The paper shows infinitely many components in Floer Hessians space.
problem Understanding the structure of Floer Hessians.
method Proving the existence of infinitely many connected components.
result Proves infinitely many connected components in Floer Hessians space.
Study flip graphs for surfaces of infinite type, finding uncountably many connected components.
problem Understanding relationships between triangulations of infinite type surfaces via flips.
method Associate triangulations to flip graphs and study sequences of simultaneous flips.
result Flip graphs for infinite type surfaces have uncountably many connected components.
Study shows infinite connection components for certain twisted spinors.
problem Understanding connections that yield non-trivial twisted harmonic spinors.
method Analyzing Dirac operators on compact spin manifolds twisted by product bundles.
result The space of connections on the twisting bundle has infinitely many connected components.
The paper defines infinite Schottky groups and their applications to infinite type surfaces.
problem Understanding group actions on infinite type surfaces.
method Definition and analysis of infinite Schottky groups and their properties.
result Every infinite type Riemann surface can be obtained as a quotient of a region of discontinuity of an infinite Schottky group.
In high dimensions, there are many distinct manifolds with nonnegative curvature.
problem Identifying distinct manifolds with nonnegative sectional curvature.
method Analyzing moduli spaces of metrics on manifolds.
result Infinite sequences of distinct manifolds have nonnegative curvature metrics with infinitely many components.
Study infinite symplectic forms on ruled surfaces.
problem Uniqueness of symplectic structures on four-manifolds.
method Provided an infinite family of non-isotopic symplectic forms.
result Symplectic forms on ruled surfaces are pairwise non-isotopic.
We show that in each dimension n≥10 there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian n-manifolds with 0≤sec≤1, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type whic…
The study finds infinitely many different geometries for odd-dimensional manifolds with positive Ricci curvature.
problem Finding different geometries for odd-dimensional manifolds with positive Ricci curvature.
method Examining closed manifolds in odd dimensions n≥5. result Large classes of manifolds admit infinitely many different geometries of positive Ricci curvature.
Defined a new graph type for compact surfaces, proving its connectedness and infinite diameter.
problem Understanding the structure of arc graphs on compact surfaces.
method Defining and analyzing the prescribed arc graph A(Σ,Γ) for compact surfaces Σ with boundary and relations Γ. result The prescribed arc graph A(Σ,Γ) is connected and infinite-diameter, with specific conditions for Gromov hyperbolicity. The paper proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
problem Proving the existence of infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
method Establishing a general sufficient condition for infinitely many topologically non-isotopic splitting 3-spheres in connected sums of 4-manifolds.
result Proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
The study distinguishes components of moduli spaces of Ricci-positive metrics on 5-manifolds.
problem Distinguishing components of moduli spaces of Ricci-positive metrics.
method Using η invariants of spinc Dirac operators to classify and find infinitely many non-diffeomorphic 5-manifolds. result Infinitely many non-diffeomorphic 5-manifolds with infinitely many components in their moduli spaces of Ricci-positive metrics.
We show that the moduli space of Ricci positive metrics on certain homotopy spheres has infinitely many connected components.
New examples show nonnegatively curved metrics with same soul but different moduli space components.
problem Finding nonnegatively curved metrics with same soul but different moduli space components.
method Examples of open manifolds with infinitely many metrics of nonnegative sectional curvature.
result Found open manifolds with souls of codimensions 3 and 2, showing different moduli space components.
A 3D space of hyperbolic manifolds is connected but not path-connected.
problem Proving connectivity and non-path-connectedness of framed hyperbolic 3-manifolds.
method Two proofs using density theorems for Kleinian groups, constructing dense sets of framings, and discussing paths.
result The space of framed infinite volume hyperbolic 3-manifolds is not path-connected.
Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X times Y. This implies that the minimal spanning forest corresponding …
Attention learns PCA on Gaussian data, proving its connection to principal component analysis.
problem Principal component analysis on Gaussian data.
method Analysis of attention mechanisms through PCA, covering finite and infinite prompt regimes.
result Attention aligns with principal eigenvectors of covariance matrices, converging to optimal solutions in the infinite-prompt limit.
