Commutative Algebra & Algebraic Geometry Seminar
The seminar will be on Fridays 4-5 PM at the CUNY Graduate Center in room 6417.
The
CUNY Graduate Center is located in 365 Fifth Avenue,
New York, NY 10016.
Organizers:
Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu
Hans Schoutens, New York City College of Technology and the Graduate Center (CUNY), hschoutens@citytech.cuny.edu
Janet Striuli, Fairfield University, jstriuli@mail.fairfield.edu
SPRING 2009
Friday, January 30
Speaker: Samar Elhitti, New York City College of Technology, CUNY.
Title: Algebraic resolution of formal ideals along a valuation, part I.
Abstract: The valuation theoretic analogue of
the problem of Resolution of Singularities is the problem of Local
Uniformization. Zariski proved local uniformization (in characteristic zero) in
1944. His proof gives a very detailed analysis of rank 1 valuation, and produces
a resolution which reflects invariants of the valuation.
A “natural” generalization of local uniformization on completions fails. In this
talk, we present a counter example which motivates our study of certain formal
ideals known as Prime Ideals of Infinite value. We extend the definition of such
ideals to arbitrary rank valuations and extend Zariski’s methods to give a proof
of local uniformization which reflects important properties of the valuation. We
simultaneously resolve the centers of all the composite valuations, and resolve
these formal ideals in question.
Friday, February 6
Speaker: Samar Elhitti, New York City College of Technology, CUNY.
Title: Algebraic resolution of formal ideals along a valuation, part II.
Friday, February 13
Speaker: Lars Winther Christensen, Texas Tech University.
Title: Auslander's Conjecture on Vanishing of Cohomology.
Abstract: Auslander conjectured that every Artin algebra satisfies a certain condition AC on vanishing of cohomology of finitely generated modules. The failure of this conjecture---by a 2003 counterexample due to Jorgensen and \c{S}ega---motivates the question addressed in this talk, namely, what is special about AC-rings? Among other things, we will see that the Auslander-Reiten conjecture holds over rings that have the AC property.
Friday, February 20
Speaker: Laurentiu Maxim, Lehman College, CUNY.
Title: Generating series for Hodge polynomials of symmetric products of varieties, part I.
Abstract: I will discuss a very general formula for the generating series of "genera with coefficients", which holds for complex quasi-projective varieties with any kind of singularities, and which includes many of the classical results in the literature as special cases. Important specializations of this result include generating series for extensions of Hirzebruch's genus to the singular setting and, in particular, generating series for Goresky-MacPherson signatures of complex projective varieties. This is joint work with J. Schuermann.
Friday, February 27
Speaker: Laurentiu Maxim, Lehman College, CUNY.
Title: Generating series for Hodge polynomials of symmetric products of varieties, part II.
Friday, March 6
Speaker: Michael Temkin, University of Pennsylvania.
Title: Inseparable local uniformization.
Note: The seminar today will be part of the Colloquiumfest.
Friday, March 13
Speaker: Fabrizio Zanello, Michigan Tech University.
Title: On the structure of pure O-sequences.
Friday, March 20
Speaker: Janet Striuli, Fairfield University.
Title: Canonical module and totally reflexive modules over Teter rings.
Friday, March 27
No meeting this week. We have the Annual Research Conference at the New York City College of Technology.
Friday, April 3
Speaker: Gunther Cornelissen, University of Utrecht, The Netherlands.
Title: Toroidal automorphic forms and the Riemann hypothesis for some function fields of curves.
Abstract: In the 1970's, Don Zagier
introduced toroidal automorphic forms to study the zeros of zeta functions. An
automorphic form on GL(2) is toroidal if all its right translates integrate to
zero over all nonsplit tori in GL(2). In the upper half plane, this corresponds
to summing over CM-points (for negative discriminant), or integrating along
geodesics (for positive discriminant).
The link with zeta functions is provided by a result of Hecke: an Eisenstein
series is toroidal if its weight is a zero of the zeta function of the
corresponding field.
