"Homological Local Mirror Symmetry." Matt is studying category-theoretic aspects of homological mirror symmetry in the case of non-compact Calabi-Yau manifolds realized as the total space of the canonical bundle of compact Fano manifolds. This project relates closely to my work with Matt Kerr and my work with Shinobu Hosono.

"Symmetric K3 Surfaces, Toric Geometry, and Picard Lattices." Motivated by the Greene-Plesser, Batyrev-Borisov, and Dolgachev versions of mirror symmetry for K3 surfaces, Ursula is investigating symmetry-enhancement of the Picard lattices of K3 surfaces, especially for K3 surfaces realized as hypersurfaces in toric Fano threefolds. Her research is closely related to my work with Brian Greene and Simon Judes.

Jacob Lewis

"Discriminants of Toric Hypersurfaces and Modular Parametrization of K3 Moduli." Jacob is applying his work on computation of discriinant loci of anticanonical elliptic curves in toric Fano surfaces to the special elliptic fibrations of my work with Adrian Clingher on mular parametrizations of K3 moduli.

Andrew Novoseltsev

"Lattice Polytopes and String Duality." Andrey has implemented Kreuzer-Skarke's Package for Analyzing Lattice Polytopes (PALP in William Stein's SAGE coputer algebra system. With this tool he is now investigating questions involving semiample (but not ample) Calabi-Yau hypersuraces and complete intersections and conjectures relating the combinatorics of lattice polytopes to the geometry and topology of the associated Calabi-Yau manifolds.

"Numerical Kähler-Einstein Metrics and Mirror Symmetry." Josh is working with me and Chris Herzog on an extension of our earlier work on numerical metrics to refine the homological mirror symmetry conjecture. The approach blends a differential-geometric proposal of Leung-Yau-Zaslow and the algebraic geometry of quivers/brane tilings. Josh is a graduate student of Robin Graham.

"Matrix Models, Genus Two Curves, and K3 Surfaces." Can is working with me and his advisor, Amer Iqbal, on a reformulation of important matrix model/topological vertex computations in terms of the recently established Clingher-Doran equivalence between moduli of genus two curves and a class of lattice-polarized K3 surfaces. Can is a graduate student in the Physics department.

"Classification of Doubly-Even Binary Linear Error-Correcting Codes." A coarse notion of topological equivalence of Adinkra graphs relates the classification of supersymmetric representations to the classification of doubly-even codes. Robert is working on the design and implementation of an algorithm in Distributed SAGE with the goal of classifying all doubly-even codes of length 32 and less. It is expected that this will form an appendix to a forthcoming paper of the DFGHIL collaboration.

Noah Giansiracusa

"Cubic Surfaces and their Moduli," Spring 2006. Noah's senior thesis was an extension of a paper begun while taking my Calabi-Yau Manifolds course during Winter 2006. Noah received an Outstanding Graduating Senior in Mathematics Award (Comprehensive Major) in 2006. He started Brown University's PhD program in Mathematics in Autumn 2006.