arXiv research

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.

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)…

Integrable dynamics explained via geometric maps and cluster algebras.

The semidirect product of a Lie algebra and a 2-term representation up to homotopy is a Lie 2-algebra. Such Lie 2-algebras include many examples arising from the Courant algebroid appearing in generalized complex geometry. In this paper, we integrate such a Lie 2-algebra to a strict Lie 2-group in the finite dimensiona…

Study of geometric structures on manifolds, focusing on integrability conditions.

The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space $\mathrm{Curv}^{\mathrm{U}(n)*}$ of dual unitarily invariant curvature measures. Building on the recent results from integral geometry in complex space forms, we describe this algebra structure explicitly as a…

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…

We give an intrinsic definition of the special geometry which arises in global N=2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special Kahler manifold is so related to an integrable system. The cotangent bundle of a special …

We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension as does the connection with the "curved flats" of Ferus and Pedit. Moreover, we …

Geometric approach links hydrodynamic integrability to compatible nets.

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or $ω\mathscr{H}$ manifolds), as a natural setting where the notion of integrability can be formulated. We prove that the existe…

Study differential and integral calculus on noncommutative C*-algebras.

Based on the classical Plücker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in $CP^3$. Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the e…

Proves a special case of the Gaussian kinematic formula using large sphere limits.

Higher Gauge Flow Models integrate higher geometry and symmetries into Generative Flow Models.

This is a revised version of the notes from the week-long course I gave at the Centre de Recerca Matematica, Barcelona, in September of 2010. The aim is to give a working overview of recent methods and results in "Blaschkean integral geometry" (i.e. the subject revolving around the kinematic formulas of Blaschke) in th…

We study the geometry of universal embedding spaces for compact almost complex manifolds of a given dimension. These spaces are complex algebraic analogues of twistor spaces that were introduced by J-P. Demailly and H. Gaussier. Their original goal was the study of a conjecture made by F. Bogomolov, asserting the "tran…

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

This dissertation explores T-duality between hyperkähler structures and branes on algebraic integrable systems.

Integrality of FJRW invariants for Lie algebras A_l, D_l, and E_6.

Smoothness of Sklyanin algebras examined in 3D and 4D cases.

Study reveals new geometric structures for magnetic field Hamiltonian systems.

We study the geometry and topology of (filtered) algebra-bundles ${\bfΨ}^{\mathbb Z}$ over a smooth manifold $X$ with typical fibre $Ψ^{\mathbb Z}(Z; V)$, the algebra of classical pseudodifferential operators of integral order on the compact manifold $Z$ acting on smooth sections of a vector bundle $V$. First a theorem…

The paper proves a Serre-Swan Theorem for coisotropic algebras.

Paper proves optimal decomposition for matrix fields, reducing convex integration steps.

Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.

New integrable matrix PDEs derived from Frölicher-Nijenhuis brackets.

New integrable systems constructed for non-diagonal Killing tensors.

On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear systems is obtained, and the integration scheme for such equations is proposed.

New framework for logarithmically divergent integrals on manifolds with corners.

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 …

Study examines corrections to heterotic geometry on SU(3) manifolds.

Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfaces, and other classical solit…

In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Plücker coordinates. Particular emphasis is put on the restriction to Lie circle g…

We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit …

We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential Topology/Geometry, including orbibundles, differential forms, integration and (equivariant) …

A Hadwiger-type theorem for the exceptional Lie groups $G_2$ and $Spin(7)$ is proved. The algebras of $G_2$ or $Spin(7)$ invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formula…

We study integrable non-degenerate Monge-Ampere equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining the equations. This is used to deform these heavenly type equations into new integrable PDE of the second order with large symmetry pseu…

Proposes a new gauge theory for fuzzy geometries using finite-dimensional algebras.

In this paper we prove that a Finsler metrics has constant flag curvature if and only if the curvature of the induced nonlinear connection satisfies an algebraic identity with respect to some arbitrary second rank tensors. Such algebraic identity appears as an obstruction to the formal integrability of some operators i…

Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. I…

We give an exposition of graded and microformal geometry, and the language of $Q$-manifolds. $Q$-manifolds are supermanifolds endowed with an odd vector field of square zero. They can be seen as a non-linear analogue of Lie algebras (in parallel with even and odd Poisson manifolds), a basis of "non-linear homological a…

We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…

New proof for quaternionic structures on specific manifolds via automorphisms.

This paper is based on the author's talk at 1997 Taniguchi Symposium ``Integrable Systems and Algebraic Geometry''. We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumptions, these two objects are …

The paper integrates quasi-Poisson manifolds into multiplicative D-valued moment maps.

The paper integrates Rota-Baxter Lie algebras into Lie group structures and geometries.

The hidden M-algebra is integrated into a super-Lie group, allowing for compactification of extra dimensions.