Diane Maclagan (Department of
Mathematics, Rutgers University)
http://www.math.rutgers.edu/~maclagan/
Equations and degenerations of the moduli space of
genus zero stable curves with n marked points
Abstract:
Curves are one of the basic objects of algebraic
geometry, and
so much attention has been paid to the moduli space of all
curves of a
given genus. This talk will focus on the moduli space of
genus zero
stable curves with n marked points, which is a
compactification of the
space M0,n of isomorphism classes of n points on the
projective
line.
After introducing this space, I will describe joint work with
Angela
Gibney on explicit equations for it, which lets us see
degenerations to
toric varieties.
November 2, 2006, 10:10-11:00 am, Lind Hall
409
Daniel Lichtblau (Wolfram
Research, Inc.)
Computer assisted mathematics: Tools and tactics for
solving hard problems
Talk Materials: IMA2006_Lichtblau_talk.pdf IMA2006_Lichtblau_talk.nb
Abstract:
In this talk I will present several problems that have caught
my attention over the past few years. We will go over
Mathematica formulations and solutions. Along the way we will
meet with a branch-and-bound loop in its natural habitat, some
rampaging Gröbner bases, a couple of tamed logic puzzles, and at least a
dozen wild beasts.
As the purpose is to illustrate a few of the many ways in which
Mathematica can be used to advantage in tackling difficult
problems, we will go into a bit of detail in selected examples.
Do not let this deter you; there will be no exam, and it is the
methods, not the problems, that are of importance. The examples
are culled from problems I have seen on Usenet groups
(primarily MathGroup), in articles, or have been asked in
person.
November 8, 2006, 11:15 am-12:15 pm, Lind Hall
409
Sorin Popescu (Department of
Mathematics, Stony Brook University)
http://www.math.sunysb.edu/~sorin/
Excess intersection theory and homotopy continuation
methods
Abstract:
I will recall first basic techniques and results in (excess)
intersection theory in algebraic geometry and then discuss
their
implications and also applications toward a numerical approach
to
primary decomposition for ideals in polynomial rings.
November 13, 2006, 11:15 am-12:15 pm, Lind Hall
409
Uwe Nagel (Department of
Mathematics, University of Kentucky)
http://www.ms.uky.edu/~uwenagel/
Complexity measures
Abstract:
It is well-known that on the one hand the costs for computing
a Gröbner basis can be prohibitively high in the worst case,
but on
the other hand computations can often be carried out
successfully in
practice. As an attempt to explain this discrepancy, several
invariants
that measure the size or the complexity of an ideal or a module
have
been introduced. The most prominent one is the
Castelnuovo-Mumford regularity, but there are also extended
degrees
introduced by Vasconcelos and, more recently, the extended
regularity
jointly proposed with Chardin. The latter two notions are
defined
axiomatically. In the talk we will discuss the three concepts
and their
relations as well as some known results and open problems.
November 15, 2006, 11:15 am-12:15 pm, Lind Hall
409
Richard Moeckel (School of Mathematics,
University of Minnesota)
http://www.math.umn.edu/~rick/
Tropical celestial mechanics
Abstract: Some interesting
problems in mechanics can be reduced to solving
systems of algebraic equations. A good example is finding
relative
equilibria of the gravitational n-body problem. These are
special
configurations of the n point masses which can rotate rigidly
such that
the outward centrifugal forces exactly cancel the gravitational
attractions. The algebraic equations are complicated enough
that it is
a long-standing open problem even to show that the number of
solutions
is finite. I will describe a solution to this question for n=4
which
makes use of some ideas from what is now called tropical
algebraic
geometry – Puiseux series solutions, initial ideals, etc. The
problem
is open for larger n.
