arXiv research

Surveying probabilistic real algebraic geometry.

Machine learning applied to algebraic geometry for physics problems.

High-order Klein geometries constructed using Lie algebras.

A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.

The paper examines smoothness in diffusion algebra.

Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.

Surveying recent work on Kähler metrics and algebraic variety stability.

Algebraic geometry replaces manifolds in differential geometry.

Paper constructs observables using multisymplectic geometry and algebraic methods.

We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of alpha-planes (algebraically special real alpha-geometry)…

The abstract discusses how Kähler-Einstein metrics relate to algebraic geometry.

We develop the idea of using an algebraic-geometry approach to classical differential geometry problems. Consider an orthogonal net constructed according to algebraic-geometric data we obtain a set of smooth orthogonal nets that are Ribaucour transformations of the initial orthogonal net.

Authors discuss complex and non-Archimedean geometry, proving a conjecture.

Survey on algebraic K- and L-theory conjecture.

The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.

Paper extends algebraic geometry results to hyperbolic link complements.

In this work, the Z$_3$-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algeb…

Foundations laid for formal manifolds in differential geometry.

We prove that the Riemannian geometry of almost Kähler manifolds can be expressed in terms of the Poisson algebra of smooth functions on the manifold. Subsequently, Kähler-Poisson algebras are introduced, and it is shown that a corresponding purely algebraic theory of geometry and curvature can be developed. As an illu…

New algebraic structures for Hermitian geometry cohomologies.

Reformulates divergence map for Turaev cobracket in non-commutative geometry.

Study connects derivations and holonomy symmetries in heterotic geometries.

This thesis details the results of four interrelated projects. The first of these presents a new proof of the theorem of Cooper, Danciger and Wienhard classifying the limits under conjugacy of the orthogonal groups in GL(n; R). The second provides a detailed investigation into Heisenberg geometry, which is the maximall…

Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.

New insights into algebraic geometry of a conjecture, leading to origami curves.

This Master Thesis is devoted to the study of $n$-plectic manifolds and the Strongly Homotopy Lie algebras, also called $L_{\infty}$-algebras, that can be associated to them. Since multisymplectic geometry and $L_{\infty}$-algebras are relevant in Theoretical Physics, and in particular in String Theory, we introduce th…

This is a survey of the author's paper arXiv:1001.0023 on "Algebraic Geometry over C-infinity rings". If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a "C-infinity ring". That is, for each smooth function f : R^n --> R there is an n-fold operation Φ_f : C^\infty(X)^n --> C^\i…

Paper generalizes toric concepts to nonrational settings.

We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator $x$ is invertible and furthermore working polynomials in $\ln x$ instead of polynomials in $x$. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual H…

Introduces noncommutative geometry for modeling quantum spacetime.

New algebraic geometry and statistical manifold connections proven.

Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.

Cone structures in quantum field theory linked to information geometry.

Surveying connections between algebraic geometry and surface topology.

This is a survey article, based on the author's lectures in the 2015 AMS Summer Research Institute in Algebraic Geometry, and to appear in the Proceedings.

We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a v…

Enhances conformal geometry in higher dimensions with infinite-dimensional algebra.

I survey methods from differential geometry, algebraic geometry and representation theory relevant for the permanent v. determinant problem from computer science, an algebraic analog of the P v. NP problem.

We show that compact complex manifolds of algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic type have infinite fundamental group. This generalizes the main Theorem in [DM] where the same result was proved for the special cases of holomorphic affine connections and holomorphic conformal structu…

Recently V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry. In this note we generalize this argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtai…

Constructs brane current algebras from QP-manifolds, generalizing string currents.

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

We propose a new definition of so called Hamiltonian forms in n-plectic geometry and show that they have a non-trivial Lie infinity-algebra structure.

Research on refined algebraic domains respecting differential geometry.

The paper explores deep learning through algebra and geometry, highlighting geometric structures and differential processes.

This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential g…

A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of $X$. We study how the …

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…