Variant of previous work on smooth algebraic functions with compact and non-compact preimages.
problem Constructing smooth algebraic functions with specific preimage properties.
method Explicit construction of smooth real algebraic functions with controlled preimage compactness.
result New results in singularity theory and real algebraic geometry.
Real algebraic structures help classify overtwisted contact 3-spheres.
problem Classifying overtwisted contact structures on 3-spheres.
method Using real algebraic functions and open book decompositions.
result Most overtwisted contact structures are real algebraic.
Study real algebraic curves on real del Pezzo surfaces using degeneration methods.
problem Classify topological types of real algebraic curves on real del Pezzo surfaces.
method Degeneration methods and real enumerative geometry.
result Obstructions and constructions of real algebraic curves with prescribed topology.
New mixed singularities help classify real algebraic links.
problem Classify real algebraic links in the 3-sphere.
method Construct new mixed singularities.
result New mixed singularities may help solve the Benedetti-Shiota conjecture.
Surveying probabilistic real algebraic geometry.
problem Classical problems in real algebraic geometry.
method Probabilistic perspective on classical topics.
result Modern approach to Hilbert's Sixteenth Problem.
New invariant distinguishes real algebraic surfaces.
problem Distinguishing real algebraic surfaces up to birational diffeomorphism.
method Introducing real (logarithmic)-Kodaira dimension.
result Constructs infinite families of non-birationally diffeomorphic surfaces.
Builds geometric structures for algebraic groups over real closed fields.
problem Characterizing and decomposing algebraic groups over specific valued fields.
method Real algebraic geometry to construct and analyze affine buildings.
result Computed stabilizers and obtained group decompositions.
The paper studies spaces of non-compact real algebraic curves and their uniformisation.
problem Understanding the spaces of non-compact real algebraic curves and their uniformisation.
method Construction of spaces of non-compact real algebraic curves and description of their connected components using Fuchsian groups.
result Any connected component of the spaces of non-compact real algebraic curves is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group.
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…
Study of Poincaré-Reeb graphs for algebraic domains.
problem Characterizing geometric shapes of algebraic domains.
method Collapsing vertical segments to form Poincaré-Reeb graphs and analyzing their properties.
result Any transversal graph with specific properties can be realized as a Poincaré-Reeb graph.
Computes the component group of arbitrary real algebraic groups.
problem Computing the component group of arbitrary real algebraic groups.
method Structure results on algebraic groups and Galois cohomology methods.
result The group of connected components π0G(R) is an elementary Abelian 2-group. Classifies real algebraic curves up to six degrees and genus one.
problem Classifying real algebraic curves.
method Topological and rigid isotopy classification for curves of degree ≤6 and genus ≤1.
result Classification of real algebraic curves up to six degrees and genus one.
Study complex slices on real algebraic varieties and their properties.
problem Understanding the properties and bounds of complex slices on real algebraic varieties.
method Analyzing intersections and cohomology classes of complex subvarieties with real subvarieties.
result Proved an upper bound for the linking number of certain slices.
The paper classifies Lie algebras with special operators.
problem Classifying 3D Lie algebras with regular semisimple algebraic Nijenhuis operators.
method Described all Nijenhuis eigenbases for each 3D Lie algebra.
result Different answers in real and complex cases, some Lie algebras admit operators, others do not.
This thesis explores algebraic cycles and moduli spaces over real numbers.
problem Understanding the cycle class map and its image in real algebraic geometry.
method Constructing integral Fourier transforms on Chow rings of abelian varieties over any field.
result Proof of integral Hodge conjecture for real abelian threefolds and moduli space properties.
New geometric structures on surfaces generalize complex and real Lie algebra properties.
problem Generalizing geometric structures associated with Lie algebras.
method Define and analyze generalizations of punctual Hilbert schemes for complex and real Lie algebras.
result Construct geometric structures homeomorphic to Hitchin components.
New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.
problem Constructing real algebraic maps with specific properties and compositions.
method Explicit construction of real algebraic hypersurfaces and maps with prescribed images and compositions.
result Explicit families of functions represented as compositions of constructed maps with canonical projections.
Research explores real algebraic realization of round fold maps of codimension -1.
problem Real algebraic realization of round fold maps of codimension -1.
method Generalizes canonical projections of unit spheres to round fold maps and discusses their real algebraic realization.
result Developed new studies in real algebraic geometry focusing on round fold maps of codimension -1.
Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.
problem Characterizing real left symmetric algebras with positive definite Koszul form.
method Analyzes the properties of left multiplication operators and symmetric bilinear forms.
result Provides a complete characterization of real left symmetric algebras with positive definite Koszul form.
Constructs real algebraic maps with specific geometric constraints.
problem Construct smooth functions with prescribed Reeb graphs.
method Explicitly constructs real algebraic maps whose images are domains surrounded by products of hyperbolas and affine spaces.
result New examples of real algebraic maps with specified geometric constraints.
Study rationality of meromorphic functions between real algebraic sets in the plane.
problem Understanding rationality of meromorphic functions mapping real algebraic sets to complex algebraic sets.
method Using Schwarz reflection functions and theory of meromorphic functions.
result For certain real algebraic sets, meromorphic functions are rational.