We consider this theory for function fields of certain curves over finite
fields. The toroidal integrals corresponds to sums over certain vector bundles
on the curve.
We compute the space of toroidal automorphic forms for the function field of
three elliptic curves over finite fields. The method is elementary: we reduce
the vanishing of toroidal integrals to an
infinite system of linear equations on some graph. For this, one has to
understand the moduli of certain vector bundles on the elliptic curve.
We deduce an `"automorphic'' proof for the Riemann hypothesis for the zeta
function of those curves.
Joint work with Oliver Lorscheid (arxiv:math/0710.2994).
Note: This is a joint talk with the Collaborative Number Theory Seminar. We will meet as usual in room 6417.
Friday, April 10
No meeting this week.
Friday, April 17
No meeting this week.
Friday, April 24
Speaker: Li Guo, Rutgers University.
Title: Commutative Rota-Baxter algebras.
Abstract: Rota-Baxter algebra is an algebraic
abstraction of the integral calculus in analog to differential algebra as an
abstraction of the differential calculus.
Since its introduction in 1960, Rota-Baxter algebra has found many applications
in mathematics and physics. The commutative algebra study of Rota-Baxter
algebras began with the work of Cartier and Rota on the construction of free
commutative Rota-Baxter algebras in the 1970s and continued in a series of
papers since the 1990s by several authors, including Guo and his coauthors.
These free objects are related to the shuffle product and its generalizations
and have been described in terms of Lyndon words. We will discuss some of these
results.
Friday, May 1
Speaker: G. Michael Guy, Queensborough Community College, CUNY.
Title: Moduli of weighted stable curves and maps and some interesting intersections.
Abstract: A moduli
space is an algebraic tool which is useful in understanding the "classification"
of all curves (and other objects, as well). We will discuss the moduli space of
(weighted) stable curves and, time permitting, (weighted) stable maps. In
addition to studying the algebraic structure of the spaces, we will discuss the
intersection of their psi classes which plays an essential role in Gromov-Witten
theory. We will give a combinatorial description for some of these
intersections, and show how some interesting calculations can be attained by
repeated application of a wall-crossing formula for weighted stable curves and
maps.
We will develop these ideas with a "beginner" in mind and hope everyone will
find ample satsifaction in this treatment. This is based on joint work with V.
Alexeev.
Friday, May 8
Speaker: Jonathan Cornick, Queensborough Community College, CUNY.
Title: Modules of Finite Projective Dimension
over Generalizations of Group Algebras.
Friday, May 15
Speaker: Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY.
Title: On the action of ultra-Frobenius on cohomology and the study of rational singularities.
Abstract: By the work of Peskine, Szpiro,
Hochster, Roberts, et al, we know how useful it is to have the Frobenius act on
(co)homology. Tight closure theory has turned this insight into a efficient and
versatile tool. Thus, using tight closure theory, Hochster and Huneke reproved
the celebrated result of Hochster-Roberts that a quotient singularity is
Cohen-Macaulay, and Karen Smith reproved the positive characteristic part of
Boutot's generalization of this result that quotient singularities are
(pseudo-)rational. Here, we mean by a quotient singularity the orbit scheme of
the action of a linearly reductive algebraic group acting rationally on a smooth
scheme. To transfer the proof to characteristic zero, however, classical tight
closure theory does not work, and this is where the ultra-Frobenius comes into
play. It is obtained as the ultraproduct of Frobenii on rings of varying
positive characteristic, and acts on a faithfully flat overring of the original
coordinate ring. Nonetheless, the previous theory still applies if instead of
working with ordinary cohomology, we now use ultra-cohomology.
Thus we can now prove that quotient singularities are rational without any deep
vanishing theorems, and even for arbitrary schemes (Boutot only proved it for
schemes of finite type over $C$). Kawamata observed that if the group is finite,
the quotient singularity is even log-terminal. We prove this in general, under
the additional assumption that the quotient singularity is $Q$-Gorenstein (this
latter condition is automatically satisfied for finite group actions).
