Study on group actions on spheres using multisymplectic geometry.
problem Existence of homotopy comoment maps for compact Lie group actions on spheres.
method Investigation of multisymplectic actions and comoments on spheres.
result Explicit constructions of comoments for interesting cases.
We establish, via geometric quantization of the supercotangent bundle sM of (M,g), a correspondence between its conformal geometry and those of the spinor bundle. In particular, the Kosmann Lie derivative of spinors is obtained by quantization of the comoment map, associated to the new Hamiltonian action of conf(M,g) o…
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.
Homotopy momentum map extends Noether's theorem in general relativity.
problem Extending Noether's theorem to spacetime vector fields.
method Using homotopy momentum map and L∞-algebras. result Extension of conserved currents to spacetime vector fields.
The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.
problem Understanding when surfaces are homotopy equivalent to graphs.
method Analyzes second-countable orientable surfaces with noncompact boundary.
result Defines a necessary and sufficient condition for surfaces to be homotopy equivalent to graphs.
Innovates a three-component link homotopy invariant.
problem Classifying three-component link maps up to homotopy.
method Developed tools and invariants for distinguishing three-component link maps.
result Found three-component link maps that are not homotopic.
Embeddings of mapping tori for end-periodic graph maps are proven.
problem Embedding mapping tori of end-periodic graph maps into finite complexes.
method Flowline-preserving homotopy equivalence and π1-injective map. result Every mapping class of Γ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.
In this paper we classify the homotopy classes of proper maps E→Rk, where E is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps Rn→Rk. We find a stability range of such maps. We conclude with some remarks…
We give an alternative to Postnikov's homotopy classification of maps from 3-dimensional CW-complexes to homogeneous spaces G/H of Lie groups. It describes homotopy classes in terms of lifts to the group G and is suitable for extending the notion of homotopy to Sobolev maps. This is required for applications to variati…
Optimizes energy of mappings from complex projective spaces.
problem Finding energy-minimizing mappings between complex projective spaces and Riemannian manifolds.
method Establishes optimal lower bounds for energy functionals and characterizes optimal mappings.
result Optimal lower bounds for energy functionals are characterized for mappings from real and complex projective spaces.
Study of area minimizing surfaces in homotopy classes of maps.
problem Existence and regularity of area minimizing surfaces in metric spaces.
method Introducing relative 1-homotopy type for Sobolev maps, using local quadratic isoperimetric inequality, and analog for closed surfaces.
result Existence and local Hölder regularity of area minimizing surfaces in proper geodesic metric spaces.
Study compares weak and homotopy moment maps in multisymplectic geometry.
problem Existence and equivariance of moment maps in multisymplectic geometry.
method Comparison of weak and homotopy moment maps.
result Analysis of existence and equivariance phenomena.
This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants -- the homotopy groups, and (2) an analog o…
A so-called special generic map is by definition a map of smooth manifolds all of whose singularities are definite fold points. It is in general an open problem posed by Saeki in 1993 to determine the set of integers p for which a given homotopy sphere admits a special generic map into Rp. By means of t…
Given a Lie group acting on a manifold M preserving a closed n+1-form ω, the notion of homotopy moment map for this action was introduced in Callies-Fregier-Rogers-Zambon [6], in terms of L∞-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This descr…
Koschorke introduced a map from the space of closed n-component links to the ordered configuration space of n-tuples of points in R3, and conjectured that this map separates homotopy links. The purpose of this paper is to construct an analogous map for string links, and to prove (1) this map in fact sep…
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.
Heegaard Floer homology study confirms composition maps match up to homotopy.
problem Verifying consistency of composition maps in Heegaard Floer homology.
method Using Auroux and Zemke's results, proving agreement up to homotopy.
result Proves consistency of composition maps in Heegaard Floer homology.