November 15, 2006, 2:15 pm-3:15 pm, Lind Hall
409
Wenyuan Wu (Department of
Applied Mathematics, University of Western Ontario)
http://publish.uwo.ca/~wwu26/
On approximate triangular decompositions in dimension
zero
Abstract:
Triangular decompositions for systems of polynomial equations
with n variables, with exact coefficients are well-developed
theoretically and in terms of implemented algorithms in
computer algebra systems. However there is much less research
about triangular decompositions for systems with approximate
coefficients. In this talk we will discuss the zero-dimensional
case, of systems having finitely many roots. Our methods depend
on having approximations for all the roots, and these are
provided by the homotopy continuation methods of Sommese,
Verschelde and Wampler. We introduce approximate
equiprojectable decompositions for such systems, which
represent a generalization of the recently developed analogous
concept for exact systems. We demonstrate experimentally the
favourable computational features of this new approach, and
give a statistical analysis of its error. Our paper is
available at
http://publish.uwo.ca/~wwu26/
November 20, 2006, 11:15 am-12:15 pm, Lind Hall
409
Chehrzad Shakiban (IMA,
University of Minnesota)
http://www.ima.umn.edu/~shakiban/
Computations in classical invariant theory of binary
forms
Slides: pdf ppt
Abstract: Recent years have witnessed a reflourishing
of interest in classical
invariant theory, both as a mathematical subject and in
applications.
The applications have required a revival of the computational
approach,
a task that is now improved by the current availability of
symbolic
manipulation computer softwares. In this talk, we will review
the basic
concepts of invariants and covariants of binary forms, and
discuss some
elementary examples. We will then use the symbolic method of
Aronhold,
and algebrochemical methods of Clifford and Sylvester for
computations
and present some applications to syzygies and transvectants of
covariants.
November 29, 2006, 11:15 am-12:15 pm, Lind Hall
409
Alicia Dickenstein
(Departamento de Matemática, Universidad de Buenos Aires)
http://mate.dm.uba.ar/~alidick/
Tropical discriminants
Abstract:
The theory of A-discriminants is a far going generalization of
the
discriminant of a univariate polynomial, proposed in the late
80's by
Gel'fand, Kapranov and Zelevinsky, who also described many of
their
combinatorial properties. We present a new approach to this
theory
using tropical geometry.
We tropicalize the Horn-Kapranov uniformization, which allows us
to
determine invariants of A-discriminants, even if the actual
equations
are too hard to be computed.
Joint work with Eva Maria
Feichtner and Bernd
Sturmfels.
December 6, 2006, 11:15 am-12:15 pm, Lind Hall
409
Niels Lauritzen (Matematisk
Institut, Aarhus Universitet)
http://home.imf.au.dk/niels/
Gröbner walks and Scarf homotopies
Abstract: We give a light
introduction to Gröbner basis conversion
and outline emerging insights on the connection to a classical
algorithm by Scarf in optimization.
December 13, 2006, 11:15 am-12:15 pm, Lind Hall
409
Gennady Lyubeznik (School of
Mathmatics, University
of Minnesota)
Some algorithmic aspects of local cohomology
Abstract: One application of
local cohomology is that it provides a
lower bound on the number of defining equations of algebraic
varieties.
To be useful for this application, local cohomology must be
efficiently
computable. We will discuss some computability issues and
resulting
lower bounds for the number of defining equations of some
interesting
varieties.
January 10, 2007, 11:15 am-12:15 pm, Lind Hall
409
Anton Leykin (IMA Postdoc, University of Minnesota)
http://www.ima.umn.edu/~leykin/
Algorithms in algebraic analysis
Abstract:
In the first part of this talk I will give an introduction to
the
algorithmic theory of D-modules. This would include the
description of the properties of the rings of differential
operators, in particular, the ones that allow for computation
of
Gröbner bases.
The second part will show the applications of D-modules to the
computation of local cohomology of a polynomial ring at a given
ideal. The nonvanishing of the local cohomology module of a
certain
degree may answer the question about the minimal number of
generators for the ideal.
The presentation is going to be accompanied by the
demonstration of
the relevant computations in the D-modules for Macaulay
2
package.