FALL 2008
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 2-3 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
Janet Striuli, Fairfield University, jstriuli@mail.fairfield.edu
Friday, September 5 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Laura Ghezzi, New York City College of Technology, CUNY.
Title: Minimal free resolutions of points in the projective space.
Friday, September 12 at Rutgers University, 2-3 PM in Hill 425.
Organizational meeting and working seminar on Hilbert coefficients (Laura Ghezzi, Jooyoun Hong and Wolmer Vasconcelos).
Friday, September 19 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Janet Striuli, Fairfield University.
Title: Representation Theory in commutative
algebra.
Friday, September 26 at Rutgers University, 2-4 PM in Hill 425.
Speakers: Laura Ghezzi and Jooyoun Hong.
Title: Sally modules.
Friday, October 3 at Rutgers University, 2-3 PM in Hill 425.
Working seminar on Hilbert coefficients (Laura Ghezzi, Jooyoun Hong and Wolmer Vasconcelos).
Friday, October 10 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Andrew Crabbe, Syracuse University.
Title: Building large indecomposable modules.
Friday, October 17 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Uma Iyer, Bronx Community College, CUNY.
Title: Volichenko differential operators.
Friday, October 24 at Rutgers University, 2-3 PM in Hill 425.
Working seminar on Hilbert coefficients (Laura Ghezzi, Jooyoun Hong
and
Wolmer
Vasconcelos).
Friday, October 31 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Claudia Miller, Syracuse University.
Title: Rigidity of the Frobenius endomorphism.
Friday, November 7 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Hamid Rahamati, University of Nebraska at Lincoln.
Title: Free resolutions of parameter ideals over some rings with finite local cohomology.
Friday, November 14 at at Rutgers University, 2-3 PM in Hill 425.
Speaker: Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY.
Title: Why the multiplicity of the cusp singularity x^2=y^3 is equal to 1+1/3+2/3.
Abstract: Recent work of
Denef-Loeser-Cluckers et al. has put the rationality of certain zeta series in a
motivic context, that is to say, at the level of Grothendieck rings. I will
propose a variant construction that enables us to deal in a similar fashion with
the Hilbert series. The classical Grothendieck group of the category of finite
modules over a local ring, unfortunately, is the free group on one generator,
and hence cannot add any "motivation" to rationality. Instead, I propose the
category of weighted modules, that is to say, pairs (M,c) with M a finite module
and c a natural number. The corresponding Grothendieck group turns out to be a
highly non-trivial object: the weighted Grothendieck group. Using the cusp as an
example, I will show how we can make sense of a motivic Hilbert series, which is
still rational and specializes to the classical Hilbert series. As the Hilbert
series also encodes the multiplicity (=2 in this case), I will make sense of its
motivic counterpart. It turns out that it is a sum of three classes,
specializing to respectively the numbers 1, 1/3 and 2/3, thus vindicating the
title of this talk.
Friday, November 21 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Manoj Kummini, Purdue University.
Title: Homological invariants of bipartite edge ideals.
Abstract: We will discuss regularity,
projective dimension and arithmetic rank of monomial ideals generated by edges
of bipartite graphs.
Friday, November 28. No meeting this week.
Friday, December 5 at the CUNY Graduate Center, 4-5 PM in room 6417.
Speaker: Roozbeh Hazrat, Queen’s University, Belfast.
Title: Reduced K-theory for Azumaya Algebras, an overview.
Abstract: The theory of Azumaya algebras
developed parallel to the theory of central simple algebras. However the latter
are algebras over fields whereas the former are algebras over rings. One wonders
how the K- theory of these objects compare to each other. We look at higher K-
theory and reduced K-theory of these objects. We ask nice questions!
SPRING 2008
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
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.
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.
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.
Friday, May 30 at Rutgers University, 1-2 PM in Hill 425.
Speaker: Laura Ghezzi, New York City College of Technology, CUNY.
Title: Big balanced Cohen-Macaulay modules.