Commutative Algebra & Algebraic Geometry Seminar
This is a joint seminar between the CUNY Graduate Center and Rutgers University (New Brunswick Campus).
The seminar will be on Fridays, and the meetings will alternate between the CUNY Graduate Center and Rutgers University. See the schedule below for more precise information. The meetings at the CUNY Graduate Center will be 4-5 PM in room 6417. The meetings at Rutgers will be 1-2 PM in Hill 425.
The CUNY Graduate Center is located in 365 Fifth Avenue,
New York, NY 10016.
The Department of Mathematics at Rutgers-New Brunswick is located in the
Hill Center on the Busch Campus of Rutgers University in Piscataway, NJ.
Click here
for directions.
Organizers:
Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu
Jooyoun Hong, Southern Connecticut State University, hongj2@southernct.edu
Hans Schoutens, New York City College of Technology and the Graduate Center (CUNY), hschoutens@citytech.cuny.edu
SCHEDULE, SPRING 2008
Friday, January 25 at Rutgers University, 1-2 PM in Hill 425.
Speaker: Wolmer Vasconcelos, Rutgers University.
Title: The Chern numbers of a local ring (I).
Friday, February 1 at the CUNY Graduate Center, 4-5 PM in room 6417.
Organizational Meeting.
Note: Today's talk in the CUNY Logic Workshop will be of interest to Commutative Algebraists and Algebraic Geometers. Click here for more information.
Friday, February 8 at Rutgers University, 1-2 PM in Hill 425.
Speaker: Michael Ziewe, Rutgers University.
Title: The lattice of subfields of K(x).
Friday, February 15 at the CUNY Graduate Center, 4-5 PM in room 6417.
The seminar is cancelled. The talk is postponed to April 18.
Note: Today Hans Schoutens will give a talk in the CUNY Logic Workshop which will be of interest to Commutative Algebraists and Algebraic Geometers. Click here for more information.
Friday, February 22 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY.
Title: Tight closure, an introduction
Abstract: I will give a short introduction to
tight closure theory, a recent method developed by Hochster and Huneke,
formalizing former work of Kunz, Szpiro, Peskine, Hochster, et al., in which
the Frobenius homomorphism (=the $p$-th power map in a ring of characteristic
$p$) and its action on (co-)homology play a prominent role for studying
singularities and other homological properties. The idea is to define a closure
operation on ideals which refines integral closure. Certain singularities can
then be characterized by the tight closedness of certain types of ideals. I give
an axiomatic treatment, which allows, in principle, to extend the theory to
other situations than positive characteristic (I will just briefly describe the
characteristic zero case). The power of this theory lies in the ease with which
one can establish deep theorems, such as the CM-ness of quotient singularities,
the Briancon-Skoda theorem, the Ein-Lazarfeld-Smith theorem on symbolic powers,
...
Note: Today's talk in the CUNY Logic Workshop will be of interest to Commutative Algebraists and Algebraic Geometers. Click here for more information.
Friday, February 29 at Rutgers University, 1-2 PM in Hill 425.
Speaker: Laura Ghezzi, New York City College of Technology, CUNY.
Title: A generalization of the Strong Castelnuovo Lemma, (I).
Abstract: We are interested in the “linear part” of the minimal free resolution of a set of distinct points in the n-dimensional projective space. This study has been initiated by Green and it lead to very deep results relating the geometric properties of the variety with the existence of a long linear strand in the resolution. The Strong Castelnuovo Lemma (SCL) shows that if the points are in general position, then there is a linear syzygy of order n-1 if and only if the points are on a rational normal curve. Cavaliere, Rossi and Valla conjectured that if the points are not necessarily in general position the possible extension of the SCL should be the following: There is a linear syzygy of order n-1 if and only if either the points are on a rational normal curve or in the union of two linear subspaces whose dimensions add up to n. In this talk we prove the Conjecture for n+4 points, which is the first open and interesting case. If time permits we also discuss the proof for any number of points.
