Invites geometers to Garside theory for mapping class groups.
problem None explicitly stated, but related to geometric group theory.
method Garside theory applied to mapping class groups.
result No specific key result mentioned in the abstract.
In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a variety of settings: smooth maps between equidimensional manifolds, CR-singularitie…
We prove that the introduction of the class of geometrically atomic bundle maps by Harvey and Lawson in their theory of singular connections is not necessary because an arbitrary map satisfies the conditions of geometric atomicity.
Study on mappings in Carnot groups, proving rigidity results.
problem Understanding mappings in Carnot groups and proving rigidity.
method Structural results for Sobolev mappings, proving rigidity or regularity.
result Establishes partial rigidity and partial regularity theorems.
Proves conjecture for hyperbolic-by-cyclic groups using geometric methods.
problem Proving Farrell-Jones Conjecture for specific group structures.
method Geometric methods and structure theory of mapping tori.
result Proved Farrell-Jones Conjecture for mapping tori of automorphisms of hyperbolic-by-cyclic groups.
Simplified Milnor-Schwarz lemma for geometric group theory.
problem Conditions for orbit maps to be quasi-isometries.
method Succinct treatment and applications to non-Archimedean groups.
result Sharpened results on mapping class groups and quasi-isometry classification.
Study normal bundle and deformation to get new pushforward maps.
problem Construct pushforward maps in various homology theories.
method Use deformation Lie groupoids to construct pushforward maps.
result Functoriality of pushforward maps recovers and generalizes previous cases.
This paper presents and explores a theory of \emph{multiholomorphic maps}. This group of ideas generalizes the theory of pseudoholomorphic curves in a direction suggested by consideration of the kinds of compatible geometric structures that appear in the realm of special holonomy as well as some of the topological and …
Geometric cohomology model uses co-oriented maps to define a product structure.
problem Constructing a geometric model for cohomology of smooth manifolds.
method Develops a cochain complex model based on co-oriented smooth maps, focusing on their pull-back product structure.
result Geometric cochains with a partially defined product structure induce the cup product in cohomology.
We construct a higher Whitehead torsion map, using algebraic K-theory of spaces, and show that it satisfies the usual properties of the classical Whitehead torsion. This is used to describe a "geometric assembly map" defined on stabilized structure spaces in purely homotopy theoretic terms.
Abstract reviews geometric theories of smooth and F-smooth systems.
problem Geometric theories of smooth and F-smooth systems.
method Reviews geometric theories of smooth and F-smooth systems.
result Discusses geometric theories of smooth and F-smooth systems.
Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly disconnected). We characterise the subsurfaces which lead to virtually abelian ge…
Maps persistence diagrams into Hilbert and Euclidean spaces with explicit distortions.
problem Embedding persistence diagrams into Euclidean spaces for statistical analysis.
method Explicit geometric maps with distortion functions.
result Controlled geometric information loss through explicit distortion functions.
We show that the computation of the Fredholm index of a fully elliptic pseudodifferential operator on an integrated Lie manifold can be reduced to the computation of the index of a Dirac operator, perturbed by a smoothing operator, canonically associated, via the so-called clutching map. To this end we adapt to our fra…
The paper characterizes harmonic maps and studies their properties on Riemannian manifolds.
problem Characterizing and analyzing harmonic maps between Riemannian manifolds.
method Analytic and geometric methods, including L2-orthogonal decomposition and energy density analysis.
result A criterion for harmonic submersions and diffeomorphisms, and new results linking harmonic symmetric bilinear forms and metrics.
Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.
problem Deformation theory of Lie groupoids and algebroids.
method Defining a morphism between deformation complexes and Hochschild complexes, applying to adiabatic groupoids.
result Induced van Est map from geometric to algebraic deformation cohomology.
Coincidences of maps between smooth manifolds are studied via a geometric approach which involves (nonstabilized) normal bordism theory and pathspaces.
Study classifies mappings of bivariate normal densities, revealing three types with distinct geometric and statistical properties.
problem Understanding the properties of two-component bivariate normal mixtures.
method Classification via A-equivalence and statistical analysis. result Three distinct types of mappings with specific geometric and statistical properties, and upper bounds for the number of modes.
New stretch maps minimize distortion in geometric group theory.
problem Finding optimal maps in geometric group theory.
method Proving minimizers using modulus of curve families and MSP.
result Stretch maps are minimizers of mean quasiconformal distortion.
Stability of biharmonic maps in critical dimension proven.
problem Stability of biharmonic maps between manifolds in critical dimension.
method Generalization of Morse stability theory to biharmonic maps, development of strong energy quantization method.
result Strong energy quantization in a wide class of problems in geometric analysis.
Every pseudo-Anosov mapping class φ defines an associated veering triangulation τφ of a punctured mapping torus. We show that generically, τφ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesic…
Develops reduction method for strong Dirac maps.
problem Generalizing Poisson momentum maps.
method General procedure for reduction along strong Dirac maps.
result Recover and introduce new Poisson, quasi-Poisson, and Dirac reduced structures.
The paper defines approximate fibrations in higher topos theory.
problem Defining approximate fibrations in a new mathematical framework.
method Introducing approximate fibrations for geometric morphisms of ∞-topoi, providing characterizations and comparing to previous definitions. result Generalization of shape-theoretic characterizations to a topos-theoretical proof.
New theory for Hamiltonian actions on special geometric structures.
problem Hamiltonian actions on cosymplectic groupoids.
method Developed a moment map theory for 0-shifted cosymplectic structures.
result Established a version of the Kirwan convexity theorem.
