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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:
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Date |
Speaker |
At |
Title (click on title to view summary) |
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Jan 21st |
Kevin Tucker (UM) |
CSU |
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Jan 28th |
Dan Bates (CSU) |
CSU |
Basic numerical algebraic geometry and how
it helps with symbolic computation |
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Feb 4th |
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Renzo and Chris out of town. |
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Feb 11th |
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Renzo and Chris out of town. |
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Feb 18th |
Hsian-Hua Tseng ( |
CSU |
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Feb 25th |
Andrew Obus ( |
CSU |
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Mar 4th |
Arend Bayer ( Everett Howe ( |
CSU |
Stability conditions for the local projective plane |
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Mar 8th Weber 117, 2pm Note unusual day and time |
Milena Hering (U.Conn) |
CSU |
The moduli space of points on the
projective line and quadratic |
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Mar 9th 4pm , Room 220 |
W. Dale Brownawell |
CU |
Linear independence in minimal group extensions |
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Mar 11th |
Jon Hauenstein ( |
CSU |
1)
Regeneration and applications
of numerical algebraic geometry |
Mar 25th
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Jeff Achter (CSU) |
CSU |
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Apr 1st |
Chris Hall (U. |
CSU |
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Apr 8th |
Jared Weinstein ( |
CSU |
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Apr 15th |
Vincent Bouchard ( (Note special time 1pm-3pm) |
CSU |
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Apr 22nd |
Sebastian Casalaina Martin (CU Boulder) |
CU |
Simultaneous stable reduction for curves with ADE singularities. |
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Apr 29th |
Satyan Devadoss (Williams/MSRI) |
CSU |
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May 6th |
Marco Aldi (UC Berkeley) |
CSU |
Logistics:
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CU Boulder Campus
map
·
CSU Campus map
Abstracts:
Jan 21: Kevin Tucker
Jumping numbers are recently defined analytic invariants of singularities on complex algebraic varieties. We will give a gentle introduction in the case of plane curves while simultaneously highlighting several recent results. These include the surprising fact that the jumping numbers of a plane curve depend only on the topological type of the curve. Finally, equisingularity invariants will be used to present a formula for the jumping numbers of a unibranch curve. Some remaining open questions will also be discussed.
In the first hour, I will provide a quick refresher on numerical algebraic
geometry, i.e., numerical methods for studying the solution sets of polynomial
systems. I am aware that most people at the talk will have seen some of
this, so I will keep this brief and focus more on aspects I have not discussed
previously. After that brief review, I will talk about a project with
Chris Peterson (CSU), Tim McCoy (CSU/Notre Dame), Andrew Sommese (Notre Dame),
and Jon Hauenstein (Texas A&M) in which we try to recover the prime
decomposition of the radical of an ideal with numerical and symbolic-numeric
tools (rather than the usual symbolic methods). This sort of algorithm is
one of the popular directions in numerical algebraic geometry these days.
For the second hour, I will cover a new set of methods for finding the real solutions of polynomial systems. There is some underlying theory that may be of interest to those not so interested in solving polynomial systems (Gale duality, in particular). I'll describe the theory, sketch out the method, and walk through one or more examples. This method is nice as it constitutes a 99% savings in computational resources for some problems. This is joint work with Frank Sottile (Texas A&M), Matt Niemerg (CSU), and Jon Hauenstein (Texas A&M).
Feb 18: Hsian-Hua Tseng
Let r be a positive integer. Given a line bundle L over a space X there is a
stack \sqrt[r]{L/X} over X which classifies r-th roots of L. This stack of r-th
roots of L, which is a gerbe over X banded by the cyclic group of order r,
appears naturally in various places such as the structure result of toric
Deligne-Mumford stacks. The goal of this talk is to discuss recent progresses
towards understanding Gromov-Witten theory of such stacks of roots (joint work
with
Feb 25: Andrew Obus
If G is a finite group, then the field
of moduli of a branched G-cover of the Riemann sphere is the intersection of
all fields of definition of the G-cover. A result of Beckmann says that
for any 3-point G-Galois cover of the Riemann sphere, if a prime p does not
divide the order of G, then p is unramified in the field of moduli of the
G-cover. Wewers generalized this: if p exactly divides the order of G, then p
is tamely ramified in the field of moduli. We will discuss extensions of this
result, contained in the speaker's thesis, involving more general groups G with
cyclic p-Sylow groups.
The first hour will be an introduction to branched covers, GAGA, and the
fundamental exact sequence.
Mar 4th: Arend Bayer
I will talk about the space of stability conditions for the total space of the
cotangent bundle on P^2. Its geometry is related to classical questions on
Chern classes of stable vector bundles P^2, to the modular group Gamma_1(3),
and, via mirror symmetry, to the moduli space of elliptic curves with
Gamma_1(3)-level structure. This is based on joint work with Emanuele Macri.
