Commutative Algebra & Algebraic Geometry Seminar

Commutative Algebra & Algebraic Geometry Seminar

SPRING 2014

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:

Samar Elhitti, New York City College of Technology (CUNY), selhitti@citytech.cuny.edu

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

Friday, February 7

Speaker: Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY.

Title: Higher canonical modules and some applications.

Abstract: Over a complete CM ring, Grothendieck duality enables us to study the local cohomology of a module. Moreover, the "dualizing" module is canonically defined, whence its eponymous name. Whereas duality fails over non CM rings, there is still a canonically defined module: the Matlis dual of the top local cohomology of the ring. However, the lower local cohomology does no longer vanish, so we should also study their Matlis duals. These are what I term the "i-th higher canonical modules" and I will discuss two applications.

(1) After having proven existence of big CM modules in equal characteristic, Hochster conjectured, with some reservation, that any complete local ring even admits a (small) maximal CM module (=module of the same depth as the dimension of the ring), but very little is known in dimensions three and higher. I will give a criterion for their existence in dimension three involving first higher canonical modules.

(2) Any CM ring is unmixed, which implies the following "balanced" behavior: any system of parameters is a regular sequence. However, it is no longer true that if a ring has depth e, then the first e elements in a system of parameters form a regular sequence (although, of course, there exists at least one such system). If this stronger property nonetheless holds, then we will say that the ring (or module) has "balanced depth" e. I will show that balanced depth can be expressed in terms of the dimensions of the higher canonical modules.

Friday, February 21

Speaker: Bart Van Steirteghem, Medgar Evers College, CUNY.

Title: Multiplying functions on affine spherical varieties.

Abstract: Spherical varieties form a remarkable class of algebraic varieties equipped with an action of a complex reductive group G. They include toric, flag and symmetric varieties. A natural invariant of an affine spherical variety X is the set S(X) of irreducible representations of G that occur in the coordinate ring O(X) of X. This talk will discuss the following question: given S(X), what are the possible multiplication laws on O(X)?

Friday, March 7

Speaker: David Finston, Brooklyn College and the Graduate Center, CUNY.

Title: Additive group actions with smooth algebraic quotients.

Friday, March 21

Speaker: Imad Jaradat, New Mexico State University and Brooklyn College, CUNY.

Title: Proper triangular G_a-action on C^4 revisited.

Abstract: Every fixed point free action of the additive group of complex numbers (G_a(C)) on complex n-space (C^n) is conjugate to a translation for n < 4, but there are fixed point free proper

actions on C^5 which are not, and proper actions which are not even locally translations. It was

recently shown by Dubouloz, Finston, and myself that every proper triangular G_a-action on

C^4 is a translation. In this talk, I will continue the explicit treatment begun in my Spring 2013

CAAG talk of a case of the theorem that inspired the methods used to handle the general case.

Friday, April 4

Speaker: Andrew Parker, New York City College of Technology, CUNY.

Title: A^1-Homotopy Theory.

Abstract: In this talk, we will lay the mathematical foundations for replacing the standard unit interval with the affine line, for the purposes of applying topological arguments in an algebro-geometrical context.

Friday, April 25

Speaker: Anthony Iarrobino, Northeastern University.

Title: When do two nilpotent matrices commute?

Abstract: The similarity class of an nXn nilpotent matrix B over a field k is given by its Jordan type, the partition P of n, specifying the sizes of the Jordan blocks. The variety N(B) parametrizing nilpotent matrices that commute with B is irreducible, so there is a partition Q= Q(P) that is the generic Jordan type for matrices A in N(B). Q(P) has parts that differ pairwise by at least two, and Q(P) is stable: Q(Q(P))=Q(P). We discuss what is known about the map P to Q(P), in particular a recursive conjecture by P. Oblak (2008), very recently shown by R. Basili after partial results by P. Oblak, T. Kosir, L. Khatami, and others.

We then discuss a “Table theorem” when Q has two parts and a “Box Conjecture” in general for the set of partitions P having a given partition Q as maximum commuting orbit: so Q=Q(P). This work is joint with Leila Khatami, Bart van Steirteghem, and Rui Zhao.

Friday, May 9

Speaker: Mufit Sezer, Bilkent University, Turkey.

Title: Modular invariant rings.

Abstract: Let R:=F[x_1,...,x_n] denote the polynomial ring in n variables over a field F. We consider a group G acting on R as degree preserving automorphisms. The ring of invariants, R^G, is the subalgebra in R of polynomials that is fixed by this action. Invariant theory is interested in the structure of R^G and in finding connections between properties of Gand R^G. A main goal is to construct R^G by computing generators. We discuss a variety of results and examples with particular attention to the modular case that is when G is finite and its order is divisible by the characteristic of F.

