21 research outputs found
Involutions on standard Young tableaux and divisors on metric graphs
We elaborate upon a bijection discovered by Cools, Draisma, Payne, and Robeva
between the set of rectangular standard Young tableaux and the set of
equivalence classes of chip configurations on certain metric graphs under the
relation of linear equivalence. We present an explicit formula for computing
the -reduced divisors (representatives of the equivalence classes)
associated to given tableaux, and use this formula to prove (i) evacuation of
tableaux corresponds (under the bijection) to reflecting the metric graph, and
(ii) conjugation of the tableaux corresponds to taking the Riemann-Roch dual of
the divisor.Comment: 21 pages, 8 figure
A Tropical Approach to the Brill-Noether Theory Over Hurwitz Spaces
The geometry of a curve can be analyzed in many ways. One way of doing this is to study the set of all divisors on a curve of prescribed rank and degree, known as a Brill-Noether variety. A sequence of results, starting in the 1980s, answered several fundamental questions about these varieties for general curves. However, many of these questions are still unanswered if we restrict to special families of curves. This dissertation has three main goals. First, we examine Brill-Noether varieties for these special families and provide combinatorial descriptions of their irreducible components. Second, we provide a natural generalization of Brill-Noether varieties, known as Splitting-Type varieties, that parameterize this decomposition. Lastly, we provide purely combinatorial descriptions of these Splitting-Type varieties and explore the geometric consequences of these descriptions. These results are based upon and extend tools and techniques from tropical geometry
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Tropical Aspects in Geometry, Topology and Physics
The workshop Tropical Aspects in Geometry, Topology and Physics was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject. The development of tropical geometry is based on deep links between problems in real and complex enumerative geometry, symplectic geometry, quantum fields theory, mirror symmetry, dynamical systems and other research areas. On the other hand, new interesting phenomena discovered in the framework of tropical geometry (like refined tropical enumerative invariants) pose the problem of a conceptual understanding of these phenomena in the “classical” geometry and mathematical physics
Toric Geometry and String Theory
In this thesis we probe various interactions between toric geometry and
string theory. First, the notion of a top was introduced by Candelas and Font
as a useful tool to investigate string dualities. These objects torically
encode the local geometry of a degeneration of an elliptic fibration. We
classify all tops and give a prescription for assigning an affine, possibly
twisted Kac-Moody algebra to any such top. Tops related to twisted Kac-Moody
algebras can be used to construct string compactifications with reduced rank of
the gauge group. Secondly, we compute all loop closed and open topological
string amplitudes on orientifolds of toric Calabi-Yau threefolds, by using
geometric transitions involving SO/Sp Chern-Simons theory, localization on the
moduli space of holomorphic maps with involution, and the topological vertex.
In particular, we count Klein bottles and projective planes with any number of
handles in some Calabi-Yau orientifolds. We determine the BPS structure of the
amplitudes, and illustrate our general results in various examples with and
without D-branes. We also present an application of our results to the BPS
structure of the coloured Kauffman polynomial of knots.
This thesis is based on hep-th/0303218 (with H. Skarke), hep-th/0405083 and
hep-th/0411227 (with B. Florea and M. Marino).Comment: Oxford University DPhil Thesis (Advisor: Philip Candelas), accepted
October 2005, 152 pp., 43 figure