In recent years a lot of attention has been paid to topological spaces which are a bit more general than smooth manifolds - orbifolds. Orbifolds are intuitively speaking manifolds with some singularities. The formal definition is also modelled on that of manifolds, an orbifold is a topological space which locally is ho…
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.
The first author's geometric Hopf invariant of a stable map F:Σ∞X→Σ∞Y is a stable Z2-equivariant map h(F):Σ∞X→Σ∞(Y∧Y) constructed by an explicit difference construction applied to (F∧F)ΔX−ΔYF. The stable Z2-equivariant homotopy c…
Quantum map counts BPS states in special theories.
problem Computing protected spin characters in class S theories.
method Geometric approach from 5D supersymmetric Yang-Mills theory.
result Explicit computation of protected spin characters in various examples.
We describe for any Riemannian manifold a certain infinitesimal neighbourhood of the diagonal. Semi-conformal maps are analyzed as those that preserve such neighbourhoods; harmonic maps are analyzed as those that preserve mirror image formation for pairs of points in such neighbourhoods.
Geometrically classifies maps from R^0|2 to any manifold, unifying theories.
problem Classifying maps from R^0|2 to any manifold without auxiliary structures.
method Relates maps to pullback of decomposable bivector bundle over S via algebraic constraints.
result Reduced manifold has fiber dimension dim(S) + 1, unifying topological and algebraic views.
Novel Morse theory for mapping cone cohomology.
problem Cohomology of mapping cones varies with closed forms.
method Introduced a Morse complex for mapping cones.
result Cohomology of cone Morse complex is isomorphic to mapping cone cohomology.
Survey discusses new ideas in geometric group theory and their applications.
problem Understanding geodesic metric spaces and their equivariant wall structures.
method Introduces and highlights the impact of injective metric spaces and cubical approximation theorem.
result Rich equivariant wall structures in various geodesic metric spaces.
We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k-1)n to dimension …
Study fourth-order geometric flow of shape operator for co-dimension one immersions.
problem Analyzing the geometry of isometric immersions in Riemannian manifolds.
method Introduce a moduli flow to decrease curvature variation energy.
result The flow decreases a natural energy measuring curvature variation.
Geometric approach to moment maps in complex geometry.
problem Constructing moment maps in complex geometry.
method Introducing universal families and equivariant differential forms.
result New geometric proofs and equations for moment maps.
Jets of mappings introduced by Ehresmann are still the most useful objects for formulating geometric frameworks of physical theories. We are proposing modifications designed to make jet theory less dependent on local coordinates. Extensions of the theory with applications to the calculus of variations and mechanics are…
Develops a new exponential map for time-varying vector fields.
problem Lack of global flows for general time-varying vector fields.
method Categorical development of spaces of vector fields and flows, allowing for systematic localisation.
result Derives the homeomorphism of the exponential map for vector fields with measurable time-dependence.
In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs (f_1, f_2) of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC(f_1, f_2) (and MC(f_1, f_2), resp.) of pathcomponents (and of poi…
The paper trivializes moment maps for various geometric structures.
problem Trivializing moment maps for different geometric structures.
method General framework of a reductive group G acting on a smooth affine variety, using Kempf-Ness theory, Morse theory, and ideas from Nakajima and Kronheimer. result Locally trivial fibration of moment maps over a regular locus of the center of the Lie algebra of a maximal compact subgroup.
Constructs maps from field theories to complexified K-theory and elliptic cohomology.
problem Identifying geometric models for Chern characters in supersymmetric field theories.
method Higher-dimensional generalization of Fei Han's method, involving super moduli spaces and derived geometry.
result Provides evidence for the Stolz--Teichner program and geometric models for Chern characters.
We give an overview of the theory of Cannon-Thurston maps which forms one of the links between the complex analytic and hyperbolic geometric study of Kleinian groups. We also briefly sketch connections to hyperbolic subgroups of hyperbolic groups and end with some open questions.
Classifies theories with eight supercharges using pseudo-periodic maps and Riemann surfaces.
problem Classifying theories with eight supercharges using mathematical tools.
method Assumes theories are given by genus g fibrations of Riemann surfaces, uses pseudo-periodic maps of negative type in mapping class group.
result Identifies dual graphs and 3d mirror quivers, unifies various SCFTs in combinatorial framework.
Defines and computes geometric pairings for discrete groups using Baum-Connes assembly map.
problem Defining and computing geometric pairings for discrete countable groups.
method Constructs explicit morphisms and the Chern-Baum-Connes assembly map.
result Explicit formulation of a Chern-Connes pairing with the periodic cyclic cohomology of the group algebra.
We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic…
A new simple proof for surface map degree inequality.
problem Degree of maps between closed surfaces.
method Elementary proof without additional techniques.
result A new proof of the inequality χ(M) ≤ d·χ(N).
Study continuity of limit sets in symmetric spaces.
problem Continuity of limit sets for geometrically finite subgroups in symmetric spaces.
method Extended geometrically finite representations theory.
result Limit sets vary continuously with respect to Hausdorff distance under strong convergence.
The paper studies heat kernel behavior on symmetric spaces.
problem Large-time behavior of heat operator traces on symmetric spaces.
method Uses representation theory and Carmona's proof of Vogan's lambda map.
result Provides an asymptotic formula for heat kernel behavior.
Develops a lifting theory for exponential maps in semi-Riemannian geometry.
problem Overcoming singularities in exponential maps to prove geodesic connectivity.
method Lifting theory for semi-Riemannian manifolds with path-continuation property.
result General path-lifting theorem extending globally under certain conditions.