FALL 2013

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:

Samar Elhitti, New York City College of Technology (CUNY), selhitti@citytech.cuny.edu

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

Friday, September 20

Speaker: Laura Ghezzi, New York City College of Technology, CUNY.

Title: Variation of the first Hilbert coefficients of parameter ideals with a common integral closure.

Abstract: We study a problem posed by Vasconcelos on the variation of the first Hilbert coefficients of parameter ideals with a common integral closure in a local ring. We obtain affirmative answers in significant cases and we give a counterexample in general. This is joint work with S. Goto, J. Hong, K. Ozeki, T.T. Phuong and W. Vasconcelos.

Friday, October 4

Speaker: David Finston, Brooklyn College and the Graduate Center, CUNY.

Title: Exotic Spheres and the Cancellation Problem.

Abstract: An exotic structure on a variety X is a variety Y that is diffeomorphic to but not algebraically isomorphic to X. Examples will be presented of exotic complex affine spheres which happen to provide counterexamples to the affine cancellation problem.

Friday, October 11

Speaker: Fatih Koksal, Texas Tech University.

Title: Transfer of injectivity under faithfully flat extensions.

Abstract (follow the link)

Friday, October 18

No meeting this week. We will be at the Union College Mathematics Conference.

Friday, November 1

Speaker: Jason McCullough, Rider University.

Title: On the Projective Dimension of Ideals of Quadrics.

Abstract: I will present recent joint work with Huneke, Mantero and Seceleanu concerning the projective dimension of homogeneous ideals generated by quadrics in a polynomial ring over a field. With the aim of improving the known bounds of Ananyan-Hochster, we originally set out to bound the projective dimension of ideals generated by just 4 quadrics. Along the way, we showed (1) that the projective dimension of R/I where I has height 2 and is generated by n quadrics has projective dimension at most 2n-2 (Moreover, this bound is tight.), (2) if I (not necessarily generated by quadrics) satisfies a multiplicity based condition, which we call maximal multiplicity, then I is Cohen-Macaulay, that is the projective dimension of R/I is the height of I, and (3) the projective dimension of an ideal generated by 4 quadrics is at most 9. I will discuss all three results and some current work.

Friday, November 15

Speaker: Janet Striuli, Fairfield University.

Title: Constructing totally reflexive modules.

Abstract: A theorem by Christensen, Piepmeyer, Striuli and Takahashi states that if the ring is not Gorenstein and there is a totally reflexive module then there are infinitely many. This opens the question on how to construct totally reflexive modules. In this talk I will give a survey on some constructions that are available.

Friday, December 6

Speaker: Sanju Vaidya, Mercy College.

Title: Puiseux Expansions in Nonzero characteristic.

Abstract: It is well known that the Puiseux field of all Puiseux expansions is an algebraic closure of the meromorphic series field if its characteristic is zero. But for the nonzero characteristic case, Chevalley proved that the Puiseux field is not algebraically closed. Using Abhyankar’s notion of the generalized Puiseux expansion, Huang constructed a generalized Puiseux field and showed that it contains an algebraic closure of the meromorphic series field. He also proved a criterion for a certain type of generalized Puiseux elements which says that they are algebraic over the meromorphic series field iff they are periodical. We investigate some functions of the generalized Puiseux field that are algebraic over the meromorphic series field; moreover we calculate their Galois groups. It turns out that Galois group of certain functions over the meromorphic series field is a semidirect product of a cyclic group and a direct sum of p cyclic groups. We also exhibit functions whose Galois groups are dihedral group, a certain type of a Burnside group and a direct sum of p cyclic groups. Additionally, we extend the said criterion of Huang to some special type of functions. It is fascinating to see how this is generalized and cited by many research papers published in various journals.

SPRING 2013

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:

Samar Elhitti, New York City College of Technology (CUNY), selhitti@citytech.cuny.edu

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

Friday, February 1

Speaker: Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY.

Title: Signal degradation in circuits caused by associated primes.

Abstract: Disclaimer: in spite of its title, this is not a talk in electrical engineering! Degradation is meant to indicate that a linear map between modules can become close to zero due to the configuration of associated primes of source and target. An easy example: if the maximal dimension of an associated prime of the domain is less than the minimal dimension of an associated prime of the source, then any morphism between them is zero. I will concentrate on the following instance of this phenomenon. Let us call an endomorphism that factors through one or more other modules (gates), a circuit. Degradation now manifests itself as follows: if the gates have no associated prime in common, then there exists a bound k (depending on the gates only), such that any composition of k circuits is zero; in particular, any circuit is nilpotent. This work relies on the concept of ordinal length and its connection with associated primes, as I will explain.

