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F. R. A. GME. N. T.
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This is a seminar series intended to involve people in the
For more information, or to provide a speaker, contact Renzo, renzo AT math.colostate.edu, or Yano, casa AT math.colorado.edu
Schedule of Talks:
|
Date |
Speaker |
At |
Title (click on title to view summary) |
|
Sep 2nd |
(Brown) |
CSU |
Logarithmic Geometry, Logarithmic Stable
Maps and Their |
|
Sep 9th |
(UBC) |
CSU |
|
|
Sep 16th |
(U.Oregon) |
CSU |
|
|
Sep 23rd |
( ( |
CSU |
Stability conditions on the local projective plane and \Gamma_1(3)-action |
|
Sep 30th |
( |
CSU |
|
|
Oct 7th |
( |
CSU |
Compactifications of reductive groups as
moduli of principal bundles |
|
Oct 12th |
Leo Murata |
CU |
A relation
between arithmetical functions and sum-of-digits |
|
Oct 14th |
( |
CSU |
|
|
Oct 18th |
( |
CSU |
Colloquium. Landau-Ginzburg/Calabi-Yau correspondence
and mirror symmetry |
|
Oct 19th |
Phillip Williams CUNY |
CU |
Resultant and conductor:
minimality and semi-stability |
|
Oct 21st |
(Naval Academy/CSU) |
CSU |
|
|
Oct 28th |
(U. Filippo Viviani (Roma 3) |
CSU |
Compactified Jacobians of singular curves |
|
Nov 4th |
MATH DAY |
|
NO SEMINAR |
|
Nov 11th |
(U.Georgia) |
CSU |
Vector bundles of conformal blocks on
$\bar{M}_{0,n}$ for sl_n, |
|
Nov 18th |
( |
CSU |
Secant varieties and the Waring problem. |
|
Dec 2nd |
( |
CSU |
Fluctuations
in the number of points of curves over finite fields |
Logistics:
·
CU Boulder Campus
map
·
CSU Campus map
Abstracts:
Sep 2: Steffen Marcus
We
will begin with a (hopefully) thorough introduction to
Logarithmic Geometry in the sense of Kato-Fontaine-Illusie. We will
then discuss Li's moduli space of relative stable maps and Chen's
moduli space of logarithmic stable maps. We will finish with a
description of current work towards identifying the evaluation spaces
for logarithmic stable maps - the goal of which is to develop a
working logarithmic GW theory. This is joint work with Abramovich,
Chen and Gillam.
Sep 9: Alan Stapledon
We construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations.
More specifically, we introduce an invariant of a compactification of such a variety called the ``tropical motivic nearby fiber''. Under suitable conditions, this invariant specializes to the Hodge-Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We deduce a formula for the Euler characteristic of a general fiber of the degeneration. The first hour will be a gentle introduction to tropical geometry.
At the risk of not getting through too much, I will make the first hour very gentle.
Sep 16: Nick Proudfoot
In
the first hour I'll give an introduction to hypertoric varieties (starting with
toric varieties), emphasizing combinatorial interpretations of their Betti
numbers. In the second hour I'll talk about "hypertoric category
O", a beautiful category of modules over a non-commutative deformation of
the coordinate ring of a hypertoric variety, or (equivalently) sheaves of
modules over a non-commutative deformation of the structure sheaf of a
hypertoric variety.
Sep 23: Emanuele Macri’
In this talk, we report on joint work with Arend Bayer on
the space of stability conditions for the canonical bundle on the projective
plane. We will describe a connected component of this space, generalizing and
completing a previous construction of Bridgeland. In particular, we will see
how this space is related to classical results of Drezet--Le Potier on stable
vector bundles on the projective plane. Using this, we can determine the group
of autoequivalences of the derived category. As a consequence, we can identify
a \Gamma_1(3)-action on the space of stability conditions, which will give a
global picture of mirror symmetry for this example.
Sep 23: Ana-Maria Castravet
We construct new exceptional divisors on the
Grothendieck-Knudsen
moduli space \bar M_{0,n}, of stable rational curves with n marked points,
from new combinatorial structures which we call hypertrees. We conjecture
that these divisors, together with the boundary, generate the cone of
effective divisors of \bar M_{0,n}. This is joint work with Jenia Tevelev.
