New uniform K-theory and Poincare duality established for manifolds.
problem Developing a new framework for K-theory and K-homology.
method Constructing uniform K-homology, defining external and cap products, proving homotopy invariance and Poincare duality.
result Established Poincare duality between uniform K-theory and uniform K-homology on spin-c manifolds.
Uniform K-stability ensures existence of special metrics on toric manifolds.
problem Existence of conformally Kähler, Einstein-Maxwell metrics on toric manifolds.
method Introducing uniform K-stability and showing its equivalence to properness of relative K-energy.
result Uniform K-stability is necessary and sufficient for the existence of f-extremal metrics on toric manifolds. K-homology classes linked to elliptic operators.
problem Defining K-homology classes for elliptic operators.
method Using uniform K-homology and principal symbols.
result Classes depend only on the operator's principal symbol.
We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. This generalization will follow as a corollary from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the un…
Uniform convexity in divisible domains leads to hyperbolic geometry.
problem Understanding the geometry of divisible convex sets in Finsler manifolds.
method Proving β-uniform convexity of a specific Finsler metric. result A strictly convex divisible domain induces a β-uniformly convex Finsler metric. Uniform K-stability proven using asymptotic results in Kähler geometry.
problem Proving uniform K-stability in Kähler geometry.
method Analyzing slopes of functionals along rays defined by test configurations.
result Coercivity of the Mabuchi functional implies uniform K-stability.
Extends index theorem to uniformly elliptic operators on manifolds.
problem Generalizing index theorem to uniformly elliptic operators.
method Local index theorem on manifolds of bounded geometry.
result Validates multigraded elliptic uniform pseudodifferential operators.
Uniform framework for type C3 geometries, answering a question and calculating automorphism groups.
problem Understanding the structure of exceptional homogeneous compact geometries of type C3.
method Uniform framework to study and analyze these geometries.
result These geometries are simply connected, answering a question by Kramer and Lytchak.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
Uniformly proves index invariance for signature operators on manifolds.
problem Proving index invariance for signature operators under uniform homotopy.
method Uniform homotopy invariance of Roe index for signature operators.
result Uniform homotopy invariance of Roe index for signature operators.
We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. The Ck,α-smoothness result is o…
New algorithms achieve uniform stability for empirical risk minimization.
problem Designing uniformly stable optimization algorithms for empirical risk minimization.
method Black-box conversion of smooth optimization algorithms and development of Mirror Descent for smooth optimization.
result Optimal algorithms with uniform stability and convergence rates for smooth optimization.
Uniform Shapiro-Lopatinski conditions ensure well-posedness of boundary value problems on manifolds with bounded geometry.
problem Boundary value problems on manifolds with boundary and bounded geometry.
method Uniform Shapiro-Lopatinski regularity condition, compactness argument, Nirenberg trick.
result Uniform Shapiro-Lopatinski condition characterizes well-posed boundary value problems.
We study the classification of ultrametric spaces based on their small scale geometry (uniform homeomorphism), large scale geometry (coarse equivalence) and both (all scale uniform equivalences). We prove that these equivalences can be characterized with parallel constructions using a combinatoric tool called common zi…
Uniform spectral gap found for stable commutator length in hyperbolic 2-orbifolds.
problem Understanding stable commutator length in 3-manifolds.
method Explicit quasimorphisms for generic case, hyperbolic geometry for exceptional case.
result Explicit uniform gap of 1/36 for all orbifolds except a sphere with three cone points.
We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more general than similar classes defined by other authors. We revisit Spakula's uniform K-…
Develops Lefschetz theory for noncompact manifolds.
problem Lefschetz fixed-point theory for noncompact manifolds.
method Introduces uniform bounded cohomology and develops obstruction theory.
result Uniform Lefschetz class vanishes if and only if map is homotopic to a strongly fixed-point free map.
