Log geometry and exploded manifolds

A Geometry Preserving Kernel over Riemannian Manifolds

Abstract- Kernel trick and projection to tangent spaces are two choices for linearizing the data points lying on Riemannian manifolds. These approaches are used to provide the prerequisites for applying standard machine learning methods on Riemannian manifolds. Classical kernels implicitly project data to high dimensional feature space without considering the intrinsic geometry of data points. ...

Log geometry and multiplier ideals

I work in combinatorics, algebraic geometry, convex geometry and commutative algebra while staying informed on certain topics in category theory and ring theory. In particular, I focus on toric varieties and singularity theory. The study of toric varieties lies at the intersection of combinatorics, algebraic geometry, convex geometry and integer programming. There is a correspondence between ce...

Manifolds , Log Structures , and Mirror Symmetry

Affine Manifolds, Log Structures, and Mirror Symmetry

We outline work in progress suggesting an algebro-geometric version of the Strominger-Yau-Zaslow conjecture. We define the notion of a toric degeneration, a special case of a maximally unipotent degeneration of Calabi-Yau manifolds. We then show how in this case the dual intersection complex has a natural structure of an affine manifold with singularities. If the degeneration is polarized, we a...

Flag Manifolds and Toric Geometry

1.1. Flag Manifolds. One of the best understood examples of algebraic varieties is the flag manifold. One is first interested in flag varieties as they are well-defined examples for many basic concepts. Besides this, however, one can find that they deserve individual attention, for they naturally arise in many circumstances, e.g. in the theory of characteristic classes of vector bundles, mirror...