Study learns convolution operators on compact Abelian groups using regularization.
problem Learning convolution operators on compact Abelian groups.
method Regularization-based approach with ridge regression estimator.
result Characterizes the accuracy of the estimator in terms of finite sample bounds.
Abelian subgroups in certain spaces are simple.
problem Characterizing Abelian subgroups in specific geometric spaces.
method Analyzing the asymptotic norm of Abelian subgroups.
result Each Abelian subgroup is isomorphic to Zk. Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
The paper proves a Whitehead theorem for fine shape spaces.
problem Proving a Whitehead theorem for fine shape spaces.
method Using Steenrod-Sitnikov homotopy groups and ind-groups.
result Fine shape morphisms are equivalences if they induce isomorphisms on π_i.
v2: An additional assumption was added in Theorem 4.8. In order to show that a connected abelian group is admissible on the site of locally compact spaces we must in addition assume that it is locally topologically divisible. This condition is used in the proof of Lemma 4.62.
Study lattices in specific Lie groups for geometric structures.
problem Existence of lattices in Lie groups with certain geometric structures.
method Analyzing left invariant locally conformal Kähler or symplectic structures.
result Existence of lattices only in dimension 4 for Kähler structures, and in any even dimension for symplectic structures.
Analytic torsion defined for non-compact Lie groups and discrete subgroups.
problem Defining and calculating analytic torsion for non-compact Lie groups and their discrete subgroups.
method Localised analytic torsion and relative analytic torsion defined for Lie groups of type I, using representations and discrete subgroups.
result Relative analytic torsion of (G,Γ) coincides with Lott L2 analytic torsion of a covering space. Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition functi…
Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.
problem Existence and classification of LCSKT structures on Lie groups and their quotients.
method Introducing LCSKT structures and studying their properties on Lie groups and their quotients.
result Existence of non-trivial LCSKT structures on 6-dimensional nilpotent Lie algebras and almost abelian Lie algebras.
We prove that for a coarse space X the ideal S(X) of small subsets of X coincides with the ideal D<(X) of subsets A⊂X of asymptotic dimension asdim(A)<asdim(X) provided that X is coarsely equivalent to an Euclidean space Rn. Also we prove that for a locally compact Abelian group X, the equali…
The note confirms a conjecture for specific Lie groups.
problem The conjecture about constant holomorphic sectional curvature in non-Kähler geometry.
method Compact quotients of Lie groups with specific properties.
result The conjecture is confirmed for almost abelian Lie algebras and those with certain abelian ideals.
Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.
problem Classify balanced Hermitian structures on almost abelian Lie algebras.
method Classify six-dimensional almost abelian Lie algebras with balanced structures, investigate flow of balanced metrics and anomaly flow.
result Prove conjecture for compact almost abelian solvmanifolds with left-invariant complex structures.
The paper extends spectral results to non-abelian groups acting on compact Riemannian manifolds.
problem Determining potential functions from spectral data for non-abelian group actions.
method Generalized Legendrian relations and spectral invariants.
result Potential functions are determined by the equivariant spectrum for certain Schrödinger operators.
A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their …
If X is a CW complex, one can assign to each point of X an ordered abelian group of finite rank whose subset of positive elements depends continuously on the points of X. A locally trivial bundle which arises in this way we denote by E(X). In the present work we establish a topological classification of such bundles in…
The 3D Index is extended to meromorphic functions on triangulated 3-manifolds.
problem Extending the 3D Index to a broader class of triangulated 3-manifolds.
method Assigning a meromorphic function to each ideal triangulation, invariant under Pachner moves, and expanding it into a Laurent series.
result The meromorphic function can be computed from gluing equations and coincides with the 3D Index for ideal triangulations with strict angle structures.
New proof shows abelian Cantor groups can act on spaces.
problem Understanding actions of Cantor groups on metric spaces.
method Examined actions of abelian Cantor groups on metric spaces.
result Cantor groups can be abelian for n>1 in space actions.
