In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In th…
The study describes the structure at infinity of submanifolds in real space forms.
problem Understanding the structure at infinity of submanifolds in real space forms.
method Using the decay of the second fundamental form of tamed immersions.
result Estimation of the number of ends in terms of volume growth of extrinsic balls.
Proves conditions for minimal surfaces in complex hyperbolic space.
problem Conditions for finite energy equivariant minimal surfaces in complex hyperbolic space.
method Analyzes peripheral holonomy and uses Higgs bundles.
result Explicit parametrization and construction of minimal surfaces.
We introduce the new class of submanifolds of co-Banach type in tame Fréchet manifolds and construct tame Fréchet submanifolds as inverse images of regular values of certain tame maps. Our method furnishes an easy way to construct tame Fréchet manifolds. The results presented are key ingredients in the construction of …
Paper characterizes tamed and weakened tamed four-manifolds using a new technique.
problem Characterizing tamed and weakened tamed four-manifolds.
method Using weakly \widetilde{\mathcal{D}}^+_J -closed technique.
result Characterization of tamed and weakened tamed four-manifolds.
The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Φ:R×X→X on pfaffian set X is tame if the graph of Φ is a pfaffian subset of R×X×X. Any compact tame set admits plenty tame flows. We prove …
Deep learning models viewed through tame geometry for convergence guarantees.
problem Understanding convergence guarantees in deep learning models.
method Introducing tame geometry concepts and tools for nonsmooth nonconvex settings.
result Illustrates tame geometry as a natural framework for AI systems, especially deep learning.
Paper proposes a new method to stabilize noisy gradient algorithms.
problem Stochastic-gradient Langevin algorithms can introduce bias when taming denominators depend on stochastic-gradient realizations.
method Proposes a structure-preserving framework for designing tamed denominators that avoid unnecessary taming and maintain the stabilizing effect of taming.
result The method avoids stationary bias and explains the stationary error split into bias and remaining error.
The study confirms Gromov's speculation and provides bounds for taming symplectic structures.
problem Understanding the relationship between taming symplectic structures and the area of pseudoholomorphic curves.
method Analyzes the numerical cone of taming symplectic structures and characterizes coarsely holomorphic curves.
result An almost complex manifold with an area bound admits a taming symplectic structure, confirming Gromov's speculation.
Acylindrical hyperbolicity proven for a specific group of automorphisms.
problem Proving the acylindrical hyperbolicity of a group of automorphisms.
method Study of a 2D simplicial complex and proving its contractibility and Gromov-hyperbolicity; finding loxodromic elements.
result Tame automorphism group is acylindrically hyperbolic.
Solenoids are ``inverse limits'' of the circle, and the classical knot theory is the theory of tame embeddings of the circle into the 3-space. We give some general study, including certain classification results, of tame embeddings of solenoids into the 3-space as the ``inverse limits'' of the tame embeddings of the ci…
RELTA-SGLD stabilizes nonconvex SGLD updates with a lighter taming scheme.
problem Stabilizing superlinear stochastic-gradient updates in nonconvex optimization.
method Threshold-based taming with relative-growth principle for stability.
result Polynomial moment stability and first-order stationary accuracy in nonconvex SGLD.
The paper proves symplectic forms on tamed almost complex 4-manifolds.
problem Finding symplectic forms on tamed almost complex 4-manifolds.
method Unified proof approach similar to Buchdahl's Kodaira conjecture proof.
result Existence of new symplectic forms compatible with given almost complex structure.
A closed discrete subset A⊂C is called tame if C∖A is quasiconformally equivalent to C∖Z. By giving several criteria for A to be tame, we shall show that Z+iZ is not tame.
Positivity of intersections in 4-manifolds leads to taming symplectic structures.
problem Taming symplectic structures in almost complex 4-manifolds.
method Proof of positivity of intersections of pseudoholomorphic curves.
result Positivity of intersections is stable and leads to taming symplectic structures.
