The paper studies homotopy inertia groups and tangential structures of manifolds.
problem Understanding the homotopy inertia groups and tangential structures of manifolds.
method Analyzing the homotopy type and cohomology of manifolds to determine homotopy inertia groups and tangential structures.
result The homotopy inertia groups of certain manifolds are shown to be trivial under specific conditions.
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivi…
The paper studies smooth structures on quaternionic projective spaces using inertia groups.
problem Understanding smooth structures on quaternionic projective spaces.
method Computation of inertia groups and their analogues using stable homotopy theory.
result The concordance inertia group is trivial in dimension 20 but non-trivial in higher dimensions.
The paper computes smooth structures on a specific product manifold.
problem Computing the number of smooth structures on a product manifold.
method Using known low-dimensional computations of stable homotopy groups of spheres, the paper determines the inertia group of the product manifold.
result The paper establishes a diffeomorphism classification of all smooth manifolds homeomorphic to CP3imesSk for 1≤k≤7. Let M2n denote a closed (n−1)-connected smoothable topological 2n-manifold. We show that the group C(M2n) of concordance classes of smoothings of M2n is isomorphic to the group of smooth homotopy spheres Θ2n for n=4 or 5, the concordance inertia group Ic(M2n)=0 for $…
The paper computes inertia groups of certain high-dimensional manifolds.
problem Diffeomorphism classification of (n−1)-connected, smooth, closed, oriented 2n-manifolds. method Surgery theory, modified surgery, and special cases of conjectures.
result Inertia groups always vanish for neq4,8,9 and certain cases of n. We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds M2n, where n=7 or 8, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a que…
We study the topology of the inertia space of a smooth G-manifold M where G is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differ…
Study of forms on inertia spaces using Grauert--Grothendieck complex.
problem Understanding basic relative forms on inertia spaces of Lie group actions.
method Use of Grauert--Grothendieck complex on differentiable spaces.
result Sheaf complex of basic relative forms is a fine resolution of Brylinski's sheaf.
Study ramification in knot groups through finite covers and their quotients.
problem Understanding ramification in knot groups and their covers.
method Formalized ramification theory for knot groups, analyzed through finite quotients, profinite completions, and cohomology.
result Characterized ramification and inertia subgroups in knot groups and their covers.
Cyclification of orbifolds explained in cohesive higher topos theory.
problem Understanding cyclification of orbifolds in geometric and algebraic contexts.
method Cohesive higher topos theory and transgression of cohomological charges.
result Cyclification of orbifolds is a fundamental base-change construction.
Proves triviality of inertia groups in high-dimensional manifolds.
problem Classifying manifolds in the metastable range.
method Understanding the second extended power functor in synthetic spectra.
result Inertia groups of high-dimensional manifolds are trivial.
The Wall surgery obstruction groups have two interesting geometrically defined subgroups, consisting of the surgery obstructions between closed manifolds, and the inertial elements. We show that the inertia group In+1(π,w) and the closed manifold subgroup Cn+1(π,w) are equal in dimensions n+1≥6, for any…
The inertia subgroup In(π) of a surgery obstruction group Ln(π) is generated by elements which act trivially on the set of homotopy triangulations $\Cal S(X)$ for some closed topological manifold Xn−1 with π1(X)=π. This group is a subgroup of the group Cn(π) which consists of the elements which can be …
Study rolling of a ball over a sphere in n-dimensional space.
problem Rolling of a Chaplygin ball over a fixed sphere in R^n.
method Modifications of L+R and LR systems on Lie groups with invariant measure.
result Hamiltonization of the reduced system for a special inertia operator.
Inertial proximal gradient algorithm shows monotonically decreasing values.
problem Optimizing functions with inertia.
method Proximal gradient algorithm with alternated inertia.
result Algorithm with alternated inertia achieves monotonically decreasing functional values.
Differentiable groupoids and their inertia spaces are studied with a de Rham theorem.
problem Understanding the de Rham cohomology of inertia spaces.
method Introducing differentiable stratified groupoids and proving a de Rham theorem.
result Inertia groupoids of proper Lie groupoids are locally contractible differentiable stratified groupoids.