Sep 30: Chris Hall
A polynomial f(x) with coefficients in the rational function
field Q(t) may be regarded as a family of polynomials over Q: for each rational
number t_0, consider the polynomial f_0(x) obtained by replacing t by t_0.
If we suppose that f(x) is irreducible over Q(t), then Hilbert's
irreducibility theorem asserts that, for 'most' rational numbers t_0, f_0(x) is
also irreducible. It is natural to ask for other properties of f(t) that
'most' f_0(t) also have, e.g. having the same Galois group, and in the first
half of this talk (which is meant to be expository) we will give an introduction
to ideas and tools in arithmetic geometry which can be used to prove Hilbert's
theorem as well as other theorems about families of polynomials.
In the second half of this talk we will replace families of polynomials with
families of varieties (e.g. abelian varieties or complete intersections) and
consider a similar theme, that is, we will look for properties of the generic
member which are inherited by 'most' members of the family. Once again
tools from arithmetic geometry play a key role, but now recent spectacular
results pertaining to expander graphs can be made to play a role as well.
We will (try to) explain the ideas involved and show how they lead to
theorems with very strong quantifications for 'most'.
Oct 7: Michael Thaddeus
Let G be a reductive group over an algebraically closed field of characteristic zero. We study a family of moduli problems where the objects are principal G-bundles on chains of projective lines, trivialized at the two endpoints. From different stability conditions, we get different compactifications of G as a Deligne-Mumford stack. In particular, we extend the notion of "wonderful" compactification to a semisimple G with nontrivial center, such as SL(n). This describes joint work in progress with Johan Martens.
Oct 12: Leo Murata
We can show that there is a one-to-one correspondence
between
the set of simple arithmetic functions and the set of sum-of-digits functions
associated with code systems. We especially take notice of the set of
sum-of-digits functions
associated with Gray codes, and consider what are the common properties of
arithmetic functions which
are connected with gray codes.
Oct 14: Christopher Hacon
In this talk I will survey recent results on the boundedness of varieties of log general type and their applications to automorphism groups of varieties of general type, moduli spaces, the ascending chain condition for log canonical tresholds's, etc
Oct 18: Yongbin Ruan
Almost twenty years ago, Batyrev-Borisov established the
classical mirror symmetry of Calabi-Yau hypersurfaces of Gorenstein
toric varieties. The recent work of Kawitz revealed that Batyrev-Borisov
theorem only covers a small fraction of known examples. With the recent
investigation of LG/CY correspondence, we are able to establish classic
mirror symmetry for all known examples of Calabi-Yau hypersurface of the
weighted projective spaces. This is a joint work with Alessandro Chiodo.
Oct 21: Amy Ksir
Question 1:
Suppose you have an algebraic curve Y with an action by a finite group G.
Then G acts on the space of global holomorphic differentials (regular
1-forms) on Y, which is a finite dimensional vector space. This
representation of G has a character (the traces of the matrices corresponding
to elements of G), whose values lie in a finite extension of Q. What are
the conditions (on G, or on G and Y) under which the character values are all
in Q itself?
Question 2: Choose a positive
integer n, and another positive integer h relatively prime to n. Notice that
some power of h (call it h^k) will be 1 mod n. Now for each j from 1 to
n, add up j (mod n) + jh (mod n) + jh^2 (mod n) + ... j h^{k-1}
(mod n). Do you get the same sum, no matter which j you started with?
Is it always a multiple of n? Can you find values of h for which
you *don't* always get the same sum?
In this talk I will explain why these questions are related, and report on
recent results with Ted Chinburg. We have a complete answer to Question
2, which it turns out answers Question 1 in the tame case. The first hour
will feature algebraic curves and representations of finite groups. The
second hour will feature cyclotomic fields and L functions.
Oct 28: Margarida Melo
The Jacobian variety of a smooth curve is an Abelian variety
that carries
important informations about the curve itself. Its properties have been
widely studied along the decades, giving rise to a significant amount of
beautiful mathematics.
