141 research outputs found
Computing toric degenerations of flag varieties
We compute toric degenerations arising from the tropicalization of the full
flag varieties and embedded in a
product of Grassmannians. For and we
compare toric degenerations arising from string polytopes and the FFLV polytope
with those obtained from the tropicalization of the flag varieties. We also
present a general procedure to find toric degenerations in the cases where the
initial ideal arising from a cone of the tropicalization of a variety is not
prime.Comment: 35 pages, 6 figure
A tour of bordered Floer theory
Heegaard Floer theory is a kind of topological quantum field theory,
assigning graded groups to closed, connected, oriented 3-manifolds and group
homomorphisms to smooth, oriented 4-dimensional cobordisms. Bordered Heegaard
Floer homology is an extension of Heegaard Floer homology to 3-manifolds with
boundary, with extended-TQFT-type gluing properties. In this survey, we explain
the formal structure and construction of bordered Floer homology and sketch how
it can be used to compute some aspects of Heegaard Floer theory.Comment: 13 pages, 7 figure
Degenerations of spherical subalgebras and spherical roots
We obtain several structure results for a class of spherical subgroups of
connected reductive complex algebraic groups that extends the class of strongly
solvable spherical subgroups. Based on these results, we construct certain
one-parameter degenerations of the Lie algebras corresponding to such
subgroups. As an application, we exhibit explicit algorithms for computing the
set of spherical roots of such a spherical subgroup.Comment: v2: 45 pages, revised extended version with new Section 6 containing
an optimization of the initial algorith
Notes on bordered Floer homology
This is a survey of bordered Heegaard Floer homology, an extension of the
Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is
placed on how bordered Heegaard Floer homology can be used for computations.Comment: 73 pages, 29 figures. Based on lectures at the Contact and Symplectic
Topology Summer School in Budapest, July 2012. v2: Fixed many small typo
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