Whitehead doubles have matching meridional rank and bridge number.
problem Determining the meridional rank of Whitehead doubles.
method Analyzing algebraically tame knots and their Whitehead doubles.
result Meridional rank and bridge number coincide for Whitehead doubles of prime knots.
Global diffeomorphism proof between tame Fréchet spaces.
problem Existence of global diffeomorphism between Fréchet spaces.
method Mountain Pass Theorem and sufficient conditions for diffeomorphism.
result Existence of global diffeomorphism between tame Fréchet spaces.
Every countable compact subset of sphere is tame.
problem Characterizing compact subsets of spheres.
method Proving homeomorphic complements imply homeomorphic subsets.
result Wild subspaces like Antoine contain Cantor sets.
Four-dimensional, oriented Lie algebras g which satisfy the tame-compatible question of Donaldson for all almost complex structures J on g are completely described. As a consequence, examples are given of (non-unimodular) four-dimensional Lie algebras with almost complex structures which are…
We give several versions of local and global inverse mapping theorem for tame non necessarily smooth, mappings. Here tame mapping means a mapping which is subanalytic or, more generally, definable in some o-minimal structure. Our sufficient conditions are formulated in terms of various properties (convexity, positivity…
Develops calculus for tamed Dirichlet spaces using measure theory.
problem Defines calculus for measure spaces with Dirichlet forms.
method Introduces first and second order calculus on tamed Dirichlet spaces.
result Defines various geometric objects on tamed Dirichlet spaces.
This paper shows how path spaces on two-level manifolds can be Hilbert manifold structures.
problem Addressing the structure of path spaces on two-level manifolds.
method Introducing the notion of tameness and constructing charts on path spaces of two-level manifolds.
result Path spaces on tame two-level manifolds have the structure of a Hilbert manifold.
We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of GL+(2,R) avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surf…
Uniformizing maps and period maps are topologically tame, leading to algebraicity of Hodge loci.
problem Understanding the algebraicity of Hodge loci in arithmetic quotients.
method Proving topological tameness of uniformizing maps and period maps, applying Peterzil-Starchenko's o-minimal GAGA theorem.
result The Hodge locus of (S,V) is a countable union of algebraic subvarieties of S. Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form ∂∂ˉ-closed, i.e. to strong Kähler with torsion (SKT) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed …
New algorithm tames non-linear growth in stochastic optimization.
problem Computational challenges in E-step of EM framework.
method Employing interacting particle systems and taming techniques to create tIPLA.
result Non-asymptotic convergence error estimates in Wasserstein-2 distance for tIPLA.
We study the existence of strong Kähler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures J on solvmanifolds G/Γ providing some negative results for some classes of solvmanifolds. In particular, we show that if either J is invariant under the action of a nilpotent complement o…
Compact space models automorphisms, proving curvature and linearizability.
problem Modeling automorphisms of affine space.
method Constructing a metric space X with a CW-complex structure and proving curvature properties.
result X is a CAT(0) space for n=3, K of characteristic zero, proving linearizability of automorphisms.
In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration.
We study Nakai-Moishezon type question and Donaldson's "tamed to compatible" question for almost complex structures on rational four manifolds. By extending Taubes' subvarieties--current--form technique to J−nef genus 0 classes, we give affirmative answers of these two questions for all tamed almost complex structu…
Combines techniques to remove tameness condition in Morse-Smale flows.
problem Tameness condition in Morse-Smale flows.
method Combines Shilnikov's ODE techniques with Latschev's ideas.
result Removes tameness hypothesis from homotopy formula.
We show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN≤κ≤0 are proper, (compact if N is compact). In addition, if N is Hadamard then M has finite topology. We also show that the fundamental tone is an obstruction fo…
Introduces tame ρ-quaternionic manifolds for new geometric constructions.
problem Characterizing and constructing quaternionic manifolds.
method Introduces tame ρ-quaternionic manifolds and constructs new connections.
result Provides new global characterizations and constructions of quaternionic manifolds.
We introduce a new technique for finding CAT(-1) surfaces in hyperbolic 3-manifolds. We use this to show that a complete hyperbolic 3-manifold with finitely generated fundamental group is geometrically and topologically tame.