Mar 4th:
I will discuss joint work with Vassil Dimitrov. While trying to find
faster ways of multiplying points on elliptic curves by integers, Dimitrov was
considering sums of integers of the form (-1)^x * 2^y * 3^z. He
conjectured that 4985 is the smallest positive integer not representable as the
sum of three such terms. This conjecture has a superficial resemblance to
some very difficult questions, and experts had suggested that it would not be
resolved without progress in the theory of linear forms in logarithms.
However, the conjecture is true, and can be proven quite quickly by
showing that the implicit Diophantine equation has no solutions modulo
1099511627760. In this talk, I will explain what is special about
1099511627760, and show how numbers like it can be used to show that there are
infinitely many integers n that cannot be written as the sum of fewer than
c*(log n)/(log log n log log log n) integers of the form (-1)^x * 2^y * 3^z,
for some positive absolute constant c.
Mar 8th: Milena Hering
The ring of invariants for the action of the automorphism
group of the
projective line on the n-fold product of the projective line is a
classical object of study. The generators of this ring were determined
by Kempe in the 19th century. However, the ideal of relations has been
only understood very recently in work of Howard, Millson, Snowden and
Vakil. They prove that the ideal of relations is generated by
quadratic equations using a degeneration to a toric variety. I will
report on joint work with
Benjamin Howard where we further study the toric varieties arising in
this degeneration. As an application we show that the second Veronese
subring of the
ring of invariants admits a presentation whose ideal admits a
quadratic Groebner basis.
Mar 11th: Jon Hauenstein
Numerically solving polynomial systems and manipulating
algebraic sets has
many applications in engineering and biology. Such systems can
be quite large, but, due to their structure, typically have few solutions.
Regeneration is a solving method that can utilize underlying structure of
the polynomial system to efficiently solve these systems. After a brief
overview of homotopy continuation and numerical algebraic geometry, this
hour will be used to discuss regeneration and some of its applications.
This work is joint with A. Sommese and C. Wampler.
In the first hour, regeneration will be applied to computing a numerical
irreducible decomposition. A key component of this algorithm is to
determine if a given (approximate) solution is isolated. This hour will
be devoted to addressing this by discussing multiplicity using dual bases
(modern form of Macaulay's inverse systems) and its application to
computing the local dimension at the given solution. This talk will
conclude by applying the local dimension test to determine the
mobility of a mechanism and using dual bases to compute Hilbert functions
and standard monomials for zero dimensional ideals. This work is joint
with D. Bates, C. Peterson, A. Sommese, and C. Wampler.
Mar 25th: Jeff Achter
These talks will be about
cubic hypersurfaces of low dimension.
A. Geometry
I'll review classical facts and constructions involving cubic surfaces and
threefolds over the complex numbers and other algebraically closed
fields. Undoubtedly, I'll slip in some facts about cubic curves, too.
B. Moduli
It has long been known that to a complex cubic surface or threefold one can
canonically associate a principally polarized abelian variety. I'll
explain a similar construction which works for cubics over a more general
arithmetic base. This answers, at least away from the prime 2, an old
question of Deligne and a recent question of Kudla and Rapoport.
Apr 1st: Chris Hall
Given a curve C over a finite field F_q and a dominant map phi : C --> P^1, one can pull back any elliptic curve E over the function field L=\bar{F}_q(P^1) to the function field M=\bar{F}_q(C). By allowing phi to vary, there are two questions that one can try to answer. First, is it possible to construct an elliptic curve over M of arbitrarily large rank? Second, is it possible to construct an elliptic curve over L of rank one whose pullback to M also has rank one. We will explain why both of these questions are interesting and show they have an affirmative answer when q is odd.
Apr 8th: Jared Weinstein
Talk I (will be highly
accessible, nothing original)
Abstract: Let F be a field. What is the maximal algebraic
extension
of F with abelian Galois group?
When F is a number field (such as the field of rational numbers), the
answer was known by the 1930s and belongs to the study of "class field
theory". When F is a nonarchimedean local field (such as the field
of
p-adic numbers), there is a simple description of all the abelian
extensions of F, given by Lubin and Tate in 1966. We will define the
notion of formal module, which is central to Lubin-Tate theory; we
will give special attention to the case of F with positive
characteristic. Then we will indicate how moduli spaces of formal
modules allows one to construct nonabelian extensions of F. In broad
strokes, we will also say what is meant by the local Langlands
correspondence.
