The paper establishes a correspondence between special Kähler manifolds and their deformations.
problem Mapping between affine and projective special Kähler manifolds.
method Formulation of a correspondence and application to r-maps in string theory.
result One-parameter deformations of projective special Kähler manifolds correspond to perturbative α'-corrections.
Two new transversality theorems for linearly perturbed mappings are proven.
problem Understanding the behavior of mappings under small perturbations.
method Proves two transversality theorems for generic linearly perturbed Cr mappings. result Establishes new theorems for the stability of mappings under small changes.
We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold M as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle N=TM carries a canonical structure of (affine) special Kähler manifold. This gives a…
We determine all complete projective special real surfaces. By the supergravity r-map, they give rise to complete projective special Kähler manifolds of dimension 6, which are distinguished by the image of their scalar curvature function. By the supergravity c-map, the latter manifolds define in turn complete quaternio…
Goldman symplectic form and complex structure compatible on SL(3,R) Hitchin component.
problem Compatibility of Goldman's symplectic form with complex structure on SL(3,R) Hitchin component. method Proof of compatibility between Goldman's symplectic form and Labourie-Loftin complex structure.
result Goldman symplectic form and complex structure determine a pseudo-Kähler structure on SL(3,R) Hitchin component. We consider four notions of maps between smooth C^r orbifolds O, P with O compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of C^r maps between …
Compactifies maximal component of surface group representations into a closed ball.
problem Compactifying maximal component of surface group representations.
method Using cores of trees to study dynamics of mapping class group action.
result Boundary points geometrically described as mixed structures.
Algorithm learns similarities to optimize bandit decisions in unknown metric space.
problem Optimizing decisions in unknown metric space with nonparametric reward functions.
method Data-driven similarities for adaptive partitioning of context-arm space.
result Regret bounds highlight algorithm's dependence on reward functions' local geometry.
The study classifies special manifolds leading to quaternionic Kähler manifolds.
problem Classifying special projective real manifolds and their quaternionic Kähler counterparts.
method Classification through projective special real manifolds and supergravity maps.
result Found a series of complete quaternionic Kähler manifolds of co-homogeneity one.
The article studies mapping properties of Radon transform and backprojection on a unit ball.
problem Polyhomogeneous mapping properties of Radon transform and backprojection operator on the unit ball.
method Constructs a double b-fibration to desingularize the point-hyperplane relation, provides formulas and sharper estimates.
result Sharper estimates on polyhomogeneous mapping properties of Radon transform and backprojection compared to classic estimates.
Smooth approximations lead to homotopy equivalences in manifold spaces.
problem Understanding homotopy equivalences in spaces of smooth maps.
method Proving weak homotopy equivalences for spaces of C^r maps and isotopies.
result Inclusions of specific spaces are weak homotopy equivalences.
Hessian geometry explains special geometries in N=2 theories.
problem Explaining special geometries in N=2 theories. method Formulating N=2 theories in terms of Hessian structures. result Hessian geometry relates special geometries of vector multiplets to hypermultiplet geometries.
The paper simplifies strongly convex problems to simplicial structures.
problem Strongly convex problems with Cr smoothness. method Generic linear perturbations and singularity theory.
result Strongly convex C1 problems are C0 simplicial. The tangent bundle TkM of order k, of a smooth Banach manifold M consists of all equivalent classes of curves that agree up to their accelerations of order k. In the previous work of the author he proved that TkM, 1≤k≤∞, admits a vector bundle structure on M if and only if M is endowed w…
Smooth approximations proved for triangulable sets.
problem Universal approximation of continuous maps to triangulable sets.
method New approximation techniques for weakly Cr triangulable sets. result Triangulable sets are Cr-approximation targets. The study describes special real manifolds and invariant admissible cubics in Vinberg cones.
problem Understanding special real manifolds and invariant admissible cubics in Vinberg cones.
method Simplified Vinberg theory using Nil-algebras to describe invariant functions and polynomials.
result Examples of continuous families of non-homogeneous special real manifolds.
The cone projection fR(z)=z/(1+∣z∣/R) maps lines to conic arcs with specific properties.
problem Mapping lines to conic arcs with specific properties.
method Using a reciprocal lens identity and radial homeomorphism.
result The Self-Directrix Theorem and Confocal--Codirectrix Theorem.