arXiv research

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

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

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

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

Constructs real algebraic maps with specific geometric constraints.

Constructs real algebraic functions with specified preimages.

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

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

Proof confirms conjecture for certain braids and their closures.

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

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 $(\…

Geometric analysis on real analytic manifolds using seminorms.

Investigates properties of moment maps and stratifications on Lie groups.

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

Study algebraic K-theory for specific groups of non-orientable surfaces.

In this paper, we explore holomorphic Segre preserving maps. First, we investigate holomorphic Segre preserving maps sending the complexification $\mathcal{M}$ of a generic real analytic submanifold $M \subseteq \C^N$ of finite type at some point $p$ into the complexification $\mathcal{M}'$ of a generic real analytic s…

Let X be a real algebraic subset of R^n and M a smooth, closed manifold. We show that all continuous maps from M to X are homotopic (in X) to C^\infty maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations, and we establish lower b…

Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this …

We construct universal geometric spaces over the real spectrum compactification $Ξ^{\mathrm{RSp}}$ of the character variety $Ξ$ of a finitely generated group $Γ$ in $\mathrm{SL}_n$, providing geometric interpretations of boundary points. For an algebraic set $Y(\mathbb{R})$ on which $\mathrm{SL}_n(\mathbb{R})$ acts by …

It is proven that a local Lie algebra in the sense of A. A. Kirillov determines the base manifold up to a diffeomorphism provided the anchor map is nowhere-vanishing. In particular, the Lie algebras of nowhere-vanishing Poisson or Jacobi brackets determine manifolds. This result has been proven for different types of d…

We consider CR submersive mappings between generic submanifolds in complex space. We show that, under suitable conditions on the manifolds, there is an integer k such that any jet of the CR mapping at a given point is a rational function of its k-jet at that point. As a consequence, it is shown that the stability group…

The paper connects polygon spaces with quotient spaces using spin actions and normed division algebras.

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some universal constant depending on $n$. He obtained similar estimates for maps with va…

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these results by defining a new complex associated to a not necessarily orientable surfa…

Let $G$, $B$, and $H$ denote a complex semi-simple algebraic group, a Borel subgroup of $G$, and a maximal complex torus in $B$, respectively. Choose a compact real form $K$ of $G$ such that $T=K\cap H$ is a maximal torus in $T$. Then there are two models for the flag space of $G$: the complex quotient $X=G/B$ and the …

It is proved that a germ of a real analytic CR map from a smooth real-analytic minimal CR manifold M to an essentially finite real-algebraic generic submanifold M' of P^N of the same CR-dimension extends as a holomorphic correspondence along M. Applications are given for pseudoconcave submanifolds of P^N.

A 2-step nilpotent Lie algebra n is called nonsingular if ad(X): n --> [n,n] is onto for any X not in [n,n]. We explore nonsingular algebras in several directions, including the classification problem (isomorphism invariants), the existence of canonical inner products (nilsolitons) and their automorphism groups (maxima…

Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gauß map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.

We construct a functor which maps conjugate pseudo-Anosov automorphisms of a surface to the so-called stably isomorphic stationary AF-algebras; the functor gives new topological invariants of three dimensional manifolds coming from the known invariants of the AF-algebras. The main invariant is a triple (L, [I], K), whe…

We study the Gauss map and the dual variety of a real-analytic immersion of a connected compact real-analytic manifold into a sphere or into a hyperbolic space. The dual variety is defined to be the set of all normal directions of the immersion. First, we show that the image of the Gauss map characterizes the manifold.…

Criterion for polystability in Lie group actions on manifolds.

In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic m…

We prove the existence of a map of spectra $τ_A \colon kA \to lA$ between connective topological K-theory and connective algebraic L-theory of a complex $C^*$-algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural e…

The paper develops stability criteria for real reductive Lie groups acting on manifolds.

We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler-Atiyah bundle constructed by quantizing …

We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements.

Extended current algebra on S^3 with new bilinear form and 2-cocycle.

Study of Pascal algebra matrices and their jet bundle map for vector bundles.

The paper extends Chern-Weil-Lecomte map to $L_{\infty}$-algebras.

Analogous exponential map defined for Hopf algebras.

Gradient maps of real reductive group actions on manifolds studied.

The paper describes orbits of parabolic subgroups in complexified actions.

The study calculates the Smith-Thom deficiency of Hilbert squares and provides conditions for maximality.

Smooth maps bound Betti numbers of zero sets.

Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.

Compute central extension of mapping class group from stated skein algebra

We give a partial characterization of bordered Floer homology in terms of sutured Floer homology. The bordered algebra and modules are direct sums of certain sutured Floer complexes. The algebra multiplication and algebra action correspond to a new gluing map on SFH. It is defined algebraically, and is a special case o…

Unified and generalized mating frameworks for Kleinian groups and rational maps.