Commutative Algebra & Algebraic Geometry Seminar
 Organizers: Samar ElHitti, New York City College of Technology, CUNY Laura Ghezzi, New York City College of Technology, CUNY Hans Schoutens, New York City College of Technology and the Graduate Center, CUNY   To view the list of speakers and abstracts from semesters not listed below, please click here.
 The seminar meets on selected Fridays from 4 - 5 PM at the CUNY Graduate Center in room 6417, located at  365 Fifth Avenue, New York, NY 10016. If you are interested in giving a talk or wish to join the mailing list, please contact me at  selhitti [AT] citytech [DOT] cuny [DOT] edu
 FALL 2016 The seminar will be held on September 9, 23, October 21, November 4, 18 and December 2   September 9, 2016 Speaker: Hans Schoutens, NYCCT and The Graduate Center, CUNY Title: Invariants of locally nilpotent derivations a la Makar-Limanov and problems in affine geometry Abstract: This talk will be a survey talk on the recent use of locally nilpotent derivations to tackle questions about affine n-space. I will discuss the Makar-Limanov invariant of a ring, and show how it can be used to approach some of these questions. Examples are the actions of the additive and multiplicative group on affine space,  the counterexamples by Gupta for the Zariski Cancelation problem in positive characteristic, … The material in this talk is related to my CUNY Logic workshop talk from last week, and to Malgorzata’s CAAG talk from last semester   September 23, 2016 Speaker: Uma N. Iyer, Bronx Community College, CUNY Title:  Extending weight representations from a Generalized Weyl Algebra to an algebra of quantum differential operators Abstract:  Indecomposable weight representations on the Generalized Weyl Algebras were classified by Drozd, Guzner, Ovsienko(1995).  We extend their representations to weight representations of an algebra of quantum differential operators (D_q) over an algebraically closed field of characteristic 0 and classify all the irreducible weight representations of D_q. This is joint work with V. Futorny   October 21, 2016 Speaker: Malgorzata Marciniak, LaGuardia Community College Title: On the real Jacobian conjecture and its connections with the Hamiltonian differential systems Abstract: This talk will be a survey of the history of the real Jacobian conjecture. Originally formulated in 1939 by Keller for an arbitrary field, has two versions for real numbers. Given topological motivations, S. Pinchuk proved in 1994 that its strong version cannot be affirmative. Recently F. Brown, J. Gine and J. Llibre found a necessary and sufficient condition on the real polynomial map to have a globally defined polynomial inverse. Their proof is motivated by results by Sabatini from 1998 about isochronous Hamiltonian centers Speaker’s notes from the seminar can be accessed here.   November 4, 2016 Speaker: Ben Blum-Smith, Courant Institute of Mathematical Sciences Title: Modular permutation invariants and quotients of spheres Abstract: An important goal in the invariant theory of finite groups is to get information on the homological complexity of invariant rings when the characteristic of the coefficient field divides the group order ("the modular case"). This is irrelevant in the nonmodular case because such invariant rings are always Cohen-Macaulay, i.e. "homologically as simple as possible". This was proven over 40 years ago, but in the modular case even the question of when invariant rings are C-M is very far from settled. We apply work of Garsia and Stanton from the 80s to connect this question for permutation groups to the topology of a cell complex that is the quotient of a sphere by the group. Then we apply recent work of Christian Lange on sphere quotients to give a sufficient criterion for the invariant ring to be C-M over every ground field   November 18, 2016 Speaker: Jason McCullough, Rider University Title: Rees-like Algebras and the Eisenbud-Goto Conjecture Abstract: Regularity is a measure of the computational complexity of a homogeneous ideal in a polynomial ring. There are examples in which the regularity growth is doubly exponential in terms of the degrees of the generators, but better bounds were conjectured for "nice" ideals. Together with Irena Peeva I discovered a construction that overturns some of the conjectured bounds for "nice" ideals - including the Eisenbud-Goto conjecture. I'll explain the construction and some of its consequences   December 2, 2016 Speaker: Giovan Battista Pignatti, CUNY Graduate Center Title: Some topological considerations on semistar operations Abstract: Semistar operations were introduced by A. Okabe and R. Matsuda in 1994 and provide an important tool to study the multiplicative structures of the ideals of a ring. I will present the results contained in an article by