arXiv research

Variant of previous work on smooth algebraic functions with compact and non-compact preimages.

Real algebraic structures help classify overtwisted contact 3-spheres.

New mixed singularities help classify real algebraic links.

Surveying probabilistic real algebraic geometry.

Builds geometric structures for algebraic groups over real closed fields.

The paper studies spaces of non-compact real algebraic curves and their uniformisation.

We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…

The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves in real toric surfaces is a classical subject that has undergone considerable evolution. On the other hand, not much is …

Study of Poincaré-Reeb graphs for algebraic domains.

Computes the component group of arbitrary real algebraic groups.

Study complex slices on real algebraic varieties and their properties.

The paper classifies Lie algebras with special operators.

This thesis explores algebraic cycles and moduli spaces over real numbers.

New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.

Research explores real algebraic realization of round fold maps of codimension -1.

Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.

Constructs real algebraic maps with specific geometric constraints.

Study rationality of meromorphic functions between real algebraic sets in the plane.

The paper constructs real algebraic functions with specific singularities and preimages.

Researchers reconstruct algebraic maps onto curves based on prescribed Reeb graphs.

We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real…

We analyze symplectic forms on six dimensional real solvable and non-nilpotent Lie algebras. More precisely, we obtain all those algebras endowed with a symplectic form that decompose as the direct sum of two ideals or are indecomposable solvable algebras with a four dimensional nilradical.

Constructs real algebraic functions with both compact and non-compact preimages.

We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.

Constructs real algebraic functions with specified preimages.

This paper provides a stratification of semi-algebraic sets in the plane with finitely many geodesic segments.

We prove that every (compact) taut submanifold in Euclidean space is real algebraic, i.e., is a connected component of a real irreducible algebraic variety in the same ambient space. This answers affirmatively a question of Nicolaas Kuiper raised in the 1980s.

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic …

All candidates to the weakly-irreducible not irreducible holonomy algebras of Lorentzian manifolds are known. In the present paper metrics that realize all these candidates as holonomy algebras are given. This completes the classification of the Lorentzian holonomy algebras. Also new examples of metrics with the holono…

A brief survey of real algebraic structures on topological spaces is given. This article is written for the Gokova Gemetry/Topology Conference proceedings.

The study refines algebraic domains with specific boundary conditions.

Study real forms and GIT quotients in algebraic varieties.

In this note we propose a method to classify homogeneous nilpotent elements in a real $Z_m$-graded semisimple Lie algebra $g$. Using this we describe the set of orbits of homogeneous elements in a real $Z_2$-graded semisimple Lie algebra. A classification of 4-vectors (resp. 4-forms) on $R^8$ can be given using this me…

This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on symmetric real curves on the projective plane. We give a classification of real s…

This paper constructs real algebraic maps that are topologically special generic maps.

Possible irreducible holonomy algebras $\g\subset\sp(2m,\Real)$ of odd Riemannian supermanifolds and irreducible subalgebras $\g\subset\gl(n,\Real)$ with non-trivial first skew-symmetric prolongations are classified. An approach to the classification of some classes of the holonomy algebras of Riemannian supermanifolds…

Automorphisms of finite order and real forms of "smooth" affine Kac-Moody algebras are studied, i.e. of 2-dimensional extensions of the algebra of smooth loops in a simple Lie algebra. It is shown that they can be parametrized by certain invariants and that in particular the classification of involutions essentially fo…

For a non-singular real algebraic projective curve, topological restrictions on a closed motion of a simple real divisor in its linear equivalence class are found.

The abstract proves the non-existence of certain real algebraic surfaces.

We define the notion of a smooth pseudo-Riemannian algebraic variety $(X,g)$ over a field $k$ of characteristic $0$, which is an algebraic analogue of the notion of Riemannian manifold and we study, from a model-theoretic perspective, the algebraic differential equation describing the geodesics on $(X,g)$. When $k$ is …

Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie …

Automorphisms of Lie algebras and their root systems are fully lifted.

Real valued homomorphisms on the algebra of smooth functions on a differential space are described. The concept of generators of this algebra is emphasized in this description.

Classifies Lie bialgebras using Darboux families.

The paper classifies orbit closures of symplectic Lie algebras.

The coamoeba of any complex algebraic plane curve $V$ is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in $(\mathbb{C}^*)^2$ is bounded in terms of the degree of the curve. We show in this Note that up to multiplication by a constant in $(\…

We study natural additional structures on real algebraic surfaces with trivial first homology mod 2 of the complexification. If the set of real points realizes the zero of the second homology mod 2 of the complexification, then the set of real points is equipped with a pair of opposite orientations and a Spin structure…

Operators on the ring of algebraically constructible functions are used to compute local obstructions for a four-dimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield $2^{43}-43$ independent characteristic numbers mod 2, which generalize the Akbulut-K…