Eduardo Gonzalez

Seidel elements and mirror transformations

The goal of this article is to give a precise relation between the mirror symmetry transformation of Givental and the Seidel elements for a smooth projective toric variety X with −KX nef. We show that the Seidel elements entirely determine the mirror transformation and mirror coordinates.

Seidel elements and potential functions of holomorphic disc counting

Let M be a symplectic manifold equipped with a Hamiltonian circle action and let L be an invariant Lagrangian submanifold of M. We study the problem of counting holomorphic disc sections of the trivial M-bundle over a disc with boundary in L through degeneration. We obtain a conjectural relationship between the potential function of L and the Seidel element associated to the circle action. When applied to a Lagrangian torus fibre of a semi-positive toric manifold, this degeneration argument reproduces a conjecture (now a theorem) of Chan-Lau-Leung-Tseng relating certain correction terms appearing in the Seidel elements with the potential function.

Detecting intermediate protein conformations using algebraic topology

BackgroundUnderstanding protein structure and dynamics is essential for understanding their function. This is a challenging task due to the high complexity of the conformational landscapes of proteins and their rugged energy levels. In particular, it is important to detect highly populated regions which could correspond to intermediate structures or local minima.ResultsWe present a hierarchical clustering and algebraic topology based method that detects regions of interest in protein conformational space. The method is based on several techniques. We use coarse grained protein conformational search, efficient robust dimensionality reduction and topological analysis via persistent homology as the main tools. We use two dimensionality reduction methods as well, robust Principal Component Analysis (PCA) and Isomap, to generate a reduced representation of the data while preserving most of the variance in the data.ConclusionsOur hierarchical clustering method was able to produce compact, well separated clusters for all the tested examples.