Algebraic Geometry SeminarMondays 10:2011:10 AM

Picture courtesy of Frank Sottile 
Date  Speaker  Title (click for abstract) 

8/27/07  Frank Sottile, TAMU  Gale duality for complete intersections 
9/3/07  Zach Teitler, TAMU  The nef cone volume of generalized Del Pezzo surfaces 
9/10/07  J.M. Landsberg, TAMU  Matchgates, statistics and compact Hermitian symmetric spaces 
9/17/07  Paulo LimaFilho, TAMU  Integral Deligne cohomology for real projective varieties 
9/24/07  Chris Hillar, TAMU  Rational sums of squares 
10/1/07  Zach Teitler, TAMU  The nef cone volume of generalized Del Pezzo surfaces, 2 
10/8/07  Maurice Rojas, TAMU  padic Shadows of Reality 
10/15/07  Laura Matusevich, TAMU  Weyl closure of hypergeometric systems 
10/22/07  Frank Sottile, TAMU  KhovanskiiRolle homotopies for real solutions 
10/29/07  J.M. Landsberg, TAMU  The Debarrede Jong conjecture on hypersurfaces with too many lines 
11/5/07  No meeting.  
11/12/07  Zach Teitler, TAMU  Hilbert functions of fat point schemes supported on linear configurations in P^2 
11/19/07  Louiza Fouli, UT Austin  The core of ideals 
11/26/07  Luke Oeding, TAMU  Principal Minors of Symmetric Matrices and Geometry 
Gale duality for polynomial systems is an elementary
reformulation of a system of polynomial equations as a system of
equations involving rational master functions in the complement of
a hyperplane arrangement. Some properties of the original system
are easier to understand in the Gale dual system. In the first part
of this talk, I will describe this Gale duality, look at some
examples of this construction, and give some elementary
consequences.
This is joint work with Frédéric
Bihan, and is described more completely in this preprint.
TOP
We compute a naturally defined measure of the size of the nef
cone of a generalized Del Pezzo surface. The resulting number
appears in a conjecture of Manin on the asymptotic behavior of the
number of rational points of bounded height on the surface. The nef
cone volume is computed using two elementary techniques: a
simplicial decomposition of the cone, and the Weyl group of a root
system associated to the configuration of (2)curves on the
surface. Over a nonclosed field this root system is not
necessarily simplylaced.
This is joint work with Ulrich Derenthal and Michael Joyce and is
described in this preprint.
TOP
I will explain how problems in statistics and computer science,
respectively the recognition problem for principal minors (see e.g.
arXiv:math/0604374) and
characterization of matchgate relations (see e.g. pages.cs.wisc.edu/~jyc/papers/icalptalk06.pdf),
are related to a uniform construction of compact Hermitian
symmetric spaces discovered with L. Manivel. (Selecta Math. (N.S.)
8 (2002), no. 1, 137159, also see arXiv:math/0203260)
TOP
We develop an 'integral Deligne cohomology theory' for real varieties which bears to Bredon cohomology the same relation that ordinary Deligne cohomology for complex varieties bears to singular cohomology. The theory has a wide range of connections ranging from equivariant topology (via Bredon equivariant cohomology), through complex differential geometry (via holomorphic line bundles with quadratic forms and holomorphic connections) to number theory (via Milnor Ktheory of number fields and regulator maps). We will present time permitting many examples. In a forthcoming work we show that the cycle map from motivic cohomology to Bredon cohomology factors through our Deligne cohomology groups.
In EsnaultVieweg's survey on Deligne they outline a theory for
real varieties, as well. Such a variant would be related to
Liechtenbaum's etale motivic cohomology and the Borel version of
equivariant cohomology. The difference between their version and
ours stems from the difference between the etale and Nisnevich
topologies.
TOP
In recent years, techniques from semidefinite programming have
produced numerical algorithms for finding representations of
positive semidefinite polynomials as sums of squares. These
algorithms have many applications in optimization, control theory,
quadratic programming, and matrix analysis. One major drawback with
these algorithms is that their output is, in general, numerical.
For many applications, however, exact polynomial identities are
needed. In this regard, Sturmfels has asked whether a
representation with real coefficients implies one over the
rationals. We discuss recent progress on this question; in
particular, we outline a positive answer to this question for
totally real number fields.
TOP
Briefly, I will continue and finish my talk of September 3rd.
I will show how to compute the nef cone volume by decomposing
the nef cone into pieces corresponding to chambers of a Weyl group
associated to the generalized Del Pezzo surface. If the surface is
defined over a nonalgebraically closed field, then there is also a
Galois action in the picture, and the Weyl group corresponds to a
root system which is not necessarily simplylaced.
TOP
We introduce and explore some of the similarities and differences between real and padic algorithmic algebraic geometry. After briefly reviewing some quantitative results on sparse systems of polynomial equations over the reals and padic rationals, we then focus on one variable: How hard is it to decide if one polynomial in one variable has a root?
We will see that the complexity of this question is still quite open over the real numbers, but admits a more decisive answer over the padic rationals. This leads us to some natural questions, in higher dimensions, over both the reals and the padics.
We assume no background in number theory or complexity
theory.
TOP
Ahypergeometric systems are parametric systems of PDE built
from toric ideals. I will outline a proof that these systems are
Weyl closed for very generic parameters, i.e. they are the
differential annihilators of their solution spaces. No background
is required for this talk, but a familiarity with toric ideals will
help.
TOP
A general hypersurface of degree d in CP^n will have a 2nd3 dimensional space of lines (embedded P^1's) on it. O. Debarre and J. de Jong independently conjectured that a hypersurface of degree d\leq n in CP^n having a larger family of lines on it cannot be smooth. I will explain joint work with O. Tommasi where we prove this conjecture in a slightly sharper form.
In the course of proving the conjecture, we developed new
methods using projective differential geometry to locate
singularities on varieties uniruled by lines which we expect to
have significant applications.
TOP
Let $R$ be a Noetherian local ring with infinite residue field $k$ and $I$ an $R$ideal. The ideal $J$ is a \textit{reduction} of $I$ if $J \subset I$ and $I^{r+1}=JI^{r}$ for some positive integer $r$. A reduction can be thought of as a simplification of the ideal $I$. The notion of a reduction for an ideal was introduced by D. Northcott and D. Rees in order to study multiplicities. Reductions are connected to the study of blowup algebras such as the Rees ring $\mathcal{R}(I)=R[It]$ of $I$, and the associated graded ring ${\rm{gr}}_{I} (R)=R[It]/IR[It]$ of $I$.
In general minimal reductions are not unique. To remedy this
lack of uniqueness, one considers the intersection of all
reductions, namely the \textit{core} of the ideal,
${\rm{core}}(I)$. This object, that appears naturally in the
context of the Brian\c conSkoda theorem, encodes information about
all possible reductions. We present some recent work on the shape
of the core of ideals.
TOP
A principal minor of a matrix is the determinant of a submatrix which has the same row and column set. Given an n by n symmetric matrix, one can calculate all of its principal minors and store this information in a vector of length 2^n. A natural question is: given a vector of length 2^n, can one find a matrix that has those principal minors? Does such a matrix always exist? If not, can we completely describe a set of conditions that will guarantee existence? In more geometric language, we ask, What are the minimal generators of the homogeneous ideal of the variety of principal minors of symmetric matrices?
There is a lot of underlying structure in this problem which
leads to some beautiful geometry and representation theory and
we'll explore some of it in this talk.
TOP
For more information, email Zach Teitler.