The paper constructs real algebraic functions with specific singularities and preimages.
problem Constructing real algebraic functions with prescribed singular points and preimages.
method Generalizing moment-like maps and Morse-Bott functions to real algebraic settings.
result Explicit construction of real algebraic functions with exactly one singular value and prescribed preimages.
Researchers reconstruct algebraic maps onto curves based on prescribed Reeb graphs.
problem Reconstructing smooth real algebraic maps onto curves with specific Reeb graphs.
method Developed a method to reconstruct functions from general finite graphs, focusing on curves.
result Reconstructed functions from prescribed Reeb graphs, providing a new approach in real algebraic geometry.
Classifies real algebraic curves on a quadric ellipsoid of specific degree.
problem Classifying real algebraic curves of bidegree (5,5) on the quadric ellipsoid.
method Reduction to curves in the second Hirzebruch surface, combining classical construction methods on toric surfaces.
result Previously known restrictions form a complete system for this bidegree.
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.
problem Finding real algebraic functions with specific preimage properties.
method Explicit construction of real algebraic functions.
result Demonstrates real algebraic functions on non-compact manifolds with non-compact preimages.
Triangulates semi-algebraic sets over p-adic fields, proving similar results to reals.
problem Triangulating semi-algebraic sets over p-adic fields.
method Derives from a triangulation theorem for semi-algebraic sets over a p-adically closed field.
result Existence of flexible retractions and splitting for semi-algebraic sets.
Constructs real algebraic functions with specified preimages.
problem Reconstructing smooth functions with prescribed preimages.
method Using real algebraic functions and techniques from singularity theory and differential topology.
result Constructs examples of real algebraic functions with specified preimages.
This paper provides a stratification of semi-algebraic sets in the plane with finitely many geodesic segments.
problem How to stratify semi-algebraic sets in the plane with finitely many geodesic segments.
method Develops a semi-algebraic stratification of a real semi-algebraic set in the plane with open cells having the finiteness property.
result Provides insights for high-dimensional stratifications of semi-algebraic sets in connection with geodesics.
We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.
Characterizes real holomorphic chains on complex manifolds.
problem Representing homology classes by algebraic cycles.
method Characterization of real holomorphic chains; application to homology classes.
result Real holomorphic chains are characterized by local properties.
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.
Study on real algebraic curves' moduli spaces using a new complex.
problem Understanding the topology of moduli spaces of real algebraic curves.
method Defined a new complex, the ABC-complex, to encode intersection patterns. result Showed that mapping class groups are virtual duality groups and deduced orbifold homotopy group results.
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…
Abstract: Study motion of divisors on curves with topological constraints.
problem Topological restrictions on divisors' motion.
method Analysis of closed motion of simple real divisors on non-singular real algebraic projective curves.
result Found topological restrictions on divisors' motion.
The study refines algebraic domains with specific boundary conditions.
problem Understanding shapes of regions bounded by real algebraic curves.
method Inductive definition of regions, respecting characteristic finite sets.
result Generalized Poincar'e-Reeb Graphs for new types of regions.
A brief survey of real algebraic structures on topological spaces is given. This article is written for the Gokova Gemetry/Topology Conference proceedings.
Study real forms and GIT quotients in algebraic varieties.
problem Linking real points of complex GIT quotients to real GIT quotients.
method Explore actions of real forms on complex algebraic varieties and prove lifting properties.
result Some real points of complex GIT quotients can be lifted to real GIT quotients under certain conditions.
New geometric transitions studied via real algebra degenerations.
problem Understanding geometric transitions not arising from limits of ambient geometries.
method New degenerations of complex hyperbolic space and construction of new geometries over real algebras.
result Generalization of geometric transitions to new constructions over real algebras.
In this note we propose a method to classify homogeneous nilpotent elements in a real Zm-graded semisimple Lie algebra g. Using this we describe the set of orbits of homogeneous elements in a real Z2-graded semisimple Lie algebra. A classification of 4-vectors (resp. 4-forms) on R8 can be given using this me…
This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
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…
Study geodesics on algebraic varieties over differential fields.
problem Characterize geodesics on pseudo-Riemannian algebraic varieties.
method Model-theoretic approach to algebraic differential equations.
result Algebraic differential equation describing geodesics is absolutely irreducible.
Real rational knots constructed by algebraic methods.
problem Constructing real rational knots of low degree.
method Reduction to algebraic problem involving pure braid group.
result Existence of real rational knots in specific degrees.
Study on invariant almost complex structures on real flag manifolds.
problem Existence of invariant almost complex structures on real flag manifolds.
method Analysis of real flag manifolds associated to split real forms of complex simple Lie algebras.
result Some real flag manifolds do not admit invariant almost complex structures.
The abstract proves the non-existence of certain real algebraic surfaces.
problem The existence of real polynomial functions with specific properties.
method Algebraic solution to a problem proposed by D. A. Panov.
result There does not exist a real polynomial function with the specified properties.
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…