Friday, February 15

Speaker: Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY.

Title: An application of Fermat's last theorem (?): computably categorical fields of infinite transcendence degree.

Abstract: Given two number fields, it is not hard to check (algorithmically) that they are isomorphic. The problem, however, becomes much harder if we consider non-algebraic or infinitely generated extensions. To capture this phenomenon, the notion of categorical computability (cc) can be used: this is a computable (=recursive) field F so that if G is another computable field which is isomorphic as a field to F, then we can actually compute an isomorphism between them. The field of rationals, Q, is clearly cc, but imagine already the obstacles that arise when trying to compute an isomorphism between Q(X) and an isomorphic copy G of it: take any element from the first field that is not in Q, so it is transcendental over Q, and take a similar element in G, and map the first to the second. If you were lucky and picked in both cases the variable X you're done. But what if the first element is not a transcendence basis, and you find out later (while computing your isomorphism) that the first element is in fact a square. So now you have to adjust your first choice by mapping it to some (transcendent) square in the other field. If you can do this after only finitely many times changing your mind, you're done. But what if we have infinite transcendence degree? Ershov proved that an algebraically closed field of finite transcendence degree is cc, but one of countable transcendence degree is not. It was expected that this is the general phenomenon. However, we (joint work with R. Miller) construct a field of infinite transcendence degree that is cc. The construction relies on the Mordell-Faltings fact that curves of high genus have only finitely many rational solutions, and by the positive solution of Fermat's conjecture, we have now a whole family for which we know the exact number of solutions.

Friday, March 1

Speaker: Jeanne Funk, La Guardia Community College, CUNY.

Title: The Witt Ring and its Fundamental Ideal.

Abstract: Historically, a great deal of information about the Witt ring of a field has been gleaned by investigating the associated graded of its fundamental ideal, which consists of classes represented by forms of even rank. This talk will focus on a generalization of the fundamental ideal to the category of sheaves, its connections to cohomology, and its use as a tool in calculating the Witt ring of a variety.

Friday, March 15

Speaker: Julio Urenda, New Mexico State University.

Title: An algorithmic approach to the embedding problem in affine 3-space.

Abstract (follow the link)

Friday, April 12

Speaker: Branden Stone, Bard College.

Title: Super-Stretched and Graded Countable Cohen-Macaulay type.

Abstract: This talk will explore the following question of C. Huneke and G. Leuschke: Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite \CM representation type? We will show a positive answer for standard graded non-Gorenstein rings as well as for graded Gorenstein rings of minimal multiplicity. Along the way we will develop the concept of super-stretched and its relation to rings of graded countable type. Further we will show the one dimensional case to the folklore conjecture: A graded Gorenstein ring of graded countable type is a hypersurface.

Friday, April 26

Speaker: Imad Jaradat, New Mexico State University.

Title: Proper triangular G_a--actions on C^4.

Abstract (follow the link)

Friday, May 17

Speaker: Lars Winther Christensen, Texas Tech University.

Title: Classification of local rings via products in Koszul homology.

Abstract: Let R be a local ring with maximal ideal M. How much information about R can one extract from HK, the homology of the Koszul complex on a minimal set of generators for M? If the homology is concentrated in at most three degrees, then R is Golod or a complete intersection, and a uniquely determined multiplicative structure on HK tells us which one it is. If there is homology in four degrees, then the situation is more complicated, and that is what I will talk about.

FALL 2012

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:

Samar Elhitti, New York City College of Technology (CUNY), selhitti@citytech.cuny.edu

Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu

Jooyoun Hong, Southern Connecticut State University, hongj2@southernct.edu

Janet Striuli, Fairfield University, jstriuli@mail.fairfield.edu

Friday, September 7

Speaker: Bart Van Steirteghem, Medgar Evers College, CUNY.

Title: The weight monoids of smooth affine spherical varieties.

Abstract: Spherical varieties form a remarkable class of algebraic varieties equipped with an action of a complex reductive group G. They include toric, flag and symmetric varieties. Smooth affine spherical varieties are the local models of multiplicity free (real) Hamiltonian manifolds. A natural invariant of an affine spherical variety X is its weight monoid S(X). It is the set of irreducible representations of G occurring in the coordinate ring of X, which is a multiplicity free G-module. In the 1990s, F. Knop conjectured that it is a complete invariant for smooth affine spherical varieties, and in 2006 I. Loseu proved this conjecture. Little is known about the image of the map S that sends a smooth affine spherical variety to its weight monoid. I will discuss joint work with G. Pezzini on characterizing those free and "G-saturated" monoids that belong to the image of S.