However, for singular curves, the situation is more involved since the
generalized Jacobian variety is not anymore an Abelian variety once, in
general, it is not compact.
The problem of compactifying it is, of course, very natural, and it is
considered to go back to the work of Igusa and Mayer-Mumford
in the 50's-60's.
Since then, several solutions appeared, differing from one another in
various aspects as the generality of the construction, the modular
description of the boundary and the functorial properties.
In this talk I will start by recalling some of these constructions and how
they relate to each other. I will then report on several new results,
partially obtained in collaboration with Filippo Viviani, which aim to
understand better the geometry of these moduli spaces as well as some
generalizations of these constructions.
Oct 28: Filippo Viviani
The classical Torelli map is the modular map from the moduli space of smooth projective curves of genus g into the moduli space of principally polarized abelian varieties of dimension g, sending a curve into its Jacobian. The Torelli theorem asserts that the Torelli map is injective on geometric points. We propose two extensions of the Torelli theorem: one for the compactified Torelli map and the other for the tropical Torelli map. The compactified Torelli map was constructed by Alexeev: it is a modular map from the Deligne-Mumford moduli space of stable curves to the Alexeev moduli space of stable semi-abelic pairs, sending a stable curve into its compactified Picard variety of degree g-1, endowed with its natural theta divisor and the action of the generalized Jacobian. In a joint work with L. Caporaso, we give a complete description of the fibers of the compactified Torelli map. On the other hand, in a joint work with S. Brannetti and M. Melo, we construct moduli spaces of tropical curves and tropical abelian varieties and a tropical Torelli map between them. In another joint work with L. Caporaso, we describe the fibers of the tropical Torelli map. I will report on the above two Torelli-type theorems, trying to enlight the relations between them.
Nov 11: David Swinarski
The WZW model of conformal field theory yields a vector
bundle
on the moduli space of pointed curves $\bar{M}_{g,n}$ depending on a
choice of a Lie algebra, a level, and a set of $n$ weights in the
corresponding Weyl alcove. Recently Fakhruddin gave formulas for the
Chern classes of these bundles when g=0 and showed they are globally
generated. I will discuss recent joint work with Arap,
Giansiracusa, Gibney, and Stankewicz showing that some of these
bundles are extremal in the nef cone, and identify the images of the
linear systems corresponding to these divisors.
Nov 18: Ada
In 1770 Waring asked if for every natural number d there
exists
an associated positive integer such that every natural number is the sum
of at most s dth powers of natural numbers.
(For example, every number is the sum of at most 4 squares, or 9 cubes,
etc.)
One can ask a similar question for (homogeneous) polynomials: given any
degree d such polynomial, when can we write it as a sum of dth powers of
linear forms?
In my talk I will speak about classical results and show how the Waring
problem is connected with the study of secant varieties.
(which will also be defined and explained, nothing more than linear
algebra is required for this talk!)
This seminar will serve as introduction for my second talk, where I will
speak about my results on secants.
The Lagrangian Grassmannian LG(n,2n) is a projective variety
parametrizing dimension n isotropic subspaces of a symplectic complex
vector space of double dimension 2n.
In this talk I will report about a joint work with Jarek Buczyinski, where
we computed the dimensions of the 3rd and 4th secant varieties of
LG(n,2n).
Our result is an application of the classic Terracini Lemma, and the key
point was finding a normal form for 4 general points in a LG(n,2n).
Dec 2: Chantal David
We study in this talk the distribution of the number
of points for two families of curves
over a _nite _eld with q elements: cyclic covers of P1 and smooth
plane curves. The Katz-
Sarnak philosophy makes predictions about the
statistics for such families in the large q
limit when the genus is _xed. We are looking at the
complementary statistics, when the
genus varies, but the _eld of de_nition is _xed. In
that case, one can obtain statistics for
the distribution of the number of points by sieving
the families of curves. This is joint
work with A. Bucur, B. Feigon and M. Lalin.
In previous semesters the seminar page was maintained by Renzo Cavalieri, Rachel Pries and Jeff Achter. You can find the Fall 09 page here, and the Spring 10 here.