Talk II (more technical)
Abstract: Let $q$ be a prime power. The "Hermitian
curve" $y^q + y
= x^{q+1}$, considered as an affine curve over the field of $q^2$
elements, has the maximum number of rational points possible for any
connected curve of the same genus. It also has a large group of
automorphisms. In this talk we're going to explore higher-dimensional
analogues of this phenomenon. There seems to be a relation to the
Lubin-Tate tower of moduli spaces of formal modules: these spaces are
horribly singular at the special fiber, but after blowing up the
singularities and reducing one finds nonsingular varieties with the
maximum number of rational points relative to their "topology".
We
will write down an explicit family of affine hypersurfaces appearing
in this way which seems to be a direct analogue of the Hermitian curve
(though we can't prove it!).
Apr 15th: Vincent Bouchard
Abstract: In this talk I will explain how the all-important
study of
the phenomenology of string theory translates into a well defined
mathematical question in algebraic geometry. I will focus on two
particular limits of string theory, namely heterotic strings and
F-theory. In heterotic strings, to obtain interesting physics one must
construct stable holomorphic vector bundles on Calabi-Yau threefolds
satisfying a detailed set of constraints. In F-theory, we are mostly
interested in studying the detailed geometry of elliptically fibered
Calabi-Yau fourfolds. I will explain what the mathematics look like
and where it comes from, propose some approaches to address these
questions, and describe new exciting results with interesting
experimental signatures.
Apr 22nd: Sebastian Casalaina Martin
Abstract: A basic question in moduli theory is to describe a
stable reduction of a given family of curves. That is given a family of
curves where the generic fiber is stable, one would like to "replace"
the fibers that are not stable in such a was as to obtain a family of stable
curves. Typically this will only be possible after a generically finite base
change, and the question is to describe such a base change as well as the total
space of the new family. In the language of stacks, this is the problem
of resolving a rational map to the moduli stack of stable curves.
The Deligne-Mumford Stable Reduction Theorem provides an explicit solution to
this problem for one parameter families of curves. As many questions can
be studied effectively via one parameter families, this theorem has been used
extensively in the literature. In the first half of the talk we will
discuss, in basic terms, concrete examples of stable reduction for one
parameter families of curves.
Recently, there has been growing interest in studying higher dimensional
families of curves as well. For instance, in a program started by B.
Hassett and S. Keel, moduli spaces of curves with singularities worse than
nodes have arisen naturally in the study of the canonical model of the moduli
space of stable curves. Stable reductions for such families provide a
means of comparing such moduli spaces to the moduli space of stable curves.
The aim of the second halp of the talk will be to present joint work with Radu
Laza where we describe explicit (local) stable reductions for a families of
curves with ADE singularities. ADE singularities, the most basic
singularities beyond nodes, are also the first to be encountered in the
Hassett-Keel program mentioned above. Our result describes the resolution
in terms of Weyl covers and wonderful blow-ups, both modifications of the base
of the family that are determined by the combinatorics of the Weyl groups of
type ADE associated to the singularities. In light of a general result of
de Jong's on the existence of stable reductions for families of curves, the
content of our result is to provide an explicit stable reduction for such a
family, determined by the combinatorics of the singularities. This is
important for the applications we have in mind (e.g. moduli spaces of low genus
curves and the Hassett-Keel program), which, time permitting, I will discuss.
Apr 29th: Satyan Devadoss
Abstract 1:
What is the space of all possible ways robots can move in a room? What happens
when we place obstacles in their path? We not only look at the important
subject of robot motions but look at the ideas behind robot collisions.
This leads to worlds of polyhedra, tilings, string theory, and
phylogenetic trees. This talk is heavily based on pictures and no background is
needed.
Abstract 2:
We consider the moduli space of Riemann surfaces with boundary and marked
points. Such spaces appear in open-closed string theory, particularly
with respect to holomorphic curves with Lagrangian submanifolds. We
consider a combinatorial framework to view the compactification of this space
based on the pair-of-pants decomposition of the surface, relating it to the
well-known phenomenon of bubbling. Our main result classifies all such
spaces that can be realized as convex polytopes. A new polytope is introduced
based on truncations of cubes, and its combinatorial and algebraic structures
are related to generalizations of associahedra and multiplihedra.
May 6th : Marco Aldi
Abstract: "We review the statement and the basic principles underlying the so-called Homological Mirror Symmetry Conjecture, first introduced by Kontsevich. We work out in some details the case of the elliptic curve, following Polishchuk and Zaslow. This is a beautiful calculation that can be generalized in a number of directions. We describe some of variations on this theme involving non-commutative geometry, twisted sheaves and non-kaehler surfaces. If time permits, I will report on some work with Heluani relating homological mirror symmetry to a duality between certain sheaves of vertex algebras".
In previous semesters the seminar page was maintained by Renzo Cavalieri, Rachel Pries and Jeff Achter. You can find the Fall 09 page here.