The paper constructs manifolds with infinite holes from a given manifold.
problem Creating manifolds with infinite holes from a given manifold.
method Constructing a sequence of (n+2)-dimensional manifolds (Mi,gi) with mRicgi>λ that approximate the original manifold (X,h) and have an infinite number of connected components. result The constructed manifolds (Xε) have dense boundary with an infinite number of connected components and no open subset topologically a manifold. Let P(S) be the space of projective structures on a closed surface S of genus g>1 and let Q(S) be the subset of P(S) of projective structures with quasifuchsian holonomy. It is known that Q(S) consists of infinitely many connected components. In this paper, we will show that the closure of any exotic compo…
Study shows infinitely many nonnegatively curved metric spaces on exotic 7-manifolds.
problem Investigating nonnegatively curved metrics on exotic 7-manifolds.
method Using Kreck-Stolz invariant and Atiyah-Patodi-Singer index theorem for orbifolds with boundary.
result Moduli space of nonnegatively curved metrics has infinitely many connected components.
Essential triangulations connect via specific moves in 3-manifolds.
problem Finding essential triangulations in 3-manifolds with infinite boundary components.
method Using 2-3, 3-2, 0-2, and 2-0 moves to connect essential triangulations.
result Essential triangulations are connected via specific moves.
New surfaces defy traditional pants decompositions.
problem Creating surfaces without bounded pants decompositions.
method Constructing quasiconformally homogeneous hyperbolic Riemann surfaces.
result Found surfaces with infinite type and compact boundary components that defy pants decompositions.
The dynamics of holomorphic 1-forms are studied, showing ergodic foliations and connected spaces.
problem Analyzing the dynamics of absolute period foliations in strata of holomorphic 1-forms.
method Using cohomology classes and isoperiodic forms, the authors show ergodicity and connectedness of spaces.
result The absolute period foliation is ergodic on the area-1 locus and has non-dense leaves in explicit suborbifolds.
Study of pants decompositions on surfaces of infinite type.
problem Understanding the structure of pants decompositions on surfaces of infinite type.
method Analyzed the disconnected pants graph and defined a coarser topology on the pants complex.
result The automorphism group of the new space is isomorphic to the mapping class group.
Let Ag,d be the (topological) cobordism category of orientable surfaces whose connected components are homeomorphic to either S1×I with one incoming and one outgoing boundary component or the surface Σg,d of genus g and d boundary components that are all incoming. In this paper, we stu…
We construct an infinite-dimensional information manifold based on exponential Orlicz spaces without using the notion of exponential convergence. We then show that convex mixtures of probability densities lie on the same connected component of this manifold, and characterize the class of densities for which this mixtur…
Study extends JB-algebra structure group results to infinite dimensions.
problem Extend results for real Jordan algebras to infinite dimensional JB-algebras.
method Prove structure groups, cone preserving groups, and automorphism groups are embedded Banach-Lie groups; describe components via cones, isotopes, and central projections; apply to special JB-algebra of self-adjoint operators.
result Full description of components of structure group and automorphism group, including their Banach-Lie algebras and connected components.
Eigenfunction level sets can have infinitely many connected components.
problem Estimating the number of connected components of eigenfunction level sets.
method Analyzing Neumann eigenfunctions on triangles, polygons, and surfaces.
result There exist eigenfunctions with infinitely many connected components in their level sets.
The Allen-Cahn equation yields bounded solutions with any compact topology level sets.
problem Finding bounded solutions with specified compact topology level sets.
method Utilizing infinite-index solutions of the Allen-Cahn equation.
result Existence of bounded entire solutions with zero level sets of any compact topology.
We study the geometry of compact Lorentzian manifolds that admit a somewhere timelike Killing vector field, and whose isometry group has infinitely many connected components. Up to a finite cover, such manifolds are products (or amalgamated products) of a flat Lorentzian torus and a compact Riemannian (resp., lightlike…
Minimal link diagrams are unique and have a fixed number of crossings.
problem Characterizing minimal link diagrams under Reidemeister moves.
method Analyzing minimal link diagrams through Reidemeister moves I and II.
result The uniqueness of minimal diagrams and their fixed number of crossings.
We show how to use topological ideas, such as compactness, to establish orderability properties of infinite groups. A new application is to provide a left-ordering for the group of PL homeomorphisms of a connected surface with boundary which are fixed on at least one boundary component. This has been extended in recent…
We consider the space $\X$ of Anosov diffeomorphisms homotopic to a fixed automorphism L of an infranilmanifold M. We show that if M is the 2-torus T2 then $\X$ is homotopy equivalent to T2. In contrast, if dimension of M is large enough, we show that $\X$ is rich in homotopy and has infin…
The study finds infinite nodal solutions for equations on positive Ricci curvature manifolds.
problem Existence of nodal solutions for equations on manifolds with positive Ricci curvature.
method Analyzes cohomogeneity one Riemannian manifolds with positive Ricci curvature and proves the existence of infinite nodal solutions for specific equations.
result Proves the existence of infinite nodal solutions for equations of the form −Δgu+λu=λuq on positive Ricci curvature manifolds. Study distinguishes components of moduli spaces for certain metrics.
problem Distinguishing components of moduli spaces of metrics with nonnegative sectional curvature.
method Used Kreck-Stolz s invariant and formulas to calculate and distinguish components.
result Moduli spaces of metrics with nonnegative sectional curvature have infinitely many path components.
