Study proves a space of manifolds is homology equivalent to a loop space.
problem Understanding the stable moduli space of certain high-dimensional manifolds.
method Incorporates surgery data via Lagrangian subspaces in a cobordism category.
result The stable moduli space is homology equivalent to an infinite loopspace.
This is an exposition of a proof of the Madsen-Weiss Theorem, which asserts that the homology of mapping class groups of surfaces, in a stable dimension range, is isomorphic to the homology of a certain infinite loopspace that arises naturally when one applies the "scanning method". The proof given here utilizes simpli…
In this paper we prove a homological stability theorem for the diffeomorphism groups of high dimensional manifolds with boundary, with respect to forming the boundary connected sum with the product Dp+1×Sq for ∣q−p∣<min{p,q}−2. In a recent joint paper with Boris Botvinnik (see arXiv:1509.03359…
Let L be an exact Lagrangian submanifold inside the cotangent bundle of a closed manifold N. We prove that if N satisfies a mild homotopy assumption then the image of π_2(L) in π_2(N) has finite index. We make no assumption on the Maslov class of L, and we make no orientability assumptions. The homotopy assumption is e…
The main construction of this paper contains a serious error, and I am withdrawing it. I owe Andrew Stacey and Ralph Cohen thanks for seeing the problem; in particular, Stacey has shown that the projections constructed in §3.1 will fail in general to have constant rank, so the family TV of vector spaces defined…
Infinite rank surface cluster algebras extend traditional concepts to surfaces with accumulation points.
problem Extending surface cluster algebras to infinite surfaces with accumulation points.
method Consider infinite mutation sequences and hyperbolic structures to define cluster variables as lambda lengths of arcs.
result Established transitivity of infinite mutation sequences on triangulations of infinite surfaces and provided expansion formulas for cluster variables.
Veech groups uniformize Teichmüller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite ends. We further show that examples exist for which each direction of an infinite end …
Infinite order loose corks created.
problem Creating loose corks of infinite order.
method Constructing an infinite order loose cork.
result Demonstrated construction of an infinite order loose cork.
We construct a infinite-dimensional manifold structure adapted to analytic Lie pseudogroups of infinite type. More precisely, we prove that any isotropy subgroup of an analytic Lie pseudogroup of infinite type is a regular infinite-dimensional Lie group, modelled on a locally convex strict inductive limit of Banach spa…
This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an ``infinite energy'' situation. We show that we can glue an infinite number of instantons, and that the resulting instantons ha…
Study on infinite neural networks for generalization.
problem Understanding generalization in infinitely-wide neural networks.
method Analytical and empirical investigations of infinite ensembles.
result Signals correlating with generalization identified.
Book explores infinite translation surfaces, challenging traditional geometry.
problem Understanding infinite-type translation surfaces.
method Detailed classification, construction, and analysis of symmetries.
result Complex dynamics in infinite-type translation flows.
Infinite diameter found in compression body graph.
problem Determining the diameter of compression body graphs.
method Analyzing the structure and connections within compression body graphs.
result The compression body graph has infinite diameter.
Study examines infinite limits of transformer dynamics, identifying key parameterizations.
problem Understanding the training dynamics of transformer models in the feature learning regime.
method Analysis of infinite scaling limits using dynamical mean field theory.
result Identified parameterizations that admit well-defined infinite width and depth limits.
Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.
problem Characterize Veech groups of infinite superelliptic curves.
method Analyzing geometric properties, differential equations, and group theory.
result Veech groups of infinite superelliptic curves are all matrices permuting branched points.
New homology for infinite multi-colored braids, completing previous work.
problem Categorification of highest-weight projectors for infinite braids.
method Defining limiting Khovanov-Rozansky homology for semi-infinite braids.
result Categorifies highest-weight projectors for a large class of braids.
Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
Study infinite-depth limits of neural networks with fixed width.
problem Understanding the behavior of neural networks as depth increases with fixed width.
method Analyzing finite-width residual networks with random Gaussian weights, focusing on the infinite-depth limit.
result The pre-activations converge to a zero-drift diffusion process, differing from the infinite-width limit.
The paper constructs Fuchsian groups for specific infinite genus surfaces.
problem Constructing Fuchsian groups for infinite genus surfaces.
method Explicit construction of infinitely generated Fuchsian groups for specific surfaces.
result Explicit construction of Fuchsian groups for non-compact, orientable surfaces.
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.
Paper constructs an infinite surface with one end and infinite genus.
problem Understanding surfaces with infinite genus and one end.
method Topological and geometric construction of the surface.
result Introduction of a flat and hyperbolic construction of the surface.
Infinitely many 3D shapes have multiple ways to be filled with special surfaces.
problem Understanding how many ways 3D shapes can be filled with special surfaces.
method Examined 3D shapes supported by planar open books and found multiple ways to fill them with special surfaces.
result Found infinitely many 3D shapes that can be filled with multiple, non-homeomorphic special surfaces.
Study infinite genus surfaces and Schottky groups for uniformization.
problem Investigate infinite genus surfaces and Schottky groups for uniformization.
method Definitions and proofs for infinite genus surfaces and Schottky groups, showing uniformization by Schottky groups.
result Infinite genus surfaces and handlebodies can be topologically and quasiconformally uniformized by Schottky groups.
We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense. We show that the graphs we construct are generally connected, infinite diameter a…
Infinite braids' categorification proven using Khovanov homology.
problem Categorifying Jones-Wenzl projectors for infinite braids.
method Limiting Khovanov chain complex and homotopy types.
result Jones-Wenzl projectors categorified for infinite braids.