Friday, March 7 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Ray Hoobler, City College and the Graduate Center, CUNY.
Title: An Introduction to Nori's Fundamental Group Scheme: The Algebraic Fundamental Group.
Abstract: This talk will discuss the
algebraic geometer's version of the Galois group. We will identify the
categorical properties of finite sets and the
necessary properties that a fibre functor to sets using a base point of a scheme
should have. The basic result then shows how the algebraic fundamental group is
constructed from automorphisms of finite sets with group actions. We will use
the Fulton-Hansen connectedness theorem to prove a Lefschetz type result for
projective varieties.
Friday, March 14 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Ray Hoobler, City College and the Graduate Center, CUNY.
Title: Tannakian Categories and the Fundamental Group Scheme.
Abstract: This talk will discuss how to
extend the construction of the algebraic fundamental group to affine group
schemes by identifying the categorical
properties of representations of an algebraic group G and the necessary
properties that a fibre functor with values in vector spaces over an
algebraically closed field should have. We will use Nori's Fundamental Group
Scheme to illustrate the properties that a Tannakian category should have. A
recent result will allow us to extend the Lefschetz result of the first talk to
this setting.
Friday, March 21. No meeting this week.
Friday, March 28 at Rutgers University, 1-2 PM in Hill 425.
Speaker: Jooyoun Hong, Southern Connecticut State University.
Title: Homology and Elimination.
Note: We will also have the Annual Research Conference at the New York City College of Technology.
Friday, April 4 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Michael Ziewe, Rutgers University.
Title: Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture.
Friday, April 11 at Rutgers University, 1-2 PM in Hill 425.
Speaker: Joe Brennan, University of Central Florida.
Title: Cut ideals of books and outerplanar graphs.
Friday, April 18 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Uma Iyer, Bronx Community College, CUNY.
Title: Quantum differential operators on the quantum plane.
Note: Today Hans Schoutens will give a talk in the CUNY Logic Workshop which will be of interest to Commutative Algebraists and Algebraic Geometers. Click here for more information.
Friday, April 25 at Rutgers University, 1-2 PM in Hill 425.
Speaker: Aihua Li, Montclair State University.
Title: Symbolic Powers of Radical Ideals.
Abstract: M. Hochster established several
criteria on when for a prime ideal P in a Noetherian integral domain R, the n^{th}
power P^n of P equals the n^{th} symbolic power P^{(n)}of P for every positive
integer n.
He used a so-called test sequence of ideals in a polynomial ring over R to
determine whether P^n = P^{(n)} for all n. We study test sequences for any ideal
in a Noetherian R and then extend Hochster's criteria to radical ideals of R.
Friday, May 2. No meeting this week.
Note: The New York City College of Technology is hosting the Second NYWIMN Conference (New York Women in Mathematics Network).
Friday, May 9 at Rutgers University, 1-2 PM in Hill 425.
Speaker: Aron Simis, Purdue University and Universidade Federal de Pernambuco, Brazil.
Title: Fitting ideals and analytic spread of modules.
Friday, May 16 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Yalın Fırat Çelikler, New York City College of Technology, CUNY.
Title: From Varieties to Inequalities.
Abstract: One of the interest areas of Model Theory is the study of the definable sets over structures (fields, rings, etc.) in certain languages. For example, if we just allow constants and operations of multiplication and addition in our language and stay away from projections, the sets we can define over fields are the boolean combinations of varieties. However, to study the valued fields, a richer language, one with the valuation function and an order relation on the value group, is more suitable. Also, as we can talk about convergence over valued fields, one can add the elements of certain power series rings (eg: Tate Algebras) to obtain an analytic language as opposed to an algebraic one which contains only polynomials as terms. I will be talking about how one can still use the classical tools of algebraic geometry to understand the definable sets in this setting despite the extra complexities of the enriched language.