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. We give chain homotopy maps of Khovanov-type link homology of a universal differential. The universal differential, discussed by Mikhail Khovanov, Marco Mackaay, Paul Turner and Pedro Vaz, contains the original Khovanov's differential and Lee's differential. We also consider the conditions of any differential ensuring …
Introduces homotopy momentum sections on multisymplectic manifolds.
problem No specific problem stated; focuses on introducing a new concept.
method Introduces a new concept of homotopy momentum sections on multisymplectic manifolds.
result Shows that a gauged nonlinear sigma model with Wess-Zumino term has homotopy momentum section structure.
Sharp bounds found for energy in projective space mappings.
problem Finding bounds for energy in mappings of real projective spaces.
method Sharp lower and upper bounds for energy in homotopy classes of mappings from real projective space to Riemannian manifolds.
result Characterization of maps that achieve the lower bound for energy and determination of the infimum of energy in a homotopy class.
Quasi-holomorphic homotopies of immersions of 3-manifolds into 5-manifolds
problem The study of homotopies of immersions of 3-manifolds into 5-manifolds
method Describing the local form of quasi-holomorphic homotopies and connections with holomorphic map germs
result A complete description of how the fundamental group of the complement of the image of an immersion changes under a quasi-holomorphic homotopy
Assume that all spaces and maps are localised at a fixed prime p. We study the possibility of generating a universal space U(X) from a space X which is universal in the category of homotopy associative, homotopy commutative H-spaces in the sense that any map f:X->Y to a homotopy associative, homotopy commutative …
This thesis extends Hamiltonian actions to multisymplectic geometry, classifying actions on spheres and constructing homotopy comomentum maps.
problem Extending Hamiltonian actions to multisymplectic geometry.
method Explicit constructions and concrete examples of homotopy comomentum maps.
result Complete classification of compact group actions on multisymplectic spheres and explicit construction of homotopy comomentum maps.
Cobordism groups of cooriented fold maps of codimension 1 are computed completely. Namely their odd torsion part coincides with that of the stable homotopy group of spheres in the same dimension, while the 2-primary part is the kernel of the Kahn-Priddy map. (The Kahn-Priddy map is an epimorhism of the stable homotopy …
The Hopf invariant is linked to null-homotopy properties of maps.
problem Understanding the relationship between the Hopf invariant and null-homotopy of maps.
method Using the generalized Hopf invariant and constructing smooth null-homotopies, the paper explores the relationship between the Hopf invariant and null-homotopy properties of maps.
result Sharp results on the relationship between the Hopf invariant and null-homotopy properties of maps, showing the necessity and sufficiency of certain conditions.
In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…
The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.
problem Understanding the space of Gauss maps of complete minimal surfaces and their homotopy types.
method Proves the Gauss map assignment is a Serre fibration and determines the homotopy type of the space of meromorphic functions.
result The space of meromorphic functions on M that are the Gauss map of a complete full conformal minimal immersion has the same homotopy type as the space of all continuous maps from M to the 2-sphere. The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, A∞ spaces, E∞ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends…
We affirmatively address the question of whether the proposed link homotopy invariant ω of Li is well-defined. It is also shown that if one wishes to adapt the homotopy invariant τ of Schneiderman-Teichner to a link homotopy invariant of link maps, the result coincides with ω.
Existence of polyharmonic maps proven for critical dimensions.
problem Existence of polyharmonic maps in critical dimensions.
method Blowup analysis and free homotopy class existence proof.
result Existence of minimizing m-polyharmonic maps for every free homotopy class. New quantity helps map homotopy classes in complex spaces.
problem Understanding homotopic classes of maps between complex spaces.
method Identified a new monotone quantity in mean curvature flows of maps between Riemannian manifolds.
result Sharp criteria for homotopic classes of maps between complex projective spaces and spheres.
Derivative map for disk diffeomorphisms induces nontrivial homotopy groups.
problem Proving nontriviality of homomorphisms induced by derivative maps.
method Combining recent results on homotopy spheres, plumbing approach, and explicit constructions.
result Non-zero homomorphism between specific homotopy groups.
