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Andrei Gabrielov

Publications and source records attributed to Andrei Gabrielov.

At least 19 recordsLinked to original sources
arXiv2023

Outer Lipschitz Classification of Normal Pairs of H\"older Triangles

A normal pair of H\"older triangles is the union of two normally embedded H\"older triangles satisfying some natural conditions on the tangency orders of their boundary arcs. It is a special case of a surface germ, a germ at the origin of a two-dimensional closed semialgebraic (or, more general, definable in a polynomially bounded o-minimal structure) subset of $R^n$. Classification of normal pairs considered in this paper is a step towards outer Lipschitz classification of definable surface germs. In the paper \cite{BG} we introduced a combinatorial invariant of the outer Lipschitz equivalence class of normal pairs, called $\sigma\tau$-pizza, and conjectured that it is complete: two normal pairs of H\"older triangles with the same $\sigma\tau$-pizzas are outer Lipschitz equivalent. In this paper we prove that conjecture and define realizability conditions for the $\sigma\tau$-pizza invariant. Moreover, only one of the two pizzas in the $\sigma\tau$-pizza invariant, together with some admissible permutations related to $\sigma$ and $\tau$, is sufficient for the existence and uniqueness, up to outer Lipschitz equivalence, of a normal pair of H\"older triangles.

math.MG↗
arXiv2022

Lipschitz Geometry of Real Semialgebraic Surfaces

We present here basic results in Lipschitz Geometry of semialgebraic surface germs. Although bi-Lipschitz classification problem of surface germs with respect to the inner metric was solved long ago, classification with respect to the outer metric remains an open problem. We review recent results related to the outer and ambient bi-Lipschitz classification of surface germs. In particular, we explain why the outer Lipschitz classification is much harder than the inner classification, and why the ambient Lipschitz Geometry of surface germs is very different from their outer Lipschitz Geometry. In particular, we show that the ambient Lipschitz Geometry of surface germs includes all of the Knot Theory.

math.MG↗
arXiv2022

Classification of generic spherical quadrilaterals

Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections. Under this condition, it is shown that the space of quadrilaterals with prescribed angles consists of finitely many open curves. Degeneration at the endpoints of these curves is also determined.

math.CV↗
arXiv2022

Lipschitz geometry of pairs of normally embedded H\"older triangles

We consider a special case of the outer bi-Lipschitz classification of real semialgebraic (or, more general, definable in a polynomially bounded o-minimal structure) surface germs, obtained as a union of two normally embedded H\"older triangles. We define a combinatorial invariant of an equivalence class of such surface germs, called $\sigma\tau$-pizza, and conjecture that, in this special case, it is a complete combinatorial invariant of outer bi-Lipschitz equivalence.

math.MG↗
arXiv2021

Lipschitz geometry and combinatorics of abnormal surface germs

We study outer Lipschitz geometry of real semialgebraic or, more general, definable in a polynomially bounded o-minimal structure over the reals, surface germs. In particular, any definable Hölder triangle is either Lipschitz normally embedded or contains some "abnormal" arcs. We show that abnormal arcs constitute finitely many "abnormal zones" in the space of all arcs, and investigate geometric and combinatorial properties of abnormal surface germs. We establish a strong relation between geometry and combinatorics of abnormal Hölder triangles.

math.MG↗
arXiv2021

Moduli spaces for Lamé functions and Abelian integrals of the second kind

The space of Lamé functions of order m is isomorphic to the space of pairs (elliptic curve, Abelian differential) where the differential has a single zero of order 2m at the origin and m double poles with vanishing residues. We describe the topology of this space: it is a Riemann surface of finite type; we find the number of components and the genus and Euler characteristic of each component. As an application we find the degrees of Cohn's polynomials confirming a conjecture by Robert Maier. As another application we partially describe the degeneration locus of the space of spherical metrics on tori with one conic singularity where the conic angle is an odd multiple of 2$π$.

math.CV↗
arXiv2021

Degrees of real Wronski maps

We study the map which sends vectors of polynomials into their Wronski determinants. This defines a projection map of a Grassmann variety which we call a Wronski map. Our main result is computation of degrees of the real Wronski maps. Connections with real algebraic geometry and control theory are described.

