Notre Dame
Algebraic Geometry/ Commutative
Algebra Seminar
Spring 2009
(3-4pm on Wednesday)
|
|||
|
Date |
Time/Place |
Speaker |
Title |
|
Jan 14 |
|
|
|
|
Jan 21 |
|
|
(no seminar) |
|
Jan 28 |
|
|
(no seminar) |
|
Feb 4 |
3pm 258 Hurley |
Uli Walther (Purdue) |
|
|
Feb 11 |
3pm 258 Hurley |
Christine Berkesch (Purdue) |
|
|
Feb 18 |
|
|
(no seminar) |
|
Feb 25 |
|
|
(no seminar) |
|
Mar 4 |
|
|
|
|
Mar 6-8 |
|
|
|
|
Mar 11 |
|
|
(Spring break, no seminar) |
|
Mar 18 (colloquium talk) |
|
Bernd Ulrich (Purdue) |
|
|
Mar 23 |
3pm 258 Hurley |
Kuei-Nuan Lin (Purdue) |
|
|
Mar 25 |
3pm 258 Hurley |
Lance Bryant (Purdue) |
|
|
Apr 1 |
|
|
|
|
Apr 6 (colloquium talk) |
4pm 258 Hurley |
Tommaso de Fernex ( |
|
|
Apr 8 |
|
|
|
|
Apr 15 |
|
|
|
|
Apr 22 |
3pm 258 Hurley |
Sandra Di Rocco (KTH |
|
|
Apr 29 |
3pm 258 Hurley |
Bonnie Smith (Notre Dame) |
|
|
May 6 |
|
|
|
Past seminars: Fall 2008, Spring 2008.
Student Alg. Geom. - Comm. Alg. Seminar.
Abstract of Talks:
Feb 4. Uli Walther (Purdue) : Hypergeometric functions: the GKZ-perspective
Abstract: Starting with some classical hypergeometric functions, we explain
how to
derive their classical univariate differential equations. A severe change
of coordinates transforms this ODE into a system of PDE's that has nice
geometric aspects. This type of system, called A-hypergeometric, was
introduced by Gelfand, Graev, Kapranov and Zelevinsky in about 1985. We
explain some basic questions regarding these systems. These are addressed
through homology, combinatorics, and toric geometry.
Feb 11. Christine Berkesch (Purdue): The rank of a hypergeometric system
Abstract: An A-hypergeometric system is a parametric system of PDEs arising
from
a toric ideal. The dimension of its solution space, called its rank,
is constant for generic parameters. I will discuss the combinatorial
nature of its rank at non-generic parameters.
Mar 23. Kuei-Nuan Lin (Purdue): Diagonal ideals of determinantal rings
Abstract: Let k be a field, X = (x_ij) an m by n matrix of variables over k,
I =
It(X),the ideal of k[{x_ij}] generated by the t by t minors of X, and R
= k[{x_ij}]/I. We consider the diagonal ideal D of R tensor R. The
special fiber ring of D is the homogeneous coordinate ring of the secant
variety of the determinantal variety Z =V (I). We know that secant
variety SecZ = V (I_2t−1(T)) where T = (t_ij) is a m by n matrix. We
extend this fact by showing that the Rees algebra of D, R(D) is
Cohen-Macaulay and moreover D is a an ideal of linear type when t = m,
which means that the natural map from the symmetric algebra Sym(D) onto
the Rees algebra R(D) is an isomorphism.
Apr 6. Tommaso de Fernex
(
Abstract: A complex
projective manifold is said to be Fano if the determinant of
its tangent bundle is
"positive". In algebraic geometry, this is the
equivalent of a differentiable manifold with positive curvature. Fano
manifolds occupy a special place in classification theory: projective
spaces belong this class, and in dimension two Fano surfaces have been
extensively studied in the literature. It is known that projective
spaces are rigid under small deformations, and even though some of the
Fano surfaces are not rigid, the essential aspects of their geometry
remain unaltered under small deformations. This appears to be a
general property of Fano manifolds. We will explain this phenomenon,
presenting some interesting
examples and discussing what is known and
what is expected. This is joint work with C. Hacon.
Apr 22. Sandra Di Rocco (KTH Stockholm): Classifying polytopes via Toric fibrations.
Abstract: A fibration between toric varieties, embedded in
projective
space, can be described by certain fibered polytopes. When the fibration
has a projective space as generic fiber, embedded linearly, the polytope
is called a strict Cayley polytope. It turns out that this class of
polytopes encodes exceptional geometrical properties of the corresponding
toric embeddings. Batyrev and Nill have recently conjectured a relation
between the degree of a convex polytope and the property of having a
Cayley-structure. A proof of the conjecture for smooth polytopes will be
presented. The result is obtained by translating the problem into toric
geometry. This is joint work with A. Dickenstein and R. Piene.
Apr 29. Bonnie Smith (Notre Dame): Strongly Stable Ideals and the Core
Abstract: Ideals in k[X_1,...,X_d] which are strongly
stable of degree two (SS2) are those which can be pictured by tableaux of a
certain shape. We will see how the tableau associated to an SS2 ideal I
gives us a primary decomposition for I, as well as other information necessary
for computing the core of I. We will then discuss residual intersections
of ideals (a generalization of linkage), and conclude by showing how results
from this topic help us to find the core of an SS2 ideal.
For problems with this page, email Nero Budur.