A new tamed stochastic gradient Hamiltonian Monte Carlo algorithm for superlinearly growing stochastic gradients.
problem Sampling and stochastic optimization problems with superlinearly growing stochastic gradients.
method Tamed Stochastic Gradient Hamiltonian Monte Carlo (tSGHMC) algorithm.
result Established a non-asymptotic error bound in Wasserstein-2 distance with a convergence rate of 1/4. Product of shellable complexes yields shellable triangulations under tameness conditions.
problem Understanding shellability in products of simplicial complexes.
method Definition and proof of shellability properties for products of complexes under tameness conditions.
result Product of shellable complexes yields shellable triangulations under certain conditions.
The paper defines a stratification for Lie groupoids in a tame topology context.
problem Presenting a tame topology counterpart to canonical stratification of Lie groupoids.
method Using Shiota's isotopy lemma and approximation theorem, the paper defines a canonical Whitney stratification of definable Lie groupoids into invariant strata.
result A canonical Whitney stratification of the Lie groupoid into definable strata invariant under the groupoid action.
Extends Hess-Schrader-Uhlenbrock inequality for 1-forms in tamed Dirichlet spaces.
problem Establishing a new inequality for 1-forms in tamed Dirichlet spaces.
method Developed a vector calculus for tamed Dirichlet spaces and applied it to establish the inequality.
result Established the Hess-Schrader-Uhlenbrock inequality for 1-forms in L2-cotangent module. Proves strong Tits alternative for 3D automorphisms over zero char fields.
problem Tits alternative for 3D tame automorphisms over zero char fields.
method Proved strong Tits alternative using 3D tame automorphisms over zero char fields.
result Strong Tits alternative proven for 3D tame automorphisms over zero char fields.
New theorem on embedding Moebius bands in 3D space.
problem Proving the impossibility of placing uncountably many disjoint Moebius bands in 3D space.
method Generalization of Grushin and Palamodov's result to tame subsets in R^N and arbitrary topological embeddings in R^3.
result The impossibility of embedding uncountably many pairwise disjoint Moebius bands in 3D space, even for arbitrary topological embeddings.
Given a complete isometric immersion φ:Pm⟶Nn in an ambient Riemannian manifold Nn with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space Mwn, we determine a set of conditions on the extrinsic curvatur…
Study on projective orbifolds with ends and their deformation theory.
problem Understanding deformation theory of projective orbifolds with ends.
method Analysis of convex real projective structures on strongly tame n-orbifolds.
result Deformation theory for nicest projective orbifolds.
We expose some ideas from mathematical logics, i.e. the background of the theory of o-minimal structures, and demonstrate how they lead to the notion of a tame integral of motion and some extensions and clarifications of previous results on obstructions to integrability of geodesic flows.
The paper explores spaces of Kähler and symplectic forms on 4-manifolds.
problem Investigating the properties of Kähler and symplectic forms on 4-manifolds.
method Analyzing the uniqueness, connectedness, and openness of spaces of Kähler forms and introducing holomorphically tamed symplectic forms.
result Formulated a parallel question for holomorphically tamed symplectic forms and related it to Kähler-type symplectic forms.
New algebra invariant distinguishes Legendrian knots in convex surfaces.
problem Distinguishing Legendrian knots in convex surfaces using invariants.
method Defined a differential graded algebra (DGA) for Legendrian knots in thickened convex surfaces, generating it from Reeb chords and counting immersed polygons.
result The stable tame isomorphism type of the DGA is invariant under Legendrian isotopy and can distinguish knots not distinguishable by classical invariants.
Marden's Tameness Conjecture predicts that every hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. It was recently established by Agol and Calegari-Gabai. We will survey the history of work on this conjecture and discuss its many applications.
We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.
Hilbert initiated the standpoint in foundations of mathematics. From this standpoint, we allow only a finite number of repetitions of elementary operations when we construct objects and morphisms. When we start from a subset of a Euclidean space. Then we assume that any element of the line has only a finite number of c…