Wave maps from circle to manifold controllable if homotopy classes match.
problem Global controllability of wave maps from circle to Riemannian manifolds.
method Characterization of controllability via homotopy classes, uniform-time global controllability between steady states, quantitative exponential stability.
result Global controllability is equivalent to homotopy class of data.
New proofs and refined theorems on bounded cohomology.
problem Properties of bounded cohomology and comparison map.
method Homotopy-theoretic properties and generalizations.
result New proofs and refined versions of vanishing and covering theorems.
Fine shape of local compacta represented by ordinary maps.
problem Representing fine shape of local compacta.
method Constructing a space ∣X∣ for each local compactum X such that fine shape classes correspond to homotopy classes of maps to ∣X∣. result Fine shape classes from any locally compact metrizable space Y to X bijectively correspond to homotopy classes of maps from Y to ∣X∣. The thesis defines and proves invariants for manifolds of bounded geometry.
problem Lipschitz-homotopy invariants for manifolds of bounded geometry.
method Definition of a controvariant functor and invariance of the Roe index and ρ-class.
result Lipschitz-homotopy invariants are defined and proven for manifolds of bounded geometry.
Paper proves existence of Dirac-harmonic maps with trivial index.
problem Finding Dirac-harmonic maps with trivial index.
method Defining a new quantity and proving its homotopy invariance.
result Existence of Dirac-harmonic maps from closed Riemann surfaces to Kähler manifolds.
The paper defines and proves the existence of train track maps on graphs of groups.
problem Understanding homotopy equivalences in graphs of groups.
method Developed the theory of train track maps on graphs of groups, defining maps and homotopy equivalences.
result Any homotopy equivalence of a graph of groups may be represented by a relative train track map under certain conditions.
Generalizes van Est map to geometric stacks and homotopy theory.
problem Computing cohomology of geometric stacks and Lie algebroids.
method Generalizes van Est map to stacks and foliations, using modules instead of representations.
result Derives new cohomology results and unifies differentiable stacks, Lie algebroids, and homotopy theory.
Breiman (2001) proposed to statisticians awareness of two cultures: 1. Parametric modeling culture, pioneered by R.A.Fisher and Jerzy Neyman; 2. Algorithmic predictive culture, pioneered by machine learning research. Parzen (2001), as a part of discussing Breiman (2001), proposed that researchers be aware of many cultu…
Paper defines weak (1, 3) homotopy for knot projections and classifies trivial knots.
problem Classifying knot projections under weak (1, 3) homotopy.
method Defines weak (1, 3) homotopy, introduces a map to knot isotopy classes, and determines trivial knots.
result Determines which knot projections are trivial under weak (1, 3) homotopy.
Study on the parity of fold map singular points, showing non-invariance for odd-dimensional manifolds.
problem Parity of connected components of fold map singular points for odd-dimensional manifolds.
method Constructive proofs using open book decompositions, round fold maps, and allowable moves.
result Parity of connected components is not a homotopy invariant for odd-dimensional manifolds.
Unpublished results of S Straus and W Browder state that two notions of homotopy equivalence for manifolds with smooth group actions - isovariant and equivariant - often coincide under a condition called the Gap Hypothesis; the proofs use deep results in geometric topology. This paper analyzes the difference between th…
We prove that any smooth foliation that admits a Riemannian foliation structure has a well-defined basic signature, and this geometrically defined invariant is actually a foliated homotopy invariant. We also show that foliated homotopic maps between Riemannian foliations induce isomorphic maps on basic Lichnerowicz coh…
This work focuses on important step in quantitative topology: given homotopic mappings from Sm to Sn of Lipschitz constant L, build the (asymptotically) simplest homotopy between them (meaning having the least Lipschitz constant). The present paper resolves this problem for the first case where Hopf invariant p…
Let f:M→N be a smooth area decreasing map between two Riemannian manifolds $(M,\gm)$ and $(N,\gn)$. Under weak and natural assumptions on the curvatures of $(M,\gm)$ and $(N,\gn)$, we prove that the mean curvature flow provides a smooth homotopy of f to a constant map.