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Speaker:
Dave Bayer
Title: Toric syzygies and graph colorings
Abstract: Free resolutions
of monomial and toric ideals can frequently be described as the
homogenized differential of a cell complex. For one
such resolution, the "hull resolution", the cell complex
is the set of
bounded faces of a convex hull associated with the ideal. For
unimodular toric varieties embedded in a product of projective
lines,
the hull resolution is minimal, and its cell complex can also
be
described as an infinite periodic hyperplane arrangement whose
vertices form a lattice.
One standard approach to studying graph colorings is as lattice
points
not contained in any hyperplane of a hyperplane arrangement associated
to the graph. Viewing this arrangement as an infinite periodic
hyperplane arrangement whose vertices form a lattice, we obtain
a
minimal free resolution of the above form, so the chambers of
this
arrangement correspond to the highest syzygies of a toric ideal.
Minty's criterion for colorability becomes a statement about
the
location of these syzygies: Their multidegrees plot as the integer
lattice points of a sequence of nested polytopes, with the innermost
polytopes containing syzygies corresponding to colorings using
the
fewest colors.
The first part is joint work with Bernd Sturmfels and Sorin
Popescu.
The second part is joint work with Sorin Popescu.
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Speaker:
Dan
Edidin
Title: Good representations and solvable groups
Abstract:
A representation $V$ of a connected algebraic group $G$ is
called "good" if $G$ acts with trivial stabilizers
on an open set $U$
whose complement $V\smallsetminus U$ is a finite union of
$G$-invariant linear subspaces. Good representations of the group
of
upper-triangular matrices play a key role in the proof of the
equivariant Riemann-Roch theorem. We show that, in characteristic
0, a
group is solvable iff it has a good representation. In characteristic
$p$, non-solvable groups such as $SL_2$ have good representations.
However, we can prove that, in every characteristic except 2,
any
group $G$ with a good representation $V$ such that $G$ acts
scheme-theoretically freely on the open set $U$ is in fact
solvable. This talk is based on joint work with W. Graham.
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Speaker:
David
Eisenbud
Title: Chow Forms and Other Wonders via Exterior Algebra
Abstract:
The Bernstein-Gel'fand-Gel'fand theorem states that the bounded
derived category of coherent sheaves on projective space is the
same as the bounded derived category of finitely generated modules
(mod free modules) over the exterior algebra. There is a simpler
and more concrete version of this correspondence, even useful
for computation, which I have exploited in recent work with Avramov,
Popescu, Schreyer, and Yuzvinsky. I will describe the basics
and also give some applications and open questions.
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Speaker:
Richard
Hain
Title: On the action of the absolute Galois group on the
fundamental group of the thrice punctured projective line.
Abstract: This
will be an exposition of joint work with Makoto
Matsumoto in which we prove the `easy' half of a conjecture of
Deligne about the action of the absloute Galois group on the
pro-$\ell$ fundamental group of the projective line minus three
points.
The conjecture concerns a certain graded Lie algebra over ${\mathbb
Q}_\ell$ associated with the action. We prove that the Lie algebra
is
generated by the graded group $\oplus_{n>0} K_n(\mathbb Z)$.
The basic tool in the proof is the theory of {\em weighted completion}
of a
profinite group which I shall explain in detail.
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Speaker:
J.W. Hoffman
Title: Topology of Siegel Modular Varieties
Abstract:
A Siegel modular variety is a quotient of the Siegel
upper half space of degree d by an arithmetic subgroup of Sp
(2d , Q). These are moduli spaces of abelian varieties of dimension
d with certain additional structures. This talk is an overview
of what is known about the cohomology groups of these varieties.
The connection with zeta
functions and automorphic forms is discussed. Recent calculations
done in collaboration with S. Weintraub for the case d = 2 are
presented.
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Speaker:
Mikhail
Kapranov
Title: Eisenstein series for Kac-Moody groups and S-duality
Abstract: The
S-duality conjecture of Vafa-Witten is a statement
about the generating function of the Euler characteristics of
the moduli spaces of bundles on an algebraic surface with varying
second Chern class. We find a relation of this function with
geometric Eisenstein series
for the Kac-Moody groups and establish a functional equation
for such series which gives, in particular, the elliptic behavior
of a relative version of the Vafa-Witten function for parabolic
bundles.
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Speaker:
Sean Keel
Title: Towards the ample cone of the moduli space of curves
Abstract: Which
line bundles on M_{g,n} are ample? I'll reduce
the problem to g=0, present a simple conjectural solution, and
then discuss evidence for the conjecture. I spoke on the same
subject at Fulton's birthday conference, but here I will stress
quite different details.
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Speaker:
Minhyong
Kim
Title: Crystalline representations and Neron models.
Abstract: We
discuss an application of notions from
p-adic Hodge theory to the theory of Neron models
for abelian varieties over local fields. More
precisely, we will give a description of the
p-part of the component group in terms of a crystalline
representation naturally associated to the abelian
variety.