New findings on infinite subgroups in negatively curved spaces.
problem Characterizing infinite discrete isometry subgroups in negatively pinched Hadamard manifolds.
method Generalization of Bonahon's characterization to negatively pinched Hadamard manifolds.
result Every geometrically infinite isometry subgroup has a continuum of nonconical limit points.
Study finds infinitely many prime knots with identical Alexander invariants.
problem Existence of distinct prime knots with the same Alexander invariant.
method Using pretzel knots with three tassels, we construct infinitely many distinct families of knots with identical Alexander invariants.
result Infinitely many distinct prime knots have the same Alexander invariant.
Infinite-genus surfaces have many isospectral hyperbolic structures.
problem Finding many isospectral hyperbolic structures on infinite-genus surfaces.
method Constructing families of isospectral hyperbolic structures on infinite-type surfaces without planar ends.
result Uncountable families of isospectral and quasiconformally distinct hyperbolic structures on infinite-genus surfaces with self-similar end spaces.
Infinite-dimensional contact geometry explored.
problem Generalizing contact geometry to infinite dimensions.
method Generalization of cosymplectic, contact, and cocontact manifolds to infinite dimensions.
result Model examples of time-dependent and dissipative Hamiltonian systems calculated.
VC dimensions of group CNNs are infinite for certain kernels and groups.
problem Estimating the generalization capacity of group convolutional neural networks.
method Identifying precise VC dimension estimates for simple sets of group CNNs.
result Two-parameter families of convolutional neural networks have an infinite VC dimension for infinite groups and certain kernels.
New infinite series of hyperbolic polytopes with special growth rates found.
problem Finding new infinite series of non-compact hyperbolic polytopes.
method Constructing infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes.
result Growth rates of the constructed polytopes are Perron numbers.
Concrete description of infinite order cork automorphism.
problem Describing the infinite order loose-cork automorphism.
method Concatenating the defining ribbon disk by an infinite order isotopy.
result Concrete description of the infinite order cork automorphism.
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.
New categorification method for infinite braids.
problem Categorifying highest-weight projectors for infinite braids.
method Limiting colored Khovanov-Rozansky homology of infinite braids.
result Partial isomorphism between HOMFLY-PT homology of braids and infinite torus knots.
Most Finsler metrics have infinite-dimensional holonomy groups.
problem Understanding the holonomy groups of Finsler metrics.
method Analyzing the set of Finsler metrics on a manifold.
result An open dense subset of Finsler metrics have infinite-dimensional holonomy groups.
Arithmetic 3-manifolds with infinite geodesics are proven.
problem Arithmeticity of 3-manifolds with infinitely many geodesics.
method Analysis of totally geodesic surfaces in hyperbolic 3-manifolds.
result Arithmetic 3-manifolds with infinite geodesics are proven.
The Conway knot can't be smoothly tied to any other knot an infinite number of times.
problem Understanding the concordance order of knots.
method Rasmussen invariant, satellite operations, null-homologous twists.
result The Conway knot has infinite concordance order.
The paper explores non-amenability in infinite-type surfaces and graphs.
problem Determining non-amenability in mapping class groups of infinite-type surfaces and graphs.
method Analyzes mapping class groups of infinite-type surfaces and graphs, provides examples and exhibits classes of groups.
result Completely determines non-amenability of mapping class groups of infinite-type surfaces and graphs.
Dynamic systems linked to infinite permutation matrices.
problem Dynamic equivalence of control systems.
method Association of infinite permutation matrices.
result Relationship between dynamic equivalences and permutation matrices.
This paper reformulates the p-adic Littlewood Conjecture using infinite loops.
problem The p-adic Littlewood Conjecture in number theory. method Introducing infinite loops mod n and linking them to the conjecture. result A real number α is a counterexample to the p-adic Littlewood Conjecture if and only if pkα is an infinite loop mod pm for all k. Derives PF-ODE for infinite-dimensional functions, improving function generation tasks.
problem Efficient inference in infinite-dimensional diffusion models.
method Derives PF-ODE in infinite-dimensional function spaces.
result Reduces function evaluations while maintaining sample quality.
Paper introduces combinatorial Ricci flows on infinite disk triangulations.
problem Discrete Ricci flows on infinite triangulations of the open disk.
method Combinatorial Ricci flows (CRFs) in Euclidean and hyperbolic geometries.
result Existence and uniqueness of CRFs, convergence results, uniformization theorem.
Strict inequality proven in infinite-dimensional Möbius group.
problem Proving strictness of Jørgensen inequality in infinite dimensions.
method Analogue of Jørgensen inequality in infinite-dimensional Möbius group.
result Strict inequality proven in infinite-dimensional Möbius group.
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.
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.
The paper describes a geometric Schottky group for a non-compact hyperbolic surface with infinite genus.
problem Describing a geometric Schottky group for a non-compact hyperbolic surface with infinite genus.
method Constructing a hyperbolic polygon with an infinite number of sides and identifying sides in pairs with Mobius transformations.
result A precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group.
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.
Infinite CNNs lose spatial correlations, but can be restored by correlated weights.
problem Infinite CNNs lose spatial correlations, which are crucial for their performance.
method Introduced correlated weights to restore spatial correlations in infinite CNNs.
result Optimal performance is achieved with a moderate level of weight correlation.