The paper proves a structure theorem for compact Kähler manifolds with semi-positive holomorphic sectional curvature.
problem Understanding the fundamental groups of compact Kähler manifolds with specific curvature properties.
method Analyzing foliations and topological properties to prove a locally trivial fibration structure.
result Compact Kähler manifolds with semi-positive holomorphic sectional curvature admit a locally trivial fibration structure.
We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear ho…
We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is an affirmative solution to Brock-Farb's Rank Conjecture which as…
New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.
problem Constructing manifolds with positive Ricci curvature and non-abelian fundamental groups.
method Constructing specific 9-dimensional manifolds with positive Ricci curvature and non-uniformly virtually abelian fundamental groups.
result Examples of manifolds with positive Ricci curvature and non-uniformly virtually abelian fundamental groups.
Consider a compact locally symmetric space M of rank r, with fundamental group Γ. The von Neumann algebra $\vn(Γ)$ is the convolution algebra of functions f∈ℓ2(Γ) which act by left convolution on ℓ2(Γ). Let Tr be a totally geodesic flat torus of dimension r in M and let $Γ_0\cong\bb Z^r$ be t…
The paper characterizes convex co-compact groups with one-dimensional boundary faces.
problem Characterizing convex co-compact groups with specific boundary properties.
method Proving relative hyperbolicity and using coarse Hilbert dimension.
result Convex co-compact groups with one-dimensional boundary faces are relatively hyperbolic.
We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral sym…
Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.
problem Characterizing harmonic almost complex structures on almost abelian Lie groups and solvmanifolds.
method Adapted Gray-Hervella classification to almost abelian Lie groups, characterized harmonic structures.
result Examples of harmonic almost complex structures in different Gray-Hervella classes on compact almost abelian solvmanifolds.
We construct a covariant functor from a category of Abelian principal bundles over globally hyperbolic spacetimes to a category of *-algebras that describes quantized principal connections. We work within an appropriate differential geometric setting by using the bundle of connections and we study the full gauge group,…
Study on G2-manifolds shows automorphism group is abelian.
problem Understanding automorphism groups of G2-manifolds.
method Analyzing compact 7-manifolds with closed non-parallel G2-structures.
result The identity component of automorphism group is abelian with dimension bounded.
Study on spectral sequence for abelian Lie group actions, with bounds and applications.
problem Understanding the spectral sequence for abelian Lie group actions.
method Provided upper bounds and examples to show these bounds are sharp.
result Sharp bounds on the degeneration page of spectral sequence.
The study of holonomy groups in flat solvmanifolds, proving finite abelian groups can be holonomy groups and describing dimensions.
problem Understanding the holonomy groups of flat solvmanifolds.
method Elementary proof and construction of examples for various dimensions.
result Finite abelian groups can be holonomy groups of flat solvmanifolds, and specific dimensions and holonomy groups are described.
We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism …
We prove that two countable locally finite-by-abelian groups G,H endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely-generated or both are infinitely generated. On the other hand, we show that each countable group…
Generalizes abelianization for framed local systems over surfaces.
problem Understanding framed local systems over punctured surfaces for various groups.
method Analysis of spectral networks, triangulations, and matrix reinterpretation of path lifting rules.
result Parametrizations of moduli spaces of decorated and framed local systems.
New method deforms function algebras on manifolds using spectral decomposition.
problem Deforming function algebras on compact Riemannian manifolds.
method Introducing a bilinear product on the finite spectral core of smooth functions using unimodular phases.
result The product extends to a Sobolev algebra and admits iteration under certain conditions.
Study of complex and Hermitian structures on specific Lie groups.
problem Classifying Lie groups with specific geometric structures.
method Analysis of left-invariant structures on almost abelian Lie groups.
result Classification of six-dimensional generalized Kähler almost abelian Lie groups.
Classifies complex symplectic structures on Lie algebras with large abelian ideals.
problem Classifying complex symplectic structures on Lie algebras with large abelian ideals.
method Two constructions of complex symplectic structures on Lie algebras with large abelian ideals, considering compact quotients of Lie groups.
result Complete classification of complex symplectic structures on almost abelian Lie algebras.