January 24, 2007, 11:15 am-12:15 pm, Lind Hall
409
Mihai Putinar (Department of
Mathematics, University of California)
http://math.ucsb.edu/~mputinar/
Moments of positivity
Abstract:
The seminar will offer an unifying overview of the
theory
of positive functionals, the spectral theorem, moment
problems and
polynomial optimization. We will treat only the commutative
case, in the
following order:
- The integral
- Positive definite quadratic forms and the spectral
theorem
- Orhtogonal polynomials and Jacobi matrices
- Moment problems and continued fractions
- Polynomial optimization
- Open questions
We encourage the participants to have a look at Stieltjes'
classical memoir on continued fractions, available at:
http://www.numdam.org/numdam-bin/fitem?id=AFST_1894_1_8_4_J1_0
January 31, 2007, 11:15 am-12:15 pm, Lind Hall
229 [Note room change]
Jean Bernard Lasserre (LAAS,
Centre National de la Recherche)
/http://www.laas.fr/~lasserre/
SDP and LP-relaxations in polynomial optimization:
The power of real algebraic geometry
Summary: In this seminar we
consider the general
polynomial optimization problem: that is, finding
the GLOBAL minimum of a polynomial over a compact
basic semi-algebraic set, a NP-hard problem. We will
describe how powerfull representation results in real
algebraic geometry are exploited to build up a hierarchy of
linear or semidefinite programming (LP or SDP) relaxations,
whose monotone sequence of optimal values converges to the
desired value. A comparison with the usual Kuhn-Tucker local
optimality conditions is also discussed.
February 7, 2007, 11:15 am-12:15 pm,
EE/Sci 3-180 [Note room change]
Serkan Hosten (Department of
Mathematics, San Francisco
State University)
http://math.sfsu.edu/serkan/
An introduction to algebraic statistics
Talks
(A/V) Slides: pdf
Abstract: This will be a gentle
introduction to the applications
of algebraic geometry to statistics. The main goal of the talk
is to present statistical models, i.e. sets of probability
distributions (defined parametrically most of the time), as
algebraic
varieties. I will give examples where defining equations
of such statistical model varieties have been successfully
computed:
various graphical models and models for DNA sequence evolution.
I will also talk about the algebraic degree of maximum
likelihood
estimation with old and new examples.
February 14, 2007, 11:15 am-12:15 pm, Lind Hall
229
Stephen E. Fienberg
(Department of Statistics, Carnegie Mellon University)
http://www.stat.cmu.edu/~fienberg/
Statistical formulation of issues associated with multi-way
contingency tables and the links to algebraic geometry
Talks(A/V)
Slides: pdf
Abstract:
Many statistical problems arising in the context of
multi-dimensional
tables of non-negative counts (known as contingency tables)
have natural
representations in algebraic and polyhedral geometry. I will
introduce
some of these problems in the context of actual examples of
large sparse
tables and talk about how we have treated them and why. For
example, our
work on bounds for contingency table entries has been motivated
by
problems arising in the context of the protection of
confidential
statistical data results on decompositions related to graphical
model
representations have explicit algebraic geometry formulations.
Similarly,
results on the existence of maximum likelihood estimates for
log-linear
models are tied to polyhedral representations. It turns out
that there
are close linkages that I will describe.
February 21, 2007, 11:15 am-12:15 pm,
EE/CSci 3-180
Evelyne Hubert (Institut
National de Recherche en Informatique Automatique (INRIA)
Sophia Antipolis)
http://www-sop.inria.fr/cafe/Evelyne.Hubert/
Rational and algebraic invariants of a group
action
Slides: pdf Talks(A/V)
Abstract: We consider a rational
group action on the affine space and propose a
construction of a finite set of rational invariants and a
simple
algorithm to rewrite any rational invariant in terms of those
generators.
The construction can be extended to provide
algebraic foundations to Cartan's moving frame
method, as revised in [Fels & Olver 1999].
This is joint work with Irina Kogan, North Carolina State
University.
February 28, 2007, 11:15 am-12:15 pm,
EE/CSci 3-180
Peter J. Olver (School of
Mathematics, University of Minnesota)
http://www.math.umn.edu/~olver/
Moving frames in classical invariant theory and computer
vision
Talks(A/V)
Abstract:
Classical invariant theory was inspired by the basic problems
of equivalence and symmetry of polynomials (or forms) under the
projective group. In this talk, I will explain how a powerful
new approach to the Cartan method of moving frames can be
applied to classify algebraic and differential invariants for
very general group actions, leading, among many other
applications, to new solutions to the equivalence and symmetry
problems arising in both invariant theory, differential
geometry, and object recognition in computer vision.
