John Voight : On Moduli of Nondegenerate Curves
- String Theory ( 244 Views )We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. Such nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry; however, despite their ubiquity, the intrinsic property of nondegeneracy has not seen much detailed study. We prove that every curve of genus $g \geq 4$ over an algebraically closed field is nondegenerate in the above sense. More generally, let $\mathcal{M}_g^{\textup{nd}}$ be the locus of nondegenerate curves inside the moduli space of curves of genus $g \geq 2$. Then we show that $\dim \mathcal{M}_g^{\textup{nd}} = \min(2g+1,3g-3)$, except for $g=7$ where $\dim \mathcal{M}_7^{\textup{nd}} = 16$; thus, a generic curve of genus $g$ is nondegenerate if and only if $g \geq 4$
Allan Adams : LISA in the Sky with Dark Matter
- String Theory ( 30 Views )Space-based laser interferometers such as LISA may prove spectacularly effective not only in listening to gravitational radiation but also in the direct detection of galactic halo dark matter streaming through the solar system. This talk will describe the basic strategy and touch on some entertaining open possibilities.
Mina Aganagic : D-Branes, Mirror Symmetry and Counting Holomorphic Discs
- String Theory ( 27 Views )A class of special Lagrangian subspaces of Calabi-Yau manifolds is considered and their mirrors, certain holomoprhic varieties of the mirror geometry, are identified. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical Abel-Jacobi map on the mirror. We discuss flat coordinates on the moduli spaces, and a certain quantum ambiguity which influences disc counting. Highly non-trivial new predictions for disc counting are obtained.