C.A. Finocchiaro and D. Spirito where the authors endow the set of semistar operations over an integral domain A with a topology and investigate the connections between the algebraic and the topological properties of this new space. I will focus in particular on the space SStar_f(A) of semistar operations of finite type over A proving that it is spectral. For this purpose I will use a characterization of spectral spaces which involves ultrafilters and some results from M. Hochster’s PhD thesis     SPRING 2016   February 5, 2016 Speaker: Malgorzata Marciniak, LaGuardia Community College Title: The group of algebraic automorphisms of C^n Abstract: For n=2 the group of algebraic automorphisms of C^n has a structure of an amalgam over two of its subgroups: triangular and affine morphisms which is a theorem by Jung and van der Kulk. For n=3 the group is not an amalgam but its subgroup generated by triangular and affine morphisms can be described. I will present results related to some subgroups of the group of algebraic automorphisms of C^4   February 19, 2016 Speaker: Hans Schoutens, NYCCT and The Graduate Center, CUNY Title: Differential Algebra NOT a la Grothendieck Abstract: The Weyl algebra is the (non-commutative) $C[x]$-algebra generated by the partial derivatives. It most salient features are that it is simple (=has no two-sided ideals) and left (and right) Noetherian. This then leads to the theory of holonomic modules. The construction also works for smooth $C$-algebras other than the polynomial ring, but this time, the partial derivatives are only defined locally, and so one must use all (possibly non-commuting) derivations to make the definition. Nonetheless, the resulting algebra has the same nice properties as the Weyl algebra. The theory, however, becomes much more complicated, both in the non-smooth case over $C$ as in the (smooth) case in positive characteristic. Grothendieck then proposed an abstract notion of the ring of differential operators as the submodule of all endomorphisms that commute in some finite number of steps with all scalars. And indeed, this just gives the old definition in the smooth case and yields in many singular cases in characteristic zero again a simple algebra, yet in the latter case, many other good properties are lost (like Noetherianity, the notion of holonomy,…). In characteristic $p$, the situation is even worse: a differential operator is now any endomorphism of the ring which is linear with respect to the subring of $q$-th powers, for some prime power $q$. So what exactly was wrong with the “old” version of a differential operator, that is to say, an element in the algebra generated by all derivations? I will argue that in fact, some good things can be still said in this case. For instance, simplicity is most likely to be equivalent with being smooth (this is still partially conjectural). There is even a good notion of holonomy, and perhaps even a version of the Bernstein-Sato polynomial in the singular case. I will, in fact, take a slightly more general point of view, by defining the ring of differential operators of a $k$-algebra $R$ as a “universal” object, given by the quotient of the free (non-commutative) $R$-algebra generated by  all derivations of $R$ modulo the “obvious” commutator relations. It turns out that in characteristic zero, this just gives the “old” version, and I conjecture that in positive characteristic, there is one more additional relation, coming from the fact that a $p$-th power of a derivation is again a derivation. Of course, I do not pretend to say that Grothendieck was wrong in his definition, but I am only trying to investigate what he didn’t feel like doing   March 4, 2016 Speaker:  Andrew Stout, Bronx Community College, CUNY Title: The auto Igusa-zeta function of an algebraic plane curve singularity is rational Abstract: This talk is concerned with the study of $k$-algebra endomorphisms of truncated local rings $O_{X,p}/m_p^n$ where $X$ is a variety and $p$ is a point on $X$. It will be shown that if $X$ is a curve on a smooth surface, then once we know the endomorphisms of $O_{X,p}/m_p^n$ for n=1,2,...,e, and for some natural number e, then we will know what the rest of the endomorphisms will look like. Whether this is the case when $X$ is more complicated is completely unknown. This problem can be phrased more precisely in terms  rationality of a certain motivic generating function $\zeta(t)$ called the auto Igusa-zet function, which was introduced into the literature by Hans Schoutens. Last time, I gave a talk here I made several conjectures concerning $\zeta(t)$ supported by computational evidence. Indeed, a consequence of the rationality of $\zeta(t)$ for plane curves is that all of those conjectures will be true. In particular, $\zeta(t)$ is a perfect local invariant for plane curves. Again, the situation for higher dimensional varieties is completely unknown. Although I have given a talk on this subject before at CAAG, nothing from that talk will be assumed. Moreover, the talk will be accessible to a wide audience.      