Friday, September 28

Speaker: David Swinarski, Fordham University.

Title: State polytopes and geometric invariant theory.

Abstract: I will review the construction of the state polytope of an ideal from commutative algebra and how it gives information about geometric invariant theory in algebraic geometry. I will discuss a recent paper with Ian Morrison and related work by Alper, Fedorchuk, and Smyth.

Friday, October 12

Speaker: Dave Hren, New Mexico State University.

Title: Fibers of Complete Scalar Extensions.

Abstract: Let $\left(R,\mathfrak{m}\right)$ be a local, Noetherian domain with quotient field $Q$. Heinzer, Rotthaus, and Sally showed that $\dim\widehat{R}\otimes_{R}Q=\dim R-1$if and only if $R$ is birationally dominated by a residually finite DVR by establishing a $1-1$ correspondence between prime ideals in the generic formal fiber of $R$ of height $\dim R-1$ and residually finite DVR overrings. In this talk, we discuss a generalization of their result using the notion of a complete scalar extension introduced by Schoutens.

Friday, October 26

Speaker: Manoj Kummini, Chennai Mathematics Institute, India.

Title: Poset embeddings of Hilbert functions.

Abstract: We study the posets of Hilbert functions of ideals in standard graded algebras and look at their embeddings into the poset of ideals. We will look at their behavior under polynomial extensions. This is joint work with G. Caviglia.

Friday, November 9

Speaker: David Finston, Brooklyn College.

Title: Quotient structures for additive group actions and a non-quasiprojective variety.

Abstract: While the space of orbits of a proper holomorphic action of a complex Lie group on a complex manifold always admits a natural structure of a complex manifold, the analogous result does not hold for algebraic actions. An example will be given of a proper action of the additive group of complex numbers on affine 5-space whose quotient is not a scheme. In fact the quotient exists as an algebraic space which itself is a Z/2Z quotient of a necessarily non-quasiprojective variety.

Friday, November 30

Speaker: Federico Galetto, Northeastern University.

Title: Representations with finitely many orbits and free resolutions.

Abstract: The representations of reductive groups with finitely many orbits are parametrized by graded simple Lie algebras. For the exceptional Lie algebras, Kraskiewicz and Weyman exhibit the expected minimal free resolutions for the coordinate ring of the normalization of the orbit closures. I will present an interactive method to verify their conjectures using Macaulay2. The resolutions are then used to investigate geometric properties of the orbit closures and to study modules supported on the orbit closures.

SPRING 2012

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:

Samar Elhitti, New York City College of Technology (CUNY), selhitti@citytech.cuny.edu

Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu

Jooyoun Hong, Southern Connecticut State University, hongj2@southernct.edu

Janet Striuli, Fairfield University, jstriuli@mail.fairfield.edu

Friday, February 10

Speaker: Oana Veliche, Northeastern University.

Title: On computing the Auslander index of a complete intersection numerical semigroup ring.

Friday, February 24

Speaker: Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY.

Title: Ordinal-valued invariants.

Abstract: The usual invariants in commutative algebra or algebraic geometry are integer valued, measuring a certain complexity of the ring (module, scheme, ...) and therefore are useless if they outcome is infinite. Ordinals is one way of extending natural numbers to include infinite values, measuring the order type of a certain phenomenon. For instance, length measures the order type of the lattice of ideals of a Noetherian ring (of submodules in a Noetherian module). The key fact of length is its additivity on short exact sequences, but it is easy to construct examples where this must fail for arbitrary ordinals. One complication is that there are several potential addition relations on ordinals, some of which are not even commutative. At best we can get semi-additivity. Ordinal length has also an interpretation in terms of local cohomology, and I will use this to prove a result on the extensional exchange property.

SPECIAL SEMINAR (Model Theory, Number Theory, Algebraic Geometry Umbrella Series): Wednesday, March 7 @ 4:30 PM in room C201

Speaker: Ivan Tomasic, Queen Mary University of London.

Title: A Twisted Theorem of Cheboratev.

Abstract: See http://nylogic.org/MONTAGU/Spring2012/IvanTomasic

Friday, March 9

Speaker: Feza Arslan, Mimar Sinan Fine Arts University, Istanbul.

Title: Arf closure and a conjecture.

