arXiv research

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

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

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

Study complex slices on real algebraic varieties and their properties.

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…

The study refines algebraic domains with specific boundary conditions.

Constructs real algebraic maps with specific geometric constraints.

In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the bool…

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…

We prove a triangulation theorem for semi-algebraic sets over a p-adically closed field, quite similar to its real counterpart. We derive from it several applications like the existence of flexible retractions and splitting for semi-algebraic sets.

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

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…

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

Constructs real algebraic functions with specified preimages.

Study real forms and GIT quotients in algebraic varieties.

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

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

The paper classifies orbit closures of symplectic Lie algebras.

This paper concerns the topology of isospectral real manifolds of certain Jacobi elements associated with real split semisimple Lie algebras. The manifolds are related to the compactified level sets of the generalized (nonperiodic) Toda lattice equations defined on the semisimple Lie algebras. We then give a cellular d…

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

We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recen…

Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.

Maximal Laplacian algebras applied to invariant theory solved inverse problems.

Let Y be a complex algebraic curve and let [Y]={X_1,...,X_n} be the set of all real algebraic curves X_i with complexification X_i(C)=Y, such that the real points X_i(R) divide X_i(C). We find all such families [Y]. According to Harnak theorem a number |X_i| of connected components of X_i(R) satifies by the inequality …

The paper classifies orbits of semisimple elements in real semisimple Lie algebras.

These lecture notes provide a self-contained introduction to the mathematical methods required in a Bachelor degree programme in Business, Economics, or Management. In particular, the topics covered comprise real-valued vector and matrix algebra, systems of linear algebraic equations, Leontief's stationary input-output…

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

Looking for an efficient algorithm for the computation of the homology groups of an algebraic set or even a semi-algebraic set is an important problem in the effective real algebraic geometry. Recently, Peter Burgisser, Felipe Cucker and Pierre Lairez wrote a paper [1], which made a step forward by giving an algorithm …

Real algebraic structures help classify overtwisted contact 3-spheres.

T-curves are piecewise linear curves which have been used with success since the beginning of the 1990's to construct new real algebraic curves with prescribed topology mainly on the real projective plane. In fact T-curves can be used on any real projective toric surface. We generalize here the construction of the latt…

AlgebraNets use alternative algebras for neural networks, improving performance on image and language tasks.

New method realizes planar graphs as Reeb graphs of algebraic functions.

New mixed singularities help classify real algebraic links.

In this article we give an explicit algorithm which will determine, in a discrete and computable way, whether a finite piecewise Euclidean complex is non-positively curved. In particular, given such a complex we show how to define a boolean combination of polynomial equations and inequalities in real variables, i.e. a …

Surveying probabilistic real algebraic geometry.

We investigate some geometric properties of the real algebraic variety $Δ$ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in $Δ$. We exhibit conne…

Let $R$ be an algebraic curvature tensor on a vector space of signature $(p,q)$ defining a spacelike Jordan Osserman Jacobi operator $\JJ_R$. We show that the eigenvalues of $\JJ_R$ are real and that $\JJ_R$ is diagonalizable if $p<q$.

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a non-degenerate invariant symmetric bilinear form. We show that any metric Lie algebra without simple ideals has the structure of a so called balanced quadratic extension of a…

The problem of determining the volume of a tubular neighbourhood has a long and rich history. Bounds on the volume of neighbourhoods of algebraic sets have turned out to play an important role in the probabilistic analysis of condition numbers in numerical analysis. We present a self-contained derivation of bounds on t…

Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projecti…

Functional for Spin(7) forms defined on compact manifolds.

Research on refined algebraic domains respecting differential geometry.

Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-o…

We study topological structures of the sets $(0,1/2)^3 \cap Ω$ and $(0,1/2)^3 \setminus Ω$, where~$Ω$ is one special algebraic surface defined by a symmetric polynomial in variables $a_1,a_2,a_3$ of degree~$12$. These problems arise in studying of general properties of degenerate singular points of dynamical systems ob…

Builds geometric structures for algebraic groups over real closed fields.

We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, …

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