Study contact structures and Sasakian geometry on manifolds.
problem Existence and classification of contact structures on manifolds.
method Use S1-equivariant symplectic homology to prove existence of contact structures. result Infinitely many inequivalent contact structures on various manifolds.
In a recent paper we constructed a family of foliated 2-complexes of thin type whose typical leaves have two topological ends. Here we present simpler examples of such complexes that are, in addition, symmetric with respect to an involution and have the smallest possible rank. This allows for constructing a 3-periodic …
Abstract: Extends Drinfeld correspondence to infinite-dimensional Lie groups.
problem Establishing Drinfeld correspondence in infinite dimensions.
method Extending Drinfeld correspondence to Poisson Lie groups and Lie bialgebras in infinite-dimensional settings.
result Extended Drinfeld correspondence to regular Lie groups modeled on nuclear Fréchet and Silva spaces.
Study pairs of subspaces with or without a common complement in Hilbert spaces.
problem Characterize pairs of subspaces with or without a common complement in Hilbert spaces.
method Analyze pairs of subspaces (S, T) in the Grassmann manifold Gr(H) of a Hilbert space H, identifying Delta and Gamma based on the existence of a common complement.
result Delta is open and its connected components are parametrized by dimension and codimension. Gamma is a C^\infty submanifold characterized by dimensions and semi-Fredholm indices.
Analyzes geodesic lengths in sparse networks, deriving a distribution.
problem Understanding connectivity and robustness in networked systems.
method Analytic derivation of geodesic length distribution in sparse networks.
result Simple closed-form expression for geodesic length distribution.
Study Reeb components with complex leaf structure and their symmetries.
problem Understanding symmetries of Reeb components with complex leaf structure.
method Review Hopf construction and analyze leafwise holomorphic automorphisms.
result Contains an infinite dimensional vector space of automorphisms.
For any n\ge 2, we give infinitely many unsplittable links of n components in the 3-sphere which admit non-trivial surgery yielding the 3-sphere again and whose components are mutually distinct hyperbolic knots. Berge and Kawauchi gave 2-component hyperbolic links with those two properties. We can also give infinitely …
Essential triangulations of certain manifolds are connected via specific moves.
problem Connecting essential triangulations of certain manifolds.
method Essential triangulations are connected via 2-3 and 3-2 moves alone, ignoring those for which no 2-3 move preserves essentiality.
result Essential triangulations of certain manifolds are connected via 2-3 and 3-2 moves alone.
This paper tackles the topology of infinite-dimensional fiber spaces.
problem Classifying and understanding the topological structure of infinite-dimensional fiber spaces.
method Algebraic and differential topology, focusing on the homotopy type of the moduli space of fibrations.
result Explicit homotopy calculations for low-dimensional cases, completing the solution for dimensions up to three.
Study on knot 74 surgeries reveals infinite residue characteristics and infinite order points.
problem Arithmetic properties of Dehn surgery points on knot 74. method Analyzing the canonical component of the SL2(C)-character variety. result Infinite set of ramified places and infinite order points in the Mordell-Weil group.
Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and Diff0 the connected component of its diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler manifol…
Study shows hyperelliptic components in Torelli space have infinite homology.
problem Characterizing the homotopy type of hyperelliptic components in Torelli space.
method Properties of hyperelliptic Torelli group and genus 3 theta functions.
result Second rational homology of hyperelliptic components is infinite-dimensional.
Study critical points of Laplace eigenfunctions in polygons.
problem Characterize critical points of Laplace eigenfunctions in polygonal domains.
method Analyze components of the critical set with codimension 1.
result For simply connected polygons, if a second Neumann eigenfunction has infinitely many critical points, the polygon must be a rectangle.
Paper finds first examples of unlinked knots that can't be separated.
problem Separating knots without changing their length and thickness.
method Constructs infinite families of 2-component gordian unlinks and n-component links for n≥2. result Found infinite families of 2-component gordian unlinks that cannot be separated.