Uniform lattices in certain semi-simple groups contain Anosov surface subgroups.
problem Understanding surface subgroups in uniform lattices of semi-simple groups.
method Introducing K-Sullivan maps and using coarse geometry of flag manifolds. result Quantitative version of surface subgroup theorem, showing closeness to smooth round circles.
The paper introduces new functors for cohomology groups of manifolds.
problem Behavior of cohomology groups under uniform maps.
method Introducing contravariant functors between manifold categories and vector space categories.
result Uniform homotopy invariance of cohomology groups.
Establishes geodesic stability for Kähler metrics, proving existence of constant scalar curvature.
problem Existence of constant scalar curvature Kähler metrics.
method Exploring metric geometry of Mabuchi geodesic rays and uniform convexity properties of Kähler metrics space.
result Essentially optimal form of Donaldson's geodesic stability conjecture proved.
Paper proves uniform continuity bounds for complex Monge-Ampère solutions.
problem Estimating the continuity of solutions to complex Monge-Ampère equations.
method PDE-based approach from fully non-linear equations in Kähler geometry.
result Uniform and sharp estimate for the modulus of continuity.
New uniformity definition on noncompact manifolds without metrics.
problem No Riemannian metric for uniformity on noncompact manifolds.
method Definition of uniformity without metrics, equivalent to bounded geometry.
result Equivalent definition of uniformity on noncompact manifolds.
Study shows uniform doubling property for SU(2) geometries.
problem Estimating properties of left-invariant geometries on SU(2).
method Analyzes all left-invariant geometries on SU(2) and proves uniform doubling.
result Left-invariant geometries on SU(2) are uniformly doubling.
In this paper a new connection between the discrete conformal geometry problem of disk pattern construction and the continuous conformal geometry problem of metric uniformization is presented. In a nutshell, we discuss how to construct disk patterns by optimizing an objective function, which turns out to be intimately …
Paper solves open problem in complex Finsler geometry.
problem Existence of non-Kähler weakly Kähler Finsler metrics.
method Constructs a family of weakly Kähler Finsler metrics.
result Proves uniformization theorem for unitary invariant complex Randers metrics.
New approach to nematic fields on surfaces, relaxing uniformity to quasi-uniformity.
problem Identifying least distorted nematic fields on generic surfaces.
method Relaxing the notion of uniformity into quasi-uniformity and proving parallel transport by geodesics.
result All quasi-uniform fields are parallel transported by the geodesics of the surface.
Uniform bounds for complex equations using Monge-Ampère method.
problem Bounding solutions to complex equations.
method Auxiliary Monge-Ampère equation method.
result Uniform bounds remain valid even as background metrics degenerate.
We consider a parabolic-like systems of differential equations involving geometrical quantities to examine uniformization theorems for two- and three-dimensional closed orientable manifolds. We find that in the two-dimensional case there is a simple gauge theoretic flow for a connection built from a Riemannian structur…
Survey on uniformization of metric surfaces, including fractal and topological manifolds.
problem Uniformization of metric surfaces homeomorphic to 2D topological manifolds.
method Various uniformization theorems, including quasisymmetric and quasiconformal approaches.
result Uniformization results for metric spheres and arbitrary metric surfaces.
Uniform linear bounds on volume changes in 3D hyperbolic spaces.
problem Volume variation in hyperbolic 3-manifolds.
method Uniform linear bounds proof for drilling and filling operations.
result Uniform linear bounds on volume variation proved.
Study skinning maps in non-acylindrical 3-manifolds, proving existence of cores.
problem Generalizing Thurston's theorem for non-acylindrical 3-manifolds.
method Analyze divergent sequences in deformation space, prove existence of cores.
result Existence of cores satisfying uniform geometry in non-acylindrical 3-manifolds.
We prove that Riemannian metrics with a uniform weak norm can be smoothed to having arbitrarily high regularity. This generalizes all previous smoothing results. As a consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform Betti number estimate is also obtained.
Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.
problem Bounding the diameter of Calabi-Yau fibrations near singular fibers.
method Uniform diameter bound proof for Calabi-Yau fibrations with canonical singular fibers.
result Uniform diameter bounds for all fibres in suitable rescaling.
Maximal acceleration metrics limit spacetime curvature.
problem Bounding spacetime curvature under maximal acceleration.
method Developed a geometric framework for maximal acceleration metrics and associated connections, proving curvature bounds.
result Uniform bounds on curvature components follow from uniform bounds on maximal acceleration.
New characterizations of partial positivity using Hörmander's L2-estimate.
problem Characterizing partial positivity in complex geometry.
method Using a twisted version of Hörmander's L2-estimate. result New characterizations of partial positivity, including uniform q-positivity and RC-positivity. We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.
Established a Hardy inequality on Finsler manifolds.
problem Hardy inequality on Finsler manifolds.
method Used geometric properties of Finsler structures to prove the inequality.
result Depends on reversibility constant and uniformity constant of Finsler structure.
A general method for analytic inversion in integral geometry is proposed. All classical and some new reconstruction formulas of Radon-John type are obtained by this method. No harmonic analysis and PDE is used.
In this thesis a connection between the worlds of discrete and continuous conformal geometry is explored. Specifically, a disk pattern production theroem is proved using an energy which measures how ``uniform'' the angle data of a triangulation is, see also math.DG/0002150. Then this energy is averaged over all the Del…
Sharp stability of isometries on Heisenberg group proven.
problem Quantitative stability of isometries on the Heisenberg group.
method Proving quasi-isometries close to isometries with specific closeness orders.
result Quasi-isometries of Heisenberg group are close to isometries with specific closeness orders.
Study L2-cohomology in unbounded geometry manifolds.
problem Invariance of L2-cohomology under quasi-isometries on unbounded ends. method Uniform homotopy equivalence, quasi-isometry on unbounded ends, mapping cone for L2-cohomology. result Invariance of L2-cohomology groups under quasi-isometry on unbounded ends. The paper classifies 1-dimensional uniform measures in various dimensions.
problem Classifying uniformly distributed measures of dimension 1 in general codimension.
method Analyzing measures with connected 1-dimensional support and providing a partial classification for general measures.
result Uniform measures with connected 1-dimensional support are homogeneous measures.
Paper proves the maximum Hausdorff dimension of singular set for n-uniform measures.
problem Classifying uniform measures and understanding their singular sets.
method Using the Kowalski-Preiss cone to construct an n-uniform measure and proving its singular set's Hausdorff dimension.
result The Hausdorff dimension of the singular set of any n-uniform measure is at most n-3.
Study of metric anomalies in uniform elastic solids without stress.
problem Understanding metric anomalies in uniform elastic solids.
method Introducing a quasi-plastic deformation framework and deriving a general form of metric anomalies.
result Derivation of a general form of metric anomalies yielding zero stress in uniform solids.
Uniform estimates for elliptic problems near polygonal domains.
problem Proving uniform solvability estimates for elliptic problems near polygonal domains.
method Suitable conformal modification of the metric to make the union of domains a manifold with boundary and relative bounded geometry.
result Rounding off the corners of the limit polygonal domain.
New measures found in 3-uniform geometry.
problem Understanding non-flat uniform measures in geometric measure theory.
method Combining combinatorial methods and distance symmetry properties.
result Infinite family of 3-uniform measures constructed.
We prove effective uniformization for nearly round 2-spheres and investigate their stability.
problem Proving effective uniformization for nearly round 2-spheres and their stability.
method Utilizing an identity related to the third-order differential of the conformal factor, and an isometric embedding of a round sphere into Euclidean space using an orthogonal basis of the first eigenspace of the Laplacian operator.
result We provide a simplified proof of effective uniformization and its stability.
We address the problem of local geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are described in a uniform manner by the Cartan method of equivalence. This includes conformal, Weyl and metric geometries in three and six dimen…