math.AG↗
arXiv2020

Lipschitz geometry of surface germs in $\mathbb{R}^4$: metric knots

A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in $\mathbb{R}^4$ is a topological knot (or link) in $S^3$. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in $\mathbb{R}^4$ and the knot theory. Namely, for any knot $K$, we construct a surface $X_K$ in $\mathbb{R}^4$ such that: the link at the origin of $X_{K}$ is a trivial knot; the germs $X_K$ are outer bi-Lipschitz equivalent for all $K$; two germs $X_{K}$ and $X_{K'}$ are ambient bi-Lipschitz equivalent only if the knots $K$ and $K'$ are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in $\mathbb{R}^4$, even when they are topologically trivial and outer bi-Lipschitz equivalent.

math.AG↗
arXiv2020

The space of Schwarz-Klein triangles

We describe the space of spherical triangles (in the sense of Schwarz and Klein). It is a smooth connected orientable 3-manifold, homotopy equivalent to the 1-skeleton of the cubic partition of the closed first octant in $\mathbf{R}^3$. The angles and sides are real analytic functions on this manifold which embed it to $\mathbf{R}^6$.

math.GT↗
arXiv2020

Surface singularities in $R^4$: first steps towards Lipschitz knot theory

A link of an isolated singularity of a two-dimensional semialgebraic surface in $R^4$ is a knot (or a link) in $S^3$. Thus the ambient Lipschitz classification of surface singularities in $R^4$ can be interpreted as a bi-Lipschitz refinement of the topological classification of knots (or links) in $S^3$. We show that, given a knot $K$ in $S^3$, there are infinitely many distinct ambient Lipschitz equivalence classes of outer metric Lipschitz equivalent singularities in $R^4$ with the links topologically equivalent to $K$.

math.AG↗
arXiv2017

PT-symmetric eigenvalues for homogeneous potentials

We consider one-dimensional Schrödinger equations with homogeneous potential, under appropriate PT-symmetric boundary conditions. We prove the phenomenon which was discovered by Bender and Boettcher by numerical computation: as the degree of the potential changes, the real spectrum suddenly becomes non-real in the sense that all but finitely many eigenvalues become non-real. We find the limit arguments of these non-real eigenvalues as they tend to infinity.

math-ph↗
arXiv2017

Ambient Lipschitz equivalence of real surface singularities

We present a series of examples of pairs of singular semialgebraic surfaces (real semialgebraic sets of dimension two) in ${\mathbb R}^3$ and ${\mathbb R}^4$ which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent, but not ambient Lipschitz equivalent. For each singular semialgebraic surface $S\subset {\mathbb R}^4$, we construct infinitely many semialgebraic surfaces which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent to $S$, but pairwise ambient Lipschitz non-equivalent.

math.AG↗
arXiv2016

Circular pentagons and real solutions of Painleve VI equations

We study real solutions of a class of Painleve VI equations. To each such solution we associate a geometric object, a one-parametric family of circular pentagons. We describe an algorithm which permits to compute the numbers of zeros, poles, 1-points and fixed points of the solution on the interval x>1 and their mutual position. The monodromy of the associated linear equation and parameters of the Painleve VI equation are easily recovered from the family of pentagons.

math.CV↗
arXiv2016

Topological lower bounds for arithmetic networks

We prove a complexity lower bound on deciding membership in a semialgebraic set for arithmetic networks in terms of the sum of Betti numbers with respect to "ordinary" (singular) homology. This result complements a similar lower bound by Montana, Morais and Pardo for locally close semialgebraic sets in terms of the sum of Borel-Moore Betti numbers. We also prove a lower bound in terms of the sum of Betti numbers of the projection of a semialgebraic set to a coordinate subspace.

cs.CC↗
arXiv2016

Spherical rectangles

We study spherical quadrilaterals whose angles are odd multiples of pi/2, and the equivalent accessory parameter problem for the Heun equation. We obtain a classification of these quadrilaterals up to isometry. For given angles, there are finitely many one-dimensional continuous families which we enumerate. In each family the conformal modulus is either bounded from above or bounded from below, but not both, and the numbers of families of these two types are equal. The results can be translated to classification of Heun's equations with real parameters, whose exponent differences are odd multiples of 1/2, with unitary monodromy.

math.CV↗
arXiv2015

On topological lower bounds for algebraic computation trees

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of the set, divided by m+1. This result complements the well known lower bound by Yao for locally closed semialgebraic sets in terms of the total Borel-Moore Betti number. We also prove that the height is bounded from below by the logarithm of m-th Betti number of a projection of the set onto a coordinate subspace, divided by (m+1)^2. We illustrate these general results by examples of lower complexity bounds for some specific computational problems.

cs.CC↗