Related papers: Block Jacobi matrices, Barycentric limits and Mani…
Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a…
We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.
These are lecture notes on non-K\"ahler complex threefolds presented at the MATRIX program ``The geometry of moduli spaces in string theory''. We review some basics of Calabi-Yau geometry in Section 1, describe topological features of the…
The paper studies the global convergence of the block Jacobi me\-thod for symmetric matrices. Given a symmetric matrix $A$ of order $n$, the method generates a sequence of matrices by the rule $A^{(k+1)}=U_k^TA^{(k)}U_k$, $k\geq0$, where…
This paper is devoted to the study of some connections between coadjoint orbits in infinite dimensional Lie algebras, isospectral deformations and linearization of dynamical systems. We explain how results from deformation theory,…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the…
The study of solutions with fixed energy of certain classes of Lagrangian (or Hamiltonian) systems is reduced, via the classical Maupertuis--Jacobi variational principle, to the study of geodesics in Riemannian manifolds. We are interested…
The connection between Jacobi fields and odular structures of affine manifold is established. It is shown that the Jacobi fields generate the natural geoodular structure of affinely connected manifolds.
Birational Calabi-Yau threefolds in the same deformation family provide a `weak' counterexample to the global Torelli problem, as long as they are not isomorphic. In this paper, it is shown that deformations of certain desingularized…
We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.
Additive deformations of bialgebras in the sense of J. Wirth, i.e. deformations of the multiplication map fulfilling a certain compatibility condition w.r.t. the coalgebra structure, can be generalized to braided bialgebras. The theorems…
The standard formulation of Jacobi manifolds in terms of differential operators on line bundles, although effective at capturing most of the relevant geometric features, lacks a clear algebraic interpretation similar to how Poisson algebras…
This is the second part of a series of two papers dedicated to a systematic study of holomorphic Jacobi structures. In the first part, we introduced and study the concept of a holomorphic Jacobi manifold in a very natural way as well as…
We introduce new obstructions to rationality for geometrically rational threefolds arising from the geometry of curves and their cycle maps.
In this paper we consider a class of higher dimensional differential systems in $\mathbb R^n$ which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues. First we characterize the existence of…
In this paper, we investigate the geometries associated with 3-forms of various orbital types on a symplectic 6-manifold. We demonstrate that certain unstable 3-forms, which naturally emerge from specific degenerations of Calabi-Yau…
In this paper we give a brief introduction to criteria for metrisability of a manifold and to some aspects of non-metrisable manifolds. Bias towards work currently being done by the author and his colleagues at the University of Auckland…
We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of…
In this paper we study the conjugate locus in convex manifolds. Our main tool is Jacobi fields, which we use to define a special coordinate system on the unit sphere of the tangent space; this provides a natural coordinate system to study…