Abstract: In this talk, after a quick introduction to Arf theory (Arf rings and Arf closure), we present a fast algorithm for computing the Arf closure. We also give a conjecture about the Hilbert functions of local rings having the same Arf closure.

Friday, March 23

Speaker: Jooyoun Hong, Southern Connecticut State University.

Title: Variation of Hilbert coefficients.

NOTE: This seminar is canceled

Friday, April 20

Speaker: Charles Li, CUNY Graduate Center.

Title: Dual graphs of valuations.

Friday, May 4

Speaker: Roya Beheshti-Zavareh, Washington University in St. Louis and Columbia University.

Title: Spaces of rational curves on general hypersurfaces.

Abstract: I will discuss some aspects of the geometry of moduli spaces of rational curves on general Fano hypersurfaces in projective space and other homogeneous varieties. The focus of my talk will be on some questions regarding the dimension and irreducibility of these moduli spaces. This talk is partly based on joint work with N. Mohan Kumar.

FALL 2011

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:

Samar Elhitti, New York City College of Technology (CUNY), selhitti@citytech.cuny.edu

Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu

Jooyoun Hong, Southern Connecticut State University, hongj2@southernct.edu

Janet Striuli, Fairfield University, jstriuli@mail.fairfield.edu

Friday, September 9

Speaker: Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY.

Title: The Grothendieck ring of a motivic site.

Abstract: One of the oldest problems in, if not the progenitor of, algebraic geometry is counting the number of solutions of a system of equations. Of course, this is useless when there are infinitely many solutions, and so more subtle invariants have been introduced, like dimension, multiplicity, etc. Among these, Euler characteristics behave exactly as cardinality, and it is a beautiful result from model-theory that, in the definable category, the Euler characteristic together with the dimension forms a complete invariant over the reals. Over algebraically closed fields we are not so lucky, and to capture this counting principle, Grothendieck defined a universal Euler characteristic, appropriately named the “Grothendieck ring of varieties”. Grothendieck also introduced more general types of geometrical objects called “schemes”, to capture the phenomenon of multiplicity. But somehow neither him nor anybody else has ever tried to combine these two constructions. The main obstruction against trying to do this is that unions and complements of schemes do not exist; the main incentive for trying nonetheless is that we would then be able to perform motivic integration on schemes.

I will show that we can formally adjoin objects that behave like unions of schemes, referring to them by the arguable misnomer “motives”. Complements do not always exist, but for instance the complement of an open subscheme is, as a motive, the same as the formal completion along the complement. Thus motives include formal schemes. The trick behind the whole construction is, first, to view a scheme as a functor of rational points (this is not a new idea and is the point of departure of topos theory), and then secondly to only consider rational points with values in an Artinian local ring. (An alternative to this view point, which I discussed some time ago in the Logic Workshop, is to view a scheme as the class of a quantifier formula modulo the theory of Artinian local rings.) This formalism leads to the notion of a “motivic site”, of which we can now take its Grothendieck ring. The smallest site including open covers and formal schemes, I call the “formal motivic site”, and its Grothendieck ring admits a natural homomorphism into the classical Grothendieck ring.

Friday, September 23

Speaker: Michael Burr, Fordham University.

Title: Asymptotic Purity for Very General Hypersurfaces of Products of Projective Spaces.

Abstract: For a complex irreducible projective variety, the asymptotic cohomological functions were introduced by Kuronya and Demailly to measure the growth rate of the cohomology of high tensor powers of an invertible sheaf. These functions have proven to be useful in understanding the positivity of divisors as well as other geometric properties of the variety. In this talk I will define a strong vanishing property, called asymptotic purity, and prove that very general hypersurfaces of P^n x P^n of bidegree (k,k) have this property. These examples provide evidence for the truth of a conjecture of Bogomolov concerning asymptotic purity.

Friday, October 14

Speaker: Marju Purin, Manhattan College.

Title: On a Generalization of the Auslander-Reiten Conjecture.

Abstract: Maurice Auslander and Idun Reiten stated the following conjecture in a paper from 1975: “If M is an R-module with Ext^i(M,M+R)=0 for all i>0, then M is a projective module.” This conjecture remains open for “most” classes of rings, including for commutative rings. A natural generalization is the following statement: “If M is an R-module with Ext^i(M,M+R)=0 for all i>>0, then M has finite projective dimension.” Also, this conjecture remains open for commutative rings. In our talk we discuss these conjectures, give a version of the latter conjecture for any triangulated category, and use it to show that the generalized version of the Auslander-Reiten Conjecture is stable under any derived equivalence of Noetherian rings.