March 18, 2016 Speaker: Olgur Celikbas, University of Connecticut Title: Gorenstein dimension of integrally closed ideals Abstract: Let (R,m) be a local ring and let I be a proper integrally closed m-primary ideal of R (e.g., I=m). In this talk I will discuss a recent joint work with Sean Sather-Wagstaff which improves on a result of Goto and Hayasaka, and establishes a characterization of Gorenstein rings: I has finite Gorenstein dimension if and only if R is Gorenstein     April 1, 2016 Speaker: Jason McCullough, Rider University Title: Projective Dimension: Quadrics and Unmixed Ideals Abstract: Stillman's Question asks for a bound on the projective dimension of a homogeneous ideal I in a polynomial ring S in terms of the degrees of the minimal generators of I. In a previous talk, I presented our result that a codimension 2 ideal generated by n quadrics has projective dimension at most 2n-2, which is a sharp bound. In trying to extend Engheta's method for bounding the projective dimension of 3 cubics to the situation of 4 quadrics, we tried to find a characterization of unmixed ideals in low height and multiplicity. In this talk I will present two results: (1) a construction of a family of unmixed ideals of arbitrary height and multiplicity (both at least 2, not both equal to 2) with unbounded projective dimension and (2) a tight bound (6) on the projective dimension of ideals generated by 4 quadrics. The latter result requires a finite characterization of unmixed ideals up to degree 2 that circumvents result (1) above. Both results are joint work with Craig Huneke, Paolo Mantero and Alexandra Seceleanu   April 8, 2016 — Please note time change: 3:30 -- 4:45PM. Cross-listed with Workshop at the Kolchin Seminar Speaker: Askold Khovanskii, University of Toronto Title: Topological Galois Theory Abstract   April 15, 2016, 4—5 PM Speaker: Luchezar Avramov, University of Nebraska Title:  Representation Theory And Betti Tables Over Short Gorenstein Algebras Abstract     FALL 2015   September 18, 2015 Speaker: Laura Ghezzi, NYCCT Title: Sally modules and reduction number of ideals Abstract:  In this joint project with S. Goto, J. Hong and W. Vasconcelos, we study the relationship between the reduction number of a primary ideal of a local ring relative to one of its minimal reductions and the multiplicity of the corresponding Sally module. In this talk we use the fiber of the Sally modules of almost complete intersection ideals to connect its structure to the Cohen-Macaulayness of the special fiber ring   October 2, 2015 Speaker: Jooyoun Hong, Southern Connecticut State University Title: Sally modules and reduction number of ideals Abstract:  This is a continuation of the talk given by L. Ghezzi on September 18th and based on the joint work with L. Ghezzi, S. Goto, and W.V. Vasconcelos. In this talk, we show how a reduction number of a primary ideal in a 2-dimensinoal Buchsbaum ring can be bounded by the multiplicity of the Sally module and the Buchsbaum invariant of the ring   October 30, 2015 Speaker:  Mahdi Majidi-Zolbanin, LaGuardia Community College Title: On entropy in the category of perfect complexes with cohomology of finite length Abstract: In a recent work M. Kontsevich et al. defined entropy for a triangulated endofunctor of a triangulated category with a generator. In this talk we will compute this entropy in the category of perfect complexes with cohomology of finite length over a Cohen-Macaulay local ring, for the pull-back functor along an endomorphism of that ring   November 13, 2015: Join us for this joint event with the CUNY Logic Workshop, beginning at 3:45 pm. Speaker: Franz-Viktor Kuhlmann, University of Katowice Title: In search of a complete axiomatization for Fp((t))   November 14, and 15, 2015: Join us at the AMS Sectional Meeting at Rutgers University for these two special sessions: Special Session on Advances in Valuation Theory Special Session on Commutative Algebra   November 20, 2015 Speaker: Hans Schoutens, NYCCT Title: An axiomatic treatment of Cartier crystals Abstract: Cartier modules/crystals are objects that live in characteristic $p$ as they are essentially $p^{-1}$-linear endomorphisms. Examples are Frobenius splittings and Cartier operators on dualizing sheaves. I will review the main finiteness result due to Blickle and Schwede and then propose an axiomatic treatment as abstract functors, which could potentially also live in characteristic zero. A candidate for this axiomatic setting then would be via ultraproducts of Frobenii, but at this point, this is only wishful