March 14, 2007, 11:15 am-12:15 pm,
EE/CSci 3-180
Mordechai Katzman (Department of
Pure Mathematics, University of Sheffield)
http://www.katzman.staff.shef.ac.uk/
Counting monomials
Talks(A/V)
Abstract: The contents of this
elementary talk grew out of my need to explain to
non-mathematicians what I do for a living.
I will pose (and solve) two old chessboard enumeration problems
and a new
problem. We will solve these by counting certain monomials, and
this will
naturally lead us to the notion of Hilbert functions. With
these examples in
mind, we will try and understand the simplest of monomial
ideals, namely, edge
ideals, and discover that these are not simple at all! On the
way we will
discover a new numerical invariant of forests.
March 21, 2007, 11:15 am-12:15 pm,
EE/CSci 3-180
Thorsten Theobald (Fachbereich
Informatik und Mathematik, Goethe-Universität Frankfurt am Main)
http://www.math.uni-frankfurt.de/~theobald/
Symmetries in SDP-based relaxations for
constrained polynomial optimization
Talks(A/V)
Abstract: We consider the issue
of exploiting symmetries in the
hierarchy of semidefinite programming relaxations
recently introduced in polynomial optimization.
After providing the necessary background we focus
on problems where either the symmetric or
the cyclic group is acting on the variables
and extend the representation-theoretical
methods of Gatermann and Parrilo to constrained
polynomial optimization problems.
Moreover, we also propose methods to efficiently compute
lower and upper bounds for the subclass of problems where
the objective function and the constraints are described
in terms of power sums.
(Joint work with L. Jansson, J.B. Lasserre and C. Riener)
March 28, 2007, 11:15 am-12:15 pm,
EE/CSci 3-180
Seth Sullivant (Department of
Mathematics, Harvard University)
http://www.math.harvard.edu/~seths/
Algebraic geometry of Gaussian Bayesian networks
Talks(A/V)
Abstract: Conditional independence
models for Gaussian random variables
are algebraic varieties in the cone of positive definite
matrices. We
explore the geometry of these varieties in the case of Bayesian
networks,
with a view towards generalizing the recursive factorization
theorem.
When some of the random variables are hidden, non-independence
constraints
are need to describe the Bayesian networks. These
non-independence
constraints have potential inferential uses for studying
collections of
random variables. In the case that the underlying network is a
tree, we
give a complete description of the defining constraints of the
model and
show a surprising connection to the Grassmannian.
April 4, 2007, 11:15 am-12:15 pm,
EE/CSci 3-180
Frank Sottile (Department of
Mathematics, Texas A&M)
http://www.math.tamu.edu/~sottile/
Optimal fewnomial bounds from Gale dual polynomial
systems
Talks(A/V)
Abstract: In 1980, Askold Khovanskii established his fewnomial bound
for the number of real solutions to a system of polynomials,
showing that the complexity of the set of real solutions to
a system of polynomials depends upon the number of monomials
and not on the degree. This fundamental finiteness result
in real algebraic geometry is believed to be unrealistically large.
I will report on joint work with Frederic Bihan on a new
fewnomial bound which is substantially lower than Khovanskii's
bound and asymptotically optimal. This bound is obtained by
first reducing a given system to a Gale system, and then
bounding the number of solutions to a Gale system. Like
Khovanskii's bound, this bound is the product of an exponential
function and a polynomial in the dimension, with the exponents
in both terms depending upon the number of monomials. In our
bound, the exponents are smaller than in Khovanskii's.
I will also dicuss a continuation of this work with J Maurice
Rojas in which we show that this fewnomial bound is optimal,
in an asymptotic sense. We also use it to establish a
new and significantly smaller bound for the total Betti
number of a fewnomial hypersurface.