New Euler characteristics for groupoids generalize orbifold Euler characteristics.
problem Generalizing orbifold Euler characteristics to non-orbifold groupoids.
method Introducing two Euler characteristics for groupoids, using o-minimal structures, and relating them to orbifold Euler characteristics.
result The two new Euler characteristics coincide and generalize orbifold Euler characteristics.
The study classifies normal subgroups of mapping class groups of surfaces with Cantor subsets.
problem Understanding the structure of normal subgroups in mapping class groups of surfaces with specific subsets.
method Proves two structure theorems: purity and inertia, characterizing normal subgroups.
result Characterizes finite-type normal subgroups of mapping class groups of surfaces with Cantor subsets.
New insights into BNN optimization redefine latent weights as inertia.
problem Optimizing Binarized Neural Networks (BNNs) with latent weights.
method Interpreted latent weights as inertia and introduced Binary Optimizer (Bop).
result Demonstrated improved performance of Bop on CIFAR-10 and ImageNet.
Extended surgery theory proves diffeomorphism for simply-connected 4k-manifolds.
problem Proving diffeomorphism for simply-connected 4k-manifolds.
method Construction of an extended surgery obstruction associated to a normal bordism.
result Identifies the inertia group of a (2k-1)-connected 4k-manifold.
The paper classifies ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
problem Classifying ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
method Maximum principle applications, classification based on causal character of rulings.
result Planes are the only cylindrical stationary surfaces. For non-cylindrical surfaces, classification depends on the causal character of the rulings.
Study on well-posedness of EPDiff equations with pseudo-differential inertia.
problem Analyzing the EPDiff equations with fractional Sobolev metrics.
method Fractional order Sobolev-type metrics on diffeomorphism groups, proving well-posedness.
result Proves local and global well-posedness for EPDiff equations.
Toda flow explained as a porous medium equation.
problem Understanding the Toda flow through the lens of porous medium equations.
method Analyzing the geometry and dynamics of the porous medium equation and comparing it to the Toda flow.
result The Toda flow can be represented as a specific porous medium equation, revealing its gradient and Hamiltonian nature.
Neural network HDP improves virtual inertia control for non-inductive grids.
problem Traditional virtual inertia controllers are not suitable for non-inductive grids.
method Adaptive neural network heuristic dynamic programming (HDP) for optimal control.
result The proposed HDP controller outperforms traditional controllers in virtual inertia control.
New approach limits regret in non-stationary bandits.
problem Understanding worst case regret in time-varying bandits.
method Belief inertia argument to resist new evidence after changes.
result Linear worst case regret for classical and restarting algorithms.
Paper proves homotopy braid group properties over integers and three strands.
problem Understanding homotopy braid group properties.
method Proved linearity over integers and torsion freeness for three strands.
result Homotopy braid group on three strands is torsion free.
The paper introduces knot invariants using stratified homotopy groups.
problem Defining knot invariants in stratified spaces.
method Introducing stratified homotopy groups and proving their properties.
result Stratified Whitehead's theorem holds for these groups.
Study on coarse homotopy groups, proving equivalence and matching with usual homotopy groups.
problem Understanding coarse homotopy groups in abstract coarse structures.
method Developed geometric triangulation techniques for cones to prove the equivalence and matching of coarse homotopy groups with usual homotopy groups.
result Coarse homotopy groups of the cone of a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex.
Novel algorithms scale correspondence analysis to large datasets.
problem Scaling correspondence analysis to large, high-dimensional datasets.
method Interpreting CA in terms of principal inertia components and using deep neural networks for approximation.
result Maximally correlated embeddings of pairs of random variables in CA can be reliably approximated from data using deep neural networks.
Two classification results for stationary surfaces of least moment of inertia.
problem Classifying stationary surfaces in Euclidean space based on their energy.
method Analyzing ruled and foliated surfaces, using critical point theory.
result Classification of stationary surfaces including vector planes, elongated helicoids, and specific types of surfaces.
