Study smooth loops and loop bundles, relating to G2-structures.
problem Properties of smooth loops and their applications.
method Analyze smooth loops, introduce loop bundles, define torsion and curvature.
result Showed how loop bundles relate to G2-structures. Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, loops topology, loop measures, and signatures of discrete paths. We determine the distributions of the loop homotopy class, and of the first and second homologies, defined by the lower central…
We produce skew loops -- loops having no pair of parallel tangent lines -- homotopic to any loop in a flat torus or other quotient of R^n. The interesting case here is n=3. More subtly for any n, we characterize the homotopy classes that will contain a skew loop having a specified loop in the unit sphere as tangent ind…
The paper proves T-duality and Hori formulae for winding loop spaces.
problem Realizing T-duality and Hori formulae for loop spaces.
method Proving T-duality and Hori formulae for winding q-loop spaces.
result T-duality and Hori formulae for winding q-loop spaces are proven.
Paper establishes loop space T-duality formulae and refines earlier work.
problem Establishing T-duality on loop spaces with background flux.
method Developed loop Hori map and twisted Bismut-Chern character.
result Loop Hori map induces quasi-isomorphism on loop spaces.
The 2-loop polynomial is a polynomial presenting the 2-loop part of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.
Rational loops played a central role in Uhlenbeck's construction of harmonic maps into U(n) (chiral model in physics), and they are generated by simple elements with one pole and one zero constructed from Hermitian projections. It has been believed for long time that nilpotent loops should be added to generate rational…
Kähler manifold loop space inherits Kähler structure and is complete.
problem Characterizing the geometric properties of loop spaces of Kähler manifolds.
method Proving the Kähler structure and completeness of the loop space.
result The loop space of a Kähler manifold is complete and inherits a Kähler structure.
Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most 9-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are is…
A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in euclidean 3-space with a point of positive curvature and no skew loops are the quadrics. In particular, ellipsoids are the only closed surfaces without skew loops. We also prove results about skew loops on cylinders an…
Introduces string structures linking to loop spaces.
problem Understanding geometric string structures.
method Explains connections to loop spaces.
result String structures linked to loop spaces.
Study shows looping a 6-manifold over a 4-manifold results in a product of loops on spheres.
problem Understanding the homotopy properties of 6-manifolds over 4-manifolds.
method Analyzes the homotopy equivalence and rational homotopy of the total space of a sphere bundle over a 4-manifold.
result Looping a 6-manifold over a 4-manifold is homotopy equivalent to a product of loops on spheres.
New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.
problem Defining invariants for framed 3-manifolds with semi-simple Lie groups.
method Constructing graph complexes and cocycles, including self-loops, to define invariants.
result Higher-loop invariants can be defined by graph cocycles with or without self-loops.
Geodesic loops on tetrahedra are studied in spherical and hyperbolic spaces.
problem Characterize geodesic loops on tetrahedra in different types of spaces.
method Analytical proofs for spherical and hyperbolic spaces.
result Existence and properties of geodesic loops on tetrahedra in various spaces.
New definition of twisted 1-loop invariant using Ptolemy coordinates.
problem Defining and proving properties of twisted 1-loop invariants.
method Alternative definition via Jacobian of Ptolemy coordinates.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial for hyperbolic once-punctured torus bundles.
Training-free looped transformers improve model performance without additional training.
problem Improving model performance without additional training or fine-tuning.
method A lightweight inference-time wrapper loops a contiguous mid-stack block of layers of a frozen checkpoint without additional fine-tuning.
result Our method improves model performance across various model families.
Study path spaces and their homology, extending loop products and coproducts.
problem Understanding the homology of path spaces in closed manifolds.
method Morse-Bott theory and homology operations.
result Complete computation of extended loop product and coproduct on spheres.
Study the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds.
problem Global topology of loops of contactomorphisms and Legendrians in non-orderable manifolds.
method Filtering loops by positivity and analyzing subspaces of the filtration.
result Homotopy groups of the space of loops are subgroups of the positive loops subspace.
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. New basis and Schur-Weyl duality for loop Hecke algebra defined.
problem Define a new basis for the loop Hecke algebra.
method Use higher linear rewriting theory and combinatorics of Dyck paths.
result Yields a conjecture of Damiani-Martin-Rowell and provides a representation theoretic interpretation.
A central extension of the loop group of a Lie group is called transgressive, if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the Lie group. Transgressive loop group extensions are those that can be explored by finite-dimensional, higher-categorical geometry ov…
Study finds loops with specific curvature exist using Hardy's inequality.
problem Existence of closed planar loops with prescribed curvature.
method Variational approach, Hardy's inequality and associated functional space.
result Existence of loops with specific curvature proven.
We show the Chas-Sullivan product (on the homology of the free loop space of a Riemannian manifold) is related to the Morse index of its closed geodesics. We construct related products in the cohomology of the free loop space and of the based loop space, and show they are nontrivial.
Study Brownian loops on hyperbolic surfaces, linking to Selberg zeta function.
problem Understanding Brownian loops on hyperbolic surfaces and their relation to Selberg zeta function.
method Computed mass of loops and related to Selberg zeta function for geometrically finite surfaces.
result Relate total loop mass to Selberg zeta function, providing probabilistic interpretations of determinants.
Loop group method varies with base point choice.
problem Dependence of loop group method on base point choice.
method Analyzes how loop group method for harmonic maps varies with base point.
result Loop group method results depend on base point selection.
We introduce various versions of spin structures on free loop spaces of smooth manifolds, based on a classical notion due to Killingback, and additionally coupled to two relations between loops: thin homotopies and loop fusion. The central result of this article is an equivalence between these enhanced versions of spin…
Study vortex loops as coadjoint orbits of diffeomorphisms.
problem Understanding vortex loops in terms of coadjoint orbits.
method Analyzing vortex loops as coadjoint orbits of area-preserving diffeomorphisms.
result Vortex loops are coadjoint orbits of the diffeomorphism group.