thinking   December 4, 2015 Speaker:  Herivelto Martins Borges Filho, University of Sao Paulo at Sao Carlos, Brazil Title: Frobenius nonclassical curves and minimal value set polynomials Abstract     SPRING 2015   February 13, 2015 Speaker: Dario Spirito, Università di Roma Tre Title: The Zariski topology on sets of semistar operations   February 27, 2015 Speaker: Rad Dimitric, CUNY-CSI Title: The theory of Slenderness Abstract: I will give an outline of what I think is one of the  most beautiful theories of mathematics, that fuses together algebra, topology, set theory and potentially many other areas waiting to be connected. This will also serve as a quick snapshot of my forthcoming monograph "Slender modules and rings" with Cambridge University Press   March 13, 2015 Speaker: Malgorzata Marciniak, LaGuardia Community College, CUNY Title: Analytic continuation problems in complex toric varieties Abstract: Toric varieties with their combinatorial structure make a wonderful area for solving analytic problems. During my talk I will shortly introduce necessary notation related to fans that are associated with toric structures and present few results about Hartogs and Hartogs-Bochner extensions on toric surfaces   May 1, 2015 Speaker: Hussein Mourtada, Institut de Mathématiques de Jussieu-Paris Rive Gauche Title: Jet schemes and generating sequences of some divisorial valuations Abstract:  I will talk on the one hand about the notion of a generating sequence of a valuation, and on the other hand about the relation between jet schemes and divisorial valuations. I will then describe  how this relation allows one to construct generating sequences of some divisorial valuations; this provides a constructive approach to a conjecture of Teissier on resolution of singularities     FALL 2014   September 5, 2014 Speaker: Andrew Stout, Bronx Community College and NYU Title: The auto Igusa-zeta series of an algebraic curve Abstract:  We study endomorphisms of Noetherian complete local rings in the context of motivic integration. Using the notion of an auto-arc space (similar to Weil restriction), we introduce the reduced auto Igusa-zeta series at a point, which appears to measure the degree to which a variety is not smooth at a point. We conjecture a closed formula in the case of irreducible curves with  one singular point. Finally, we show that the auto Poincar\'{e} series is rational in the case of the cuspidal cubic and the node and connect this with questions concerning new types of motivic integrals. These new types of motivic integrals offer a bridge between some of Schoutens' work and the more traditional approach to motivic integration worked out by Kontsevich, Denef, Loeser, et al   September 19, 2014 There is no meeting on this date. We will meet next on October 3, 2014.   October 3, 2014 Speaker:  Nikita Miasnikov Title: Asymptotic Invariants and flatness of local endomorphisms Abstract:  We will discuss how Kunz's Theorem can be combined with Nagata's Flatness Criterion to obtain an approach for attacking the long-standing Zariski-Lipman Conjecture.  While this approach is not new and has not so far solved the conjecture,  new is our focus on a certain numerical aspect of the situation at hand   October 10, 2014 Speaker:  Mahdi Majidi-Zolbanin, LaGuardia Community College Title: On additivity of local entropy under flat extensions Abstract:  Let f: (R,m) --> S be a local homomorphism of Noetherian local rings. Consider two endomorphisms of finite length (i.e., with zero-dimensional closed fibers) $phi: R --> R$ and $\psi: S --> S$, satisfying $\psi\circ f=f\circ\phi$. Then $\psi$ induces a finite length endomorphism $\overline{\psi}: S/f(m)S --> S/f(m)S$. When f is flat, under the assumption that S is Cohen-Macaulay we prove an additivity formula: $h_{loc}(\psi)=h_{loc}(\phi)+h_{loc}(\overline{\psi})$ for local entropy   October 24, 2014 Speaker: Lars Winther Christensen, Texas Tech University Title: Stable homology Abstract:  Tate homology and Tate cohomology was originally defined for modules over finite group algebras. The cohomological theory has a beautiful generalization---stable cohomology---to the setting of associative rings. The properties of the corresponding generalization of the homological theory have been poorly understood, and I will report on recent progress in this direction. The talk is based on joint work with Olgur Celikbas, Li Liang, and Grep Piepmeyer   November 7, 2014 Speaker: Bernadette Boyle,  Sacred Heart University Title: The Unimodality of Pure O-sequences Abstract:  Since the 1970s, great interest has been taken in the study of pure O-sequences, which are in bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these by their shape. It has been shown that all monomial complete intersections, Artinian algebras in two variables and