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
This paper explains why Adam generalizes worse than SGD by analyzing its components.
problem Understanding why Adam generalizes worse than Stochastic Gradient Descent (SGD).
method Diffusion theoretical framework to disentangle the effects of Adaptive Learning Rate and Momentum.
result Adaptive Learning Rate helps escape saddle points but not select flat minima, while Momentum provides a drift effect to help pass through saddle points.
We show that Orr's space's third homotopy group is infinitely generated.
problem Non-triviality of Orr's space's third homotopy group.
method Analyzing the structure of the third homotopy group of Orr's space.
result The third homotopy group is infinitely generated.
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.
In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This give…
Smooth actions of infinite groups linked to homotopy theory.
problem Connecting infinite-dimensional smooth groups to homotopy theory.
method Two computations: diffeological homotopy groups and localization of a strict category.
result Natural constructions yield homotopically coherent group actions of G.
Study homotopy groups of spaces of long links and knots, finding new generators.
problem Understanding homotopy groups of spaces of long links and knots.
method Graphing map increases dimensions, split injections from homotopy groups of spheres, and analyzing knotting effects.
result Generators for homotopy groups in a new degree for spaces of equidimensional long links.
Study gauge groups over high-dimensional manifolds, showing homotopy decompositions.
problem Classifying gauge groups over high-dimensional manifolds.
method Combination of manifold topology and homotopy theory techniques.
result Various homotopy decompositions of gauge groups are shown.
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.
Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.
problem Characterize the Lipschitz homotopy groups of contact 3-manifolds.
method Sub-Riemannian geometry, geometric measure theory, biLipschitz equivalence, purely unrectifiable sets.
result Contact 3-manifolds are K(π,1) spaces with uncountably generated first homotopy groups. Planes and spheres are the only stationary surfaces with constant Gauss curvature.
problem Finding surfaces with constant Gauss curvature that are stationary under a specific energy function.
method Proving the uniqueness of stationary surfaces by considering different curvature conditions.
result Planes and spheres are the only stationary surfaces with constant Gauss curvature.
New examples of manifolds with similar homotopy but different simple homotopy types.
problem Characterizing groups for which high-dimensional manifolds can be homotopy equivalent but not simple homotopy equivalent.
method Construction of doubles of thickenings and use of a formula for Whitehead torsion.
result Examples of high-dimensional manifolds exist for any finitely presented group with a nontrivial Whitehead group involution.
Groups of homotopy equivalences of graphs help realize compact subgroups.
problem Realizing compact subgroups of homotopy equivalences of graphs.
method Introduced a Polish group topology on the group of proper homotopy equivalences and proved the Nielsen Realization theorem.
result Compact subgroups of homotopy equivalences can be realized by simplicial isomorphisms of graphs.
New knots have same group, quandle, but different homotopy invariants.
problem Determining knot homotopy type from group, quandle, and module.
method Construction of 2-knots with isomorphic knot groups and fundamental quandles but distinct first Postnikov invariants.
result Fundamental quandle does not determine the first Postnikov invariant of 2-knots.
The paper classifies homotopy ribbon discs with specific groups.
problem Classifying homotopy ribbon discs with given fundamental groups.
method Using geometric and algebraic properties of groups, particularly Farrell-Jones conjecture.
result Classification of homotopy ribbon discs for specific knot groups and Baumslag-Solitar groups.
Homotopy types of 4-manifolds tied to their fundamental groups.
problem Determining the homotopy type of 4-manifolds based on their fundamental groups.
method Uses the fundamental group, second homotopy group, first Stiefel-Whitney class, and equivariant intersection pairing.
result Homotopy type of 4-manifolds is determined by given group properties.
The study creates symplectic examples with non-trivial homotopy groups.
problem Constructing symplectic manifolds with specific homotopy groups.
method Examples of symplectic manifolds with non-trivial second homotopy groups in various dimensions.
result Examples of symplectic manifolds with non-trivial homotopy groups in dimensions 4 and greater than 6.