Conditional independence
models for Gaussian random variables
are algebraic varieties in the cone of positive definite
matrices. We
explore the geometry of these varieties in the case of Bayesian
networks,
April 11, 2007, 11:15 am-12:15 pm,
EE/CSci 3-180
Saugata Basu (School of
Mathematics, Georgia Institute of Technology)
http://www.math.gatech.edu/~saugata/
Combinatorial complexity in o-minimal geometry
Talks(A/V)
Abstract:
We prove tight bounds on the combinatorial and topological
complexity
of sets defined in terms of n definable sets belonging to
some fixed
definable family of sets in an o-minimal structure. This
generalizes the
combinatorial parts of similar bounds known in the case of
semi-algebraic and
semi-Pfaffian sets, and as a result vastly increases the
applicability
of results on combinatorial and topological complexity of
arrangements studied in discrete and computational geometry. As
a sample
application, we extend a Ramsey-type theorem due to Alon et al.
originally
proved for semi-algebraic sets of fixed description complexity
to this more general setting.
The talk will be self-contained and I will go over the basic
definitions of
o-minimality for those who are unfamiliar with the notion.
April 25, 2007, 11:15am-12:15 pm,
Lind Hall 229
Gregorio Malajovich (Departamento
de Matemática Aplicada, Universidade Federal do Rio de Janeiro)
http://www.labma.ufrj.br/~gregorio/
On sparse polynomial systems, mixed volumes and condition
numbers
Abstract: pdf
April 26, 2007, 10:15-11:10 am, EE/CSci 3-180
J. Maurice Rojas (Department of
Mathematics, Texas A&M
University)
http://www.math.tamu.edu/~rojas/
Random polynomial systems and balanced metrics on toric
varieties
Abstract: Suppose c0,...,cd are independent identically distributed
real Gaussians with mean 0 and variance 1. Around the 1940s,
Kac and Rice
proved that the expected number of real roots of the
polynomial
c0 + c1 x + ... + cd
xdi, then the
expected
number of real roots is EXACTLY the square root of d. Aside
from the cute
square root phenomenon, Kostlan also observed that the
distribution function
of the real roots is constant with respect to the usual
metric on the real projective line.
The question of what a "natural" probability measure for
general
multivariate polynomials then arises. We exhibit two
(equivalent)
combinatorial constructions that conjecturally yield such a
measure. We
show how our conjecture is true in certain interesting
special cases, thus
recovering earlier work of Shub, Smale, and McLennan. We
also relate our
conjecture to earlier asymptotic results of Shiffman and
Zelditch on random sections of holomorphic line bundles.
This talk will deal concretely with polynomials and Newton
polytopes, so no background on probability or algebraic
geometry is assumed.
May 2, 2007, 11:15am-12:15 pm,
Lind Hall 409
Patricia Hersh (Department of
Mathematics, Indiana University)
http://mypage.iu.edu/~phersh/
A homological obstruction to weak order on trees
Abstract:
When sorting data on a network of computers, it is natural to
ask which
data swaps between neighbors constitute progress. In a linear
array,
the answer is simple, by virtue of the fact that permutations
admit
pleasant notions of inversions and weak order. I will discuss
how the
topology of chessboard complexes constrains the extent to which
these
ideas may carry over to other trees; it turns out that there
are
homological obstructions telling us that a tree does not admit
an
inversion function unless each node has at least as much
capacity as
its degree minus one. On the other hand, we construct an
inversion
function and weak order for all trees that do meet this
capacity
requirement, and we prove a connectivity bound conjectured by
Babson
and Reiner for 'Coxeter-like complexes' along the way.
May 9, 2007, 11:15am-12:15 pm,
Lind Hall 409
Alexander Yong (School of
Mathematics, University of Minnesota)
http://www.math.umn.edu/~ayong/
Schubert combinatorics and geometry
Abstract: The topic of Schubert
varieties of homogeneous spaces G/P
is at the interface between algebraic geometry and
combinatorics.
I'll describe work on two themes.