Friday, October 28

Speaker: Mahdi Majidi-Zolbanin, La Guardia Community College, CUNY.

Title: Entropy in Local Algebra.

Abstract: In this talk we will introduce and discuss the properties of a notion of “algebraic entropy” for self-maps of relative dimension zero of Noetherian local rings. We will show that it shares many standard properties of topological entropy. For finite self-maps we will explore the connection between the degree of the map and its algebraic entropy, when the ring is a Cohen-Macaulay domain. We will also talk about some possible applications of algebraic entropy.

Friday, November 11

Speaker: Heidi Hulsizer, Hampden-Sydney College.

Title: Resolution of Determinantal Ideals.

Abstract: Determinantal ideals have been studied for over a century and they continue to be of interest to mathematicians. The structures of several resolutions (complexes) of determinantal ideals have been determined. The hope was that the structure of all resolutions of this form could be found and that they would not depend on the characteristic. This possibility was crushed when, in 1990, Mitsuyasu Hashimoto showed that this was impossible. We will discuss the structure for the longest possible resolution that does not depend on the characteristic of the field. To describe the resolution we use what is called tableau notation. This notation is used in other branches of mathematics and it works well to define the basic elements involved in the maps of the resolution.

Friday, December 2

Speaker: Samar ElHitti, New York City College of Technology, CUNY.

Title: Artin-Schreier Defect Extensions and Strong Monomialization.

Abstract: Kuhlmann classifies Artin-Schreier defect extensions as dependent or independent. Is one category more "harmful" than the other? For instance, how does this classification hold up to Cutkosky's Strong Monomialization? In this talk, we present partial results exploring these questions.

SPRING 2011

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:

Samar Elhitti, New York City College of Technology (CUNY), selhitti@citytech.cuny.edu

Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu

Jooyoun Hong, Southern Connecticut State University, hongj2@southernct.edu

Janet Striuli, Fairfield University, jstriuli@mail.fairfield.edu

Friday, February 4

Speaker: Laura Ghezzi, New York City College of Technology, CUNY.

Title: The first Hilbert coefficients of parameter ideals relative to a module.

Abstract: The set of the first Hilbert coefficients of parameter ideals relative to a module over a Noetherian local ring codes for considerable information about its structure. In this talk, based on joint work with S. Goto, J. Hong, K. Ozeki, T.T. Phoung and W. Vasconcelos we discuss the Cohen-Macaulay, generalized Cohen-Macaulay and Buchsbaum properties of the module.

Friday, February 18

Speaker: Kia Dalili, University of Missouri-Columbia.

Title: Regularity bounds and the HomAB problem.

Friday, March 4

Speaker: Andrew Parker, CUNY.

Title: An Obstruction Theory for Projective Modules.

Abstract: After Quillen and Suslin affirmed the Serre Conjecture on the freeness of finitely generated projective modules over polynomial rings, there was interest in what could be said in general about finitely generated projective modules over any commutative ring. It had already been proven by Serre that projective modules who's rank exceeded the Krull dimension of the base ring were free in each rank beyond the dimension. This talk will cover the results on projective modules with rank equal to the dimension of the base ring and how one can determine the existence of free summands within such a projective module.

Friday, March 18

Speaker: Lars Winther Christensen, Texas Tech University.

Title: Balancedness of Tate (co)homology via pinched complexes.

Abstract: Let R be a ring. For complexes of R-modules we introduce two constructions, which we call the pinched tensor product complex and the pinched Hom complex. They resemble the usual tensor product and Hom of complexes, but they are smaller in a sense that is illustrated by the following fact. If one starts with unbounded complexes T and U of finitely generated modules, then the pinched Hom and tensor product complexes of T and U are also complexes of finitely generated modules.

Our motivation for studying these constructions is that they enable us to resolve the balancedness question for Tate (co)homology. The talk is based on joint with David A. Jorgensen.

Friday, April 8

Speaker: Aihua Li, Montclair State University.

Title: On the construction of explicit solutions to the matrix equation X^2AX=AXA.

Abstract: We study the matrix equation AXA = X^2AX, where A is a fixed, square matrix with real entries and X is an unknown square matrix. The solution space is explicitly constructed for all 2×2 complex matrices using Grobner basis techniques. When A is a 2×2 matrix, the equation $AXA = X^2AX$ is equivalent to a system of four polynomial equations. The solution space then is the variety defined by the polynomials involved. The ideal of the underlying polynomial ring generated by the defining polynomials plays an important role in solving the system.