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Speaker:
Sandor
Kovacs
Title:
Arakelov-Parshin boundedness
Abstract:
At the ICM'62 in Stockholm Shafarevich conjectured the following:
Let
$C$ be a fixed compact Riemann surface, $\Delta\subset C$ a fixed
set
of finite points and $g>1$ a fixed natural number. Then the
number of
different smooth projective families of curves of genus $g$ over
$C\setminus \Delta$ is finite.
This was confirmed by Arakelov in 1971 following ideas of
Parshin. Their method was to divide the problem into two:
"boundedness" and "rigidity". Boundedness
means that the parameter
space of such families is of finite type, in particular it has
finitely many components, while rigidity means that there exist
no
non-trivial deformations of these families, i.e., the components
of
the parameter space are $0$-dimensional. Together they imply
that the
parameter space is finite.
Higher dimensional generalizations of the boundedness part
will be
discussed in the lecture as well as applications of the results
to
other questions such as the Catanese-Schneider conjecture on
the
minimal number of singular fibers of a family of minimal varieties
of
general type (higher dimensional analogues of curves of genus
greater
than $1$) and Shokurov's conjecture on the algebraic hyperbolicity
of
certain fine moduli spaces.
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Speaker:
Rick Miranda
Title: Linear Systems of Plane Curves
Abstract:
Fix n general points p_1,...,p_n and n positive integers
m_1,...,m_n in the plane, and consider the vector space of
all polynomials f(x,y) of degree at most d which have
multiplicity at least m_i at p_i for each i. (This means
that all partial derivatives of order < m_i vanish at p_i
for each i.) Hence "multiplicity one" simply means
that
the polynomial f vanishes at the point, "multiplicity two"
means that in addition both first partials vanish, etc.
In terms of the Taylor expansion of f about p_i, the
multiplicity is the order of the lowest order terms that appear
with nonzero coefficient.
The problem I will address in this talk is: what is the
dimension of this space of polynomials? The answer for
general multiplicities and numbers of points is, surprisingly,
still open. I will discuss what is known, various extensions
to other situations and what is known there, and indicate some
recent contributions.
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Speaker: Hal Schenck
Title: Bundles on P^2 related to line arrangements
Abstract: Associated
to a configuration of lines A in P^2 is the
module of A-derivations D(A), a much studied object in the field
of
Hyperplane Arrangements. D(A) decomposes as S(-1)\oplus D_0,
and the sheaf associated to D_0 is a rank two bundle on P^2.
We relate the
stability and jump locus of this bundle to the geometry of the
configuration of line.
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Speaker:
Alice
Silverberg
Title: Ranks of
elliptic curves in families of quadratic twists
Abstract: We
discuss a number of questions about ranks of
quadratic twists of elliptic curves. In particular,
we show that the unboundedness of the ranks of the
quadratic twists of an elliptic curve over the field
of rational numbers is equivalent to the divergence
of certain infinite series.
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Speaker:
Mike
Stillman
Title: The toric Hilbert scheme
Abstract: We
introduce the toric Hilbert scheme, which parametrizes all A-graded
ideals with the same multi-graded Hilbert function as a toric
ideal. This parameter space is interesting because it is often
much
simpler than the standard Hilbert scheme, and because of potential
applications in combinatorics.
We survey some examples, and describe what we know to be true
about these spaces. In particular, we describe local equations
for these Hilbert
schemes, and use this to show that in the codimension two case,
the toric
Hilbert scheme is smooth. This represents joint work with Irena
Peeva.
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Speaker:
Bernd
Sturmfels
Title: Rational Hypergeometric Functions
Abstract:
Multivariate hypergeometric functions associated with toric
varieties were introduced by Gel'fand, Kapranov and Zelevinsky.
Singularities of such functions are discriminants, that is, divisors
projectively dual to torus orbit closures. We show that most
of these
potential denominators never appear in rational hypergeometric
functions.
We conjecture that the denominator of any rational hypergeometric
function is a product of resultants, that is, a product of special
discriminants arising from Cayley configurations. This conjecture
is proved for toric hypersurfaces and for toric varieties of
dimension at most three. Toric residues are applied to show that
every toric resultant appears in the denominator of some rational
hypergeometric function. This is joint work with Alicia Dickenstein
and Eduardo Cattani (math.AG/9911030).
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Speaker:
Felipe
Voloch
Title: Curves with many rational points
Abstract:
In 1983, Faltings proved Mordell's 1922 conjecture to
the effect that an algebraic curve of genus at least two has
at most a finite number of rational points. However, the proof
doesn't
say much about an actual upper bound for the number of rational
points.
Recently, there has been some speculation as to what such an
upper bound should depend on. Some people have suggested that,
because of a conjecture of Lang, the bound should depend on very
little. We will discuss these suggestions and how they compare
with the experimental evidence. We will also look at similar
questions over cyclotomic and finite fields.
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