actions.
method Establishing results for finite BMS measures on flow spaces associated to geometrically finite convergence group actions.
result Results apply to flow spaces associated to relatively Anosov groups.
The paper studies extremal Kaehler metrics induced by complex space forms, proving properties in both finite and infinite dimensions.
problem Analyzing extremal Kaehler metrics induced by complex space forms in finite and infinite dimensions.
method Proving properties of extremal Kaehler metrics under specific conditions in both finite and infinite dimensional settings.
result Extremal Kaehler metrics induced by infinite dimensional elliptic complex space forms have constant non-positive holomorphic sectional curvature under stability conditions.
We describe spaces of essential finite height (measured) laminations in a surface S using a parameter space we call S, an ordered semi-ring. We show that for every finite height essential lamination L in S, there is an action of π1(S) on an S-tree dual to the lift of L to the universal co…