In the procedure for solving these equations, Grobner bases are used to transform the polynomial system into a simpler one, which makes it possible to classify all the solutions. In addition to classifying all solutions for 2 × 2 matrices, certain explicit solutions are produced in arbitrary dimensions when A is nonsingular. In higher dimensions, Grobner bases are extraordinarily computationally demanding, and so a different approach is taken. This technique can be applied to more general matrix equations, and the focus here is placed on solutions coming from a particular class of matrices.

Friday, April 29

Speaker: Feza Arslan, Mimar Sinan Fine Arts University, Istanbul.

Title: Monomial curve families supporting Rossi's conjecture.

Abstract: Rossi's conjecture saying that every Gorenstein local ring has non-decreasing Hilbert function is open even for monomial curves in affine 4-space. In this talk, by using the concept of gluing, we give methods to construct families of 1-dimensional local rings associated to monomial curves with free parameters supporting Rossi's conjecture.

Friday, May 6

Speaker: Kosmas Diveris, Syracuse University.

Title: On the eventual vanishing of self-extensions.

Abstract: The AC condition concerning the vanishing of cohomology over a ring originates from the work of Auslander. More recently Christensen and Holm have shown that several longstanding homological conjectures hold for rings having the AC condition. In this talk we define a new condition that generalizes the uniform AC condition. We show that many of the known results for AC rings hold for rings having our condition. We will also discuss some examples of rings for which our condition holds and some where it fails to hold.

FALL 2010

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:

Samar Elhitti, New York City College of Technology (CUNY), selhitti@citytech.cuny.edu

Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu

Jooyoun Hong, Southern Connecticut State University, hongj2@southernct.edu

Janet Striuli, Fairfield University, jstriuli@mail.fairfield.edu

Friday, September 24

Speaker: Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY.

Title: Homological conjectures in mixed characteristic: asymptotic versions.

Abstract: Whereas all so-called homological conjectures have been proven in equal characteristic (most recently via tight closure theory), most of them remain wide open in mixed characteristic (some notable exceptions are Roberts' New Intersection Theorem and Heitmann's Direct Summand Theorem in dimension three).

There is a general principle in number theory that all local fields behave similarly, that is to say, that if something holds over the Laurent series over a finite field, it should also hold over the p-adics. However, this correspondence is not perfect, and the first crack was observed by Terjanian with respect to Artin's $C_2$-conjecture about rational points on projective hypersurfaces of low degree. However, Ax and Kochen formulated and proved a weaker correspondence for local fields, yielding the following asymptotic version: Artin's conjecture holds for high enough residual characteristic p. I will give a brief review of this as preparation to Denef's talk the following week, who will present a new, geometric proof of this result.

I will then argue that this asymptotic approach also holds for many of the homological conjectures, by a higher dimensional version of the Ax-Kochen principle. I will use the New Improved Intersection Theorem as an example, and show that if the asymptotic growth rate is sufficiently slow, then we can even prove the full conjecture.

SPECIAL SEMINAR: Monday, September 27 @ 5:30 PM in room 4102 (joint with Logic and Model Theory)

Speaker: Jan Denef, Katholieke Universiteit Leuven, Belgium.

Title: Geometric proof of Ax-Kochen.

Abstract: See http://nylogic.org/ModelTheory/Fall2010/JanDenef

Friday, October 8

Speaker: Lars Winther Christensen, Texas Tech University.

Title: Vanishing of Tate homology and depth formulas over local rings.

Abstract: Auslander's depth formula for pairs of Tor-independent modules over a regular local ring, depth(M \otimes N) = depth(M) + depth(N) - depth(R), has been generalized in several directions over a span of 40 years. In the talk I will describe a formula that holds for every pair of Tate Tor-independent modules over a Gorenstein local ring. It subsumes all the previous generalizations of Auslander's formula. The talk is based on joint work with Dave Jorgensen.

Friday, October 22

Speaker: Jooyoun Hong, Southern Connecticut State University.

Title: The positivity of the first normalized Hilbert coefficients.

Abstract: See JHong_abstract_October2010.pdf

Friday, November 5

Speaker: Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY.

Title: Homological conjectures in mixed characteristic: asymptotic versions, part II.

Abstract: This is a continuation of the September 24 talk.

Friday, November 19

Speaker: Sebastian Casalaina-Martin, University of Colorado, Boulder.

Title: Simultaneous stable reduction for curves with ADE singularities.

Abstract: A basic question in moduli theory is to describe a stable reduction of a given family of curves. Roughly speaking, given a family of curves where the generic member is a smooth curve, one would like to "replace" those curves in the family that have bad singularities with other curves, known as stable curves, that have more mild singularities.