The first is Schubert calculus: counting points in
intersections
of Schubert varieties. A goal has been combinatorial rules for
these
computations. I'll explain the carton rule which manifests
basic
symmetries
of the numbers for the Grassmannian case; this version also has
the
advantage
of generalizing to (co)minuscule G/P.
The second concerns singularities of Schubert varieties. I'll
give a
combinatorial framework for understanding invariant of
singularities via a
notion we call interval pattern avoidance.
The first half of this talk is joint work with Hugh Thomas (U.
New
Brunswick)
while the second half is joint work with Alexander Woo (UC
Davis).
May 15, 2007, 1:15-2:15 pm,
Lind Hall 305
Sandra Di Rocco (Department of
Mathematics, KTH)
http://www.math.kth.se/~dirocco/
Discriminants, dual varieties and toric geometry
Abstract: Given an algebraic
variety, embedded in projective space, the closure of all
hyperplanes
tangent at some non singular point is called the dual variety.
A general
embedding has
dual variety of co-dimension one (in the dual projective
space) and hence
defined by
an irreducible homogeneous polynomial, called the discriminant.
The study of the exceptional embeddings, i.e. the ones having
dual variety
of lower dimension,
is a very classical problem in algebraic geometry, still open
for many
classes of varieties.
I will explain the problem and give the solution for the class
of non
singular toric varieties.
May 16, 2007, 11:15am-12:15 pm,
Lind Hall 409
Peter Bürgisser (Mathematik -
Informatik, Universität Paderborn)
http://math-www.uni-paderborn.de/agpb/members.html
Average volume, curvatures, and Euler characteristic of
random real algebraic varieties
Abstract: We determine the
expected curvature polynomial of random real projective
varieties given as the zero set of independent random
polynomials with
Gaussian distribution, whose distribution is invariant under
the action
of the orthogonal group. In particular, the expected Euler
characteristic of such random real projective varieties is
found. This
considerably extends previously known results on the number of
roots,
the volume, and the Euler characteristic of the solution set of
random
polynomial equations.
May 23, 2007, 11:15am-12:15 pm,
Lind Hall 409
Ioannis Z. Emiris (Department of
Informatics & Telecommunications,
National Kapodistrian University of Athens)
http://cgi.di.uoa.gr/~emiris/index-eng.html
On the Newton polytope of specialized resultants
Slides: pdf
Abstract: We overview basic
notions from sparse, or toric, elimination
theory and
apply them in order to predict the Newton polytope of the
sparse resultant.
We consider the case when all but a constant number of
resultant parameters
are specialized. Of independent interest is the problem of
predicting the
support of the implicit equation of a parametric curve or
surface. We bound
this support by a direct approach, based on combinatorial
geometry. The talk
will point to various open questions.
June 6, 2007, 11:15am-12:15 pm,
Lind Hall 302
Gabriela Jeronimo (Departamento
de Matemática, Facultad de Ciencias Exactas y Naturales, University of
Buenos Aires)
A symbolic approach to sparse elimination
Abstract:
Sparse elimination is concerned with systems of
polynomial
equations in which each equation is given by a polynomial
having non-zero
coefficients only for those monomials lying in a
prescribed set.
We will discuss a new symbolic procedure for solving
zero-dimensional
sparse polynomial systems by means of deformation techniques.
Roughly speaking, a deformation method to solve a
zero-dimensional
polynomial equation system works as follows: the input system
is regarded
as a member of a parametric family of zero-dimensional
systems. Then, the
solutions to a particular parametric instance which is "easy
to solve"
are computed and, finally, these solutions enable one to
recover the
solutions of the original system.
The algorithm combines the polyhedral deformation introduced
by Huber and
Sturmfels with symbolic techniques relying on the
Newton-Hensel lifting
procedure. Its running time can be estimated mainly in terms
of the input
length and two invariants related to the combinatorial
structure
underlying the problem.
June 20, 2007, Lind Hall 305
Frank Sottile (Department of
Mathematics
Texas A&M University)
http://www.math.tamu.edu/~sottile/
IMA program on applications of algebraic geometry
(Special seminar as part of the annual
PIC-IAB meeting)
Slides: small.pdf talk.pdf
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