On Tue 15th Nov we host a joint event between the 3C in G program at
Warwick and the East Midlands Seminar in Geometry. All talks take place
in the Mathematics Institute.
http://magma.maths.usyd.edu.au/~kasprzyk/seminars/emsg.html
10:00 Coffee in the Mathematics Institute Common Room
12:00 B3.01 Joe Karmazyn (Sheffield) Simultaneous resolutions and noncommutative algebras
1pm Lunch in the Mathematics Institute Common Room
(N.B. Lunch is primarily intended for seminar participants)
2pm MS.04 Elisa Postinghel (Loughborough) Tropical
compactifications, Mori Dream Spaces and Minkowski bases
3pm Tea in the Mathematics Institute Common Room
4pm B3.02 Robert Marsh (Leeds) Twists of Plücker coordinates as dimer partition functions
5:15pm B3.02 Thomas Prince (Imperial) A symplectic approach to polytope mutation
6:30pm Dinner
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ABSTRACT
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Joe Karmazyn (University of Sheffield) – Simultaneous resolutions and
noncommutative algebras
Minimal resolutions of surface quotient singularities can be studied and
understood via noncommutative algebra in a variety of ways packaged as
the McKay correspondence. Assorted higher dimensional examples, such as
flopping contractions of 3-folds, can be realised from simultaneous
resolutions associated to surface quotient singularities. I will discuss
how these simultaneous resolutions can also be understood via
noncommutative algebras, and how certain examples can be easily
calculated.
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Elisa Postinghel (Loughborough University) – Tropical compactifications,
Mori Dream Spaces and Minkowski bases
Joint work in progress with Stefano Urbinati
Given a Mori Dream Space X, we construct via tropicalisation a model
dominating all the small \(\mathbb{Q}\)-factorial modifications of X. Via this
construction we recover a Minkowski basis for the Newton-Okounkov bodies
of Cartier divisors on X and hence generators of the movable cone of X.
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Robert Marsh (University of Leeds) – Twists of Plücker coordinates as dimer partition functions
Joint work with Jeanne Scott
The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a
cluster algebra structure defined in terms of certain planar diagrams
known as Postnikov diagrams. The cluster associated to such a diagram
consists entirely of Plücker coordinates.
We introduce a twist map on Gr(k,n), related to the twist of
Berenstein-Fomin-Zelevinsky, and give an explicit Laurent expansion for
the twist of an arbitrary Plücker coordinate in terms of a scaled
matching polynomial. This matching polynomial arises from the bipartite
graph dual to the Postnikov diagram of the initial cluster, modified by
an appropriate boundary condition.
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Thomas Prince (Imperial College London) – A symplectic approach to polytope mutation
Polytope mutation is a combinatorial operation which appeared in the
study of the birational geometry of Landau-Ginzburg models ‘mirror-dual’
to Fano manifolds. We give a mirror/symplectic account of this subject.
This heavily utilizes the notion of an almost-toric Lagrangian
fibration, due to Symington. This perspective elucidates the connection
with cluster and quiver mutation (in the surface case) as well as the
connection to toric degenerations via ‘algebraization’ techniques due to
Gross-Siebert.