Typically this will only be possible after a generically finite base change, and the question is to describe such a base change as well as the total space of the new family. After discussing some basic examples, the aim of this talk will be to present joint work with Radu Laza where we describe stable reductions for families of curves with ADE singularities.

Time permitting, applications to the moduli space of stable curves and the Hassett-Keel program will also be discussed.

Friday, December 3

Speaker: Hamid Rahmati, Syracuse University.

Title: Artinian Gorenstein rings and infinite syzygies.

SPRING 2010

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.

Schedule: See http://websupport1.citytech.cuny.edu/faculty/Selhitti/index.htm

Organizers:

Samar Elhitti, New York City College of Technology (CUNY), selhitti@citytech.cuny.edu

Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu

Jooyoun Hong, Southern Connecticut State University, hongj2@southernct.edu

Janet Striuli, Fairfield University, jstriuli@mail.fairfield.edu

FALL 2009

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:

Samar Elhitti, New York City College of Technology (CUNY), selhitti@citytech.cuny.edu

Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu

Jooyoun Hong, Southern Connecticut State University, hongj2@southernct.edu

Janet Striuli, Fairfield University, jstriuli@mail.fairfield.edu

Friday, September 4

Speaker: Sabine El Khoury, American University of Beirut, Lebanon.

Title: A class of Gorenstein artinian algebras of embedding codimension four.

Friday, September 11

Speaker: Laura Ghezzi, New York City College of Technology, CUNY.

Title: Hilbert functions in local rings.

Abstract: The first part of the talk is introductory. In the second part we present recent developments on the subject, focusing on a conjecture on the vanishing of the first Hilbert coefficient e_1(Q), where Q is a parameter ideal in a Noetherian local ring.

Friday, September 18

No meeting this week.

Friday, September 25

Speaker: Jooyoun Hong, Southern Connecticut State University.

Title: Chern numbers of parameter ideals.

Friday, October 2

Speaker: Suanne Au, Le Moyne College.

Title: The equivariant K-theory of toric varieties.

Friday, October 9

Speaker: Lars Winther Christensen, Texas Tech University.

Title: Floor plans in local algebra.

Abstract: In a paper from 2003, Schoutens proved that every module over a commutative local ring can be built from the prime ideals in the singular locus using a few simple constructions, or "moves". Schoutens' work provides an upper bound for the number of moves required to build the entire module category. In the talk I will show exactly how many moves are required and relate the result to recent work by other authors. The talk is based on joint work with Jesse Burke.

Friday, October 16

Speaker: Laura Ghezzi, New York City College of Technology, CUNY.

Title: A proof of the conjecture on the vanishing of the first Hilbert coefficient of parameter ideals.

Friday, October 23

No meeting this week. We will be at the Special Session on Commutative Algebra and Applications to Algebraic Geometry at the AMS meeting in University Park, PA.

Friday, October 30

Speaker: Jooyoun Hong, Southern Connecticut State University.

Title: Chern numbers of parameter ideals, part II.

Update (10/28): The seminar is cancelled.

Friday, November 6

Speaker: Bart Van Steirteghem, Medgar Evers College, CUNY.

Title: Moduli of affine spherical varieties.

Abstract: When an algebraic group G acts on an affine variety X (over an algebraically closed field k), the coordinate ring k[X] of X is naturally a G-module, that is, a linear representation of G. A natural question is whether (or to what extent) the G-module structure of k[X] determines its k-G-algebra structure.

For the case when k has characteristic zero and G is a reductive and connected linear algebraic group, V. Alexeev and M. Brion brought geometry to this question by constructing a moduli scheme which parametrizes the G-multiplication laws "compatible" with the given G-module structure.

I will introduce this moduli scheme and the closely related invariant Hilbert scheme, also of Alexeev and Brion, with several elementary examples. Finally, I will discuss a family of examples of the moduli scheme recently obtained in joint work with S. Papadakis.

Friday, November 13

Speaker: Amanda Beecher, United States Military Academy.

Title: Describing a multigraded resolution.

Friday, November 20

Speaker: Uma Iyer, Bronx Community College, CUNY.

Title: Differential operators on free algebras.

Friday, November 27

No meeting this week.

Friday, December 4

Speaker: Leila Khatami, Northeastern University.

Title: Commuting nilpotent matrices.

Friday, December 11

Speaker: Adela Vraciu, University of South Carolina.

Title: Joint Hilbert-Kunz multiplicities.

SPRING 2009

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

Janet Striuli, Fairfield University, jstriuli@mail.fairfield.edu

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

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

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.