Sphere bundles over 4-manifolds are trivial after looping, except for two cases.
problem Understanding the triviality of sphere bundles over 4-manifolds.
method Analyzing the splitting of sphere bundles after looping.
result The loop spaces of total manifolds of sphere bundles are homotopy equivalent, except for two special cases.
This paper analyzes the impact of loops on bilevel optimization efficiency.
problem The impact of loops on the efficiency of bilevel optimization algorithms.
method Unified convergence analysis and computational complexity characterization for AID-BiO and ITD-BiO with and without loops.
result Loops in bilevel optimization can improve overall efficiency but increase per-step complexity.
The paper finds non-contractible loops of Legendrian tori from knot families.
problem Computing non-contractible loops of Legendrian tori from knot families.
method Using cord algebra of knots to compute Legendrian contact homology.
result Obtained an infinite family of non-contractible loops of Legendrian tori.
Characterizes visibility and geodesic loops in complex domains.
problem Visibility and geodesic loops in complex domains.
method Using quasi-geodesic frames to characterize visibility and geodesic loops.
result Characterizes visibility and existence of geodesic loops in Kobayashi complete hyperbolic and Gromov hyperbolic domains.
Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.
problem Existence and uniqueness of solutions to loop equations in generalized Frobenius manifolds.
method Proves existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
result Existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
Cartan calculus applied to string topology homology.
problem Understanding the structure of free loop spaces.
method Introduced Cartan calculus on loop homology, linked to string topology operations.
result Loop product and bracket behavior under Hodge decomposition.
Study finds counterexamples to simple loop conjecture in higher dimensions.
problem Simple loop conjecture for surfaces to manifolds of dimensions at least four.
method Provided specific counterexamples for every g≥2 and n≥4. result No simple loop is contained in the kernel of the map for certain surfaces and manifolds.
For a finite dimensional symplectic manifold (M,ω) with a symplectic form ω, corresponding loop space (LM=C∞(S1,M)) admits a weak symplectic form Ωω. We prove that the loop space over $\mbr^n$ admits Darboux chart for the weak symplectic structure Ωω. Further, we show that inclusion map from the symp…
Framework learns robust control policies from expert demonstrations.
problem Adversarial robustness and closed-loop generalization in feedback control policies.
method Lipschitz-constrained loss minimization for certified robustness and generalization.
result Finite sample bound on policy learning error and robust closed-loop stability.
A survey of real differential geometry and loop theory is given in order to introduce the construction of an analytic loop associated to p-adic differential manifold.
The paper studies the n-loop Kontsevich invariant of knots with same Alexander polynomial.
problem Understanding the n-loop Kontsevich invariant for knots with identical Alexander polynomials.
method Analyzes the subspace generated by the n-loop Kontsevich invariant of knots with genus ≤ g and same Alexander polynomial.
result For n ≥ 2, the subspace is finite-dimensional.
Non-trivial Clifford bundle from loop space tangent bundle.
problem Triviality obstruction of Clifford bundle on loop space.
method Constructing Clifford algebra bundle from loop space tangent bundle, showing non-triviality through Stiefel-Whitney and Pontrjagin classes.
result Clifford bundle is non-trivial, obstructed by manifold's Stiefel-Whitney and Pontrjagin classes.
Proves loop coproduct invariance under simple homotopy equivalences.
problem Invariance of loop coproduct under simple homotopy equivalences.
method Transformation formula involving Whitehead torsion.
result Loop coproduct is invariant under simple homotopy equivalences.
The loop space of the Riemann sphere consisting of all Ck or Sobolev Wk,p maps from the circle S1 to the sphere is an infinite dimensional complex manifold. We compute the Picard group of holomorphic line bundles on this loop space as an infinite dimensional complex Lie group with Lie algebra the first Dolbe…
Study 2-loop part of Johnson cokernel using trace map.
problem Identify components of Johnson cokernel in degree 6.
method Use 2-loop trace map to capture Johnson cokernels.
result Capture all components of Johnson cokernels in degree 6.
Homology of torus knots stabilizes to loop space homology.
problem Computing homology of complex Grassmannians and torus knots.
method Colored sl(N) homology and free loop space computation. result Khovanov homology of torus knots stabilizes to loop space homology.
`Loop-fusion cohomology' is defined on the continuous loop space of a manifold in terms of \vCech cochains satisfying two multiplicative conditions with respect to the fusion and figure-of-eight products on loops. The main result is that these cohomology groups, with coefficients in an abelian group, are isomorphic to …
We relate the author's Lie cobracket in the module additively generated by loops on a surface with the Connes-Kreimer Lie bracket in the module additively generated by trees. To this end we introduce a pre-Lie coalgebra and a (commutative) Hopf algebra of pointed loops on a surface. In the last version I added sections…
The paper connects Riemann surface length spectra to Brownian loop measures.
problem Understanding the length spectra of Riemann surfaces with additional cusps.
method Using the Brownian loop measure to relate length spectra of Riemann surfaces with and without additional cusps.
result Expressed the total mass of Brownian loops in terms of the length of geodesic representatives.
In this paper we compute the singular homology of the space of immersions of the circle into the n-sphere. Equipped with Chas-Sullivan's loop product these homology groups are graded commutative algebras, we also compute these algebras. We enrich Morse spectral sequences for fibrations of free loop spaces together wi…
We prove the vanishing of the space of 3-loop Jacobi diagrams of odd degree. This implies that no 3-loop finite-type invariant can distinguish between a knot and its inverse.