Lie groups with bi-invariant distance are products of abelian and compact groups.
problem Characterizing Lie groups with bi-invariant distances.
method Analyzing the structure of Lie groups and introducing a Finsler norm.
result The sectional curvature of bi-invariant distances is non-negative and vanishes only for abelian subalgebras.
Ancient solutions on bundles are found for non-abelian groups.
problem Finding ancient solutions on bundles with non-abelian structural groups.
method Generalized ancient solutions of Ricci flow on mSO(3) bundles to RP3 fibre bundles over quaternionic Kähler manifolds. result Ancient solutions of Type I, κ-noncollapsed, and positive Ricci curvature on bundles.
Let Λ be a finite abelian group. A dynamical system with transformation group Λ is a triple (A,Λ,α), consisting of a unital locally convex algebra A, the finite abelian group Λ and a group homomorphism $α:Λ\rightarrow\Aut(A)$, which induces an action of Λ on A. In this paper we present a new, geometricall…
The paper defines angle structures on 3-manifolds using abelian groups.
problem Finding conditions for labelings of tetrahedra in 3-manifolds.
method Non-trivial conditions on labelings of tetrahedra in a triangulated 3-manifold using an abelian group.
result Angle structures can be used to define a non-trivial condition on labelings of tetrahedra.
Extends homotopical theory to locally compact groups, refining their compactness properties.
problem Developing homotopical invariants for locally compact groups.
method Extending classical theory of homotopical Σ-sets to locally compact Hausdorff groups, defining Σtopn sets of characters. result Recovering and generalizing classical results on characters and compactness properties of groups.
We present a K-theoritic approach to the Guillemin-Sternberg conjecture, about the commutativity of geometric quantization and symplectic reduction, which was proved by Meinrenken and Tian-Zhang. Besides providing a new proof of this conjecture for the full non-abelian group action case, our methods lead to a generalis…
Local mixing theorem for abelian covers of hyperbolic 3-manifolds.
problem Mixing properties of frame flows on abelian covers.
method Local mixing theorem for horospherical subgroups on abelian covers.
result Classification of invariant measures for horospherical subgroups.
We classify those manifolds mentioned in the title which have finite topological type. Namely we show any such connected M is isomorphic to a hyperkaehler quotient of a flat quaternionic vector space by an abelian group. We also show that a compact connected and simply connected 3-Sasakian manifold of dimension 4n-1 wh…
We discuss the Morse-Novikov cohomology of a compact manifold, associated to a closed one--form whose free abelian group generated by its periods ⟨∫γη∣[γ]∈π1(M)⟩ is of rank 1, the focus being on locally conformally symplectic manifolds. In particular, we provide an explicit computation for t…
Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.
problem Classifying Spin(7) structures on compact 8-manifolds.
method Obstruction theory applied to Spin(7) structures on compact 8-manifolds with abelian fundamental group.
result Compact Riemannian 8-manifolds with holonomy Spin(7) have exactly two Spin(7) structures extending the induced G2 structure on the boundary.
Compact actions on manifolds with specific eigenvalue conditions.
problem Actions of semidirect products on compact manifolds with eigenvalue constraints.
method Analyzing the induced action on cohomology and using it to generalize results.
result Existence of neighborhoods of trivial actions with abelian properties.
The paper studies a flow on complex Lie groups, showing convergence to solitons.
problem The study of curvature flows on complex Lie groups.
method Positive Hermitian curvature flow on left-invariant metrics.
result The flow converges to solitons in both nilpotent and almost-abelian cases.
Characterizes hypercomplex Lie groups and their solvmanifolds.
problem Understanding hypercomplex structures on Lie groups and their solvmanifolds.
method Characterization of almost abelian Lie groups with hypercomplex structures, analysis of Obata and Bismut connections, classification of hypercomplex Lie groups, and construction of solvmanifolds.
result Classification of hypercomplex almost abelian Lie groups in dimension 8 and properties of their solvmanifolds.