UW Algebra Seminar
Abstracts
Speaker: Balazs Szendroi, University of Utrecht
Title:
Approaches to the McKay correspondence: Motivic integration and derived
categories
Date:
January 6
Abstract:
Motivic integration has been very successful in the study of the McKay
correspondence, relating invariants of a G-space M (for finite G) to
invariants of a resolution Y of the quotient M/G. However, it typically
proves an equality of numbers (such as Euler characteristics) without
constructing a natural map between spaces (such as K-theory). Natural maps
between spaces can arise from functors between categories, such as the
(derived) category of sheaves. However, equivalences of derived categories
are presently only known when a crepant resolution of the quotient M/G
exists. I will discuss an approach that hopes to dispose of this
restriction, and discuss some examples.
Speaker:
Iain Gordon, University of Glasgow
Title:
Symplectic reflection algebras and Hilbert schemes of points on the plane
Date:
January 13
Abstract:
(Joint with Toby Stafford) Symplectic reflection algebras were
introduced by Etingof and Ginzburg in 2001 and have since proved useful in
many different areas from integrable systems to combinatorics. After
trying to motivate these algebras, I will present some work which realises
them as non-commutative deformations of the Hilbert scheme of points on
the plane. In particular, this gives a direct link between their
representations and sheaves on the Hilbert scheme. Although this is enough
to confirm some conjectures, it is not as good as it should be...
Speaker:
Nikos Tziolas (Crete)
Title:
Terminal 3-fold divisorial contractions with at most 1-dimensional fibers
Date:
January 20
Abstract:
Let X be a terminal 3-fold. I will present a classification of
divisorial contractions f:Y ---> X where Y is terminal, X is
Gorenstein, and f contracts an irreducible divisor onto a curve.
Speaker:
Kristian Ranestad, Oslo
Title:
Veronese surfaces and line arrangements
Date:
January 27
Abstract:
In recent work with Tom Graber, we recovered old results of
Kapranov on d-uple Veronese surfaces containing the intersection points of
d+3 and d+4 pairwise intersecting d-planes. In particular we obtain a new
proof that the moduli of stable line arrangements of degree 5 has a
compactification isomorphic to the Del Pezzo surface of degree 5. This is
a special case of the compactified moduli of line arrangements studied
recently by Paul Hacking.
Speaker:
Nothing scheduled
Title:
Date:
February 3
Abstract:
Speaker:
Karl Schwede
Title:
Glueing schemes and schemes without closed points
Date:
February 10
Abstract:
We will look at a fibered coproduct of ringed spaces and then
inspect some special cases where this actually gives a fibered coproduct
in the category of schemes.
Intuitively this is gluing a collection of schemes along some collection
of other schemes (possibly subschemes). We will use this in several
examples and then apply the techniques to construct a
scheme without closed points. We will then discuss other methods with
which one can construct schemes without closed points.
Speaker:
Bernd Sturmfels
Title:
Positivity and Morphisms in Tropical Geometry
Date:
February 17
Abstract:
Tropicalization is a functor which replaces algebraic
varieties with polyhedral complexes, so the tropicalization of
a polynomial map F is a piecewise-linear map. In this talk we
introduce the positive part of a tropical variety, and we explain
the difference between the image of the tropicalization of F and
the tropicalization of the image of F. This material is Section 3
in the paper "Tropical Geometry of Statistical Models" (with Lior
Pachter, q-bio.QM/0311009), but I promise that this will be a pure
math lecture (with no statistics or biology creeping in anywhere).
Speaker:
Alistair Craw
Title:
D-branes on orbifolds and the McKay correspondence
Date:
February 24
Abstract:
I will review the relation between moduli spaces of D-branes in
orbifolds and crepant resolutions of orbifolds. I'll describe a new
calculation for the simplest relevant example in dimension three, namely
the orbifold C^3/(Z/3), generalising recent work by Donagi, Katz and
Sharpe. I hope to make the talk accessible to both string theorists and
mathematicians.
Speaker:
Amnon Yekutieli
Title:
Grothendieck Duality via Rigid Dualizing Complexes and Differential
Graded Algebras
Date:
March 2
Abstract:
In this talk I will present a new approach to Grothendieck duality
on schemes. Our approach is based on two main ideas: rigid
dualizing complexes and perverse coherent sheaves.
We obtain most of the important features of Grothendieck duality,
including explicit formulas, yet manage to avoid lengthy and
difficult compatibility verifications. Our results apply to
finite type schemes over a regular noetherian base ring, and
hence are suitable for arithmetic geometry.
I will only discuss the algebraic part of the construction. The
highlight of the talk will be the role of differential graded
algebras in the arithmetic situation.
This is joint work with James Zhang (Univ. of Washington).
Speaker:
Michael Van Opstall
Title:
Stable degenerations of symmetric squares of curves
Date:
March 9
Abstract:
In order to produce a compact moduli space for algebraic
surfaces, one must allow some fairly singular varieties into the
moduli problem. However, the minimal model program tells us exactly
which varieties to allow in order to get a separated and compact
moduli space. The MMP allows us to compute replacements for surfaces
occuring in families which are not ``stable''.
Given a degeneration of a smooth curve to a stable curve, one can
take the symmetric fibered square to obtain a degeneration of the
(smooth) symmetric square of the curve to a very unpleasant surface.
I show how the first step towards stablizing this family replaces
the bad surface with a Hilbert scheme parameterizing length 2 subschemes
(such a subscheme is either a tangent vector or an unordered pair of
points). Finally, a condition for stability of this Hilbert scheme is
given in terms of the geometry of the original stable curve.
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