arXiv research

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…

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 …

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of $\mathbf{GL}_+(2,\R)$ 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…

This paper shows how path spaces on two-level manifolds can be Hilbert manifold structures.

We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-…

Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.

The study examines when mapping class groups are quasi-isometric to graphs of curves.

Study of homeomorphisms on infinite type surfaces with a classification theorem.

New problems defined for maps onto graphs related to Reeb spaces.

In this paper we present some bounds of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of curves of maps, etc. Moreover, we also give some explicit bounds for semi-algebraic or semi-Pfaffian cases, which depend only on the combinatoric data representing the objects…

We give an overview on the tt*-geometry defined for isolated hypersurface singularities and tame functions via Brieskorn lattices. We discuss nilpotent orbits in this context, as well as classifying spaces of Brieskorn lattices and (limits of) period maps.

Paper characterizes tamed and weakened tamed four-manifolds using a new technique.

The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow $Φ: \mathbb{R}\times X\to X$ on pfaffian set $X$ is tame if the graph of $Φ$ is a pfaffian subset of $\mathbb{R}\times X\times X$. Any compact tame set admits plenty tame flows. We prove …

Deep learning models viewed through tame geometry for convergence guarantees.

Paper proposes a new method to stabilize noisy gradient algorithms.

In this note we present a description of wave front evolving from an algebraic hypersurface by means of a pull-back of the discriminantal loci of a tame polynomial via a polynomial mapping. As an application we give examples of wave fronts which define free/almost free divisors near the focal point.

Let $F=(F_1, F_2, ..., F_m): \mathbb{C}^n \to \mathbb{C}^m$ be a polynomial dominant mapping with $n>m$. In this paper we give the relations between the bifurcation set of $F$ and the set of values where $F$ is not M-tame as well as the set of generalized critical values of $F$. We also construct explicitly a proper su…

We study the space of linear difference equations with periodic coefficients and (anti)periodic solutions. We show that this space is isomorphic to the space of tame frieze patterns and closely related to the moduli space of configurations of points in the projective space. We define the notion of combinatorial Gale tr…

The study confirms Gromov's speculation and provides bounds for taming symplectic structures.

Maps on Sasakian manifolds limit to sub-Riemannian distance bounds.

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.

We prove that the group STame($k^3$) of special tame automorphisms of the affine 3-space is not simple, over any base field of characteristic zero. Our proof is based on the study of the geometry of a 2-dimensional simply-connected simplicial complex C on which the tame automorphism group acts naturally. We prove that …

A closed discrete subset $A\subset \mathbb{C}$ is called tame if $\mathbb{C}\setminus A$ is quasiconformally equivalent to $\mathbb{C}\setminus \mathbb{Z}$. By giving several criteria for $A$ to be tame, we shall show that $\mathbb{Z}+i\mathbb{Z}$ is not tame.

Positivity of intersections in 4-manifolds leads to taming symplectic structures.

Whitehead doubles have matching meridional rank and bridge number.

Every countable compact subset of sphere is tame.

Four-dimensional, oriented Lie algebras $\mathfrak{g}$ which satisfy the tame-compatible question of Donaldson for all almost complex structures $J$ on $\mathfrak{g}$ are completely described. As a consequence, examples are given of (non-unimodular) four-dimensional Lie algebras with almost complex structures which are…

Develops calculus for tamed Dirichlet spaces using measure theory.

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…

Origami can create complex knots, with minimum creases defining a new knot invariant.

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial \bar \partial$-closed, i.e. to strong Kähler with torsion (${\rm SKT}$) 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.

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…

Study large-scale geometry of infinite type surface mapping class groups.

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.

Classifies pure mapping class groups based on surface properties.

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.

We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_{N}\leq κ\leq 0$ 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…

Uncountably many fibrations found on genus 2 handlebody.

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.

Product of shellable complexes yields shellable triangulations under tameness conditions.

A new tamed stochastic gradient Hamiltonian Monte Carlo algorithm for superlinearly growing stochastic gradients.

The paper defines a stratification for Lie groupoids in a tame topology context.

Extends Hess-Schrader-Uhlenbrock inequality for 1-forms in tamed Dirichlet spaces.

This paper proves that on any tamed closed almost complex four-manifold $(M,J)$ whose dimension of $J$-anti-invariant cohomology is equal to the self-dual second Betti number minus one, there exists a new symplectic form compatible with the given almost complex structure $J$. In particular, if the self-dual second Bett…

Proves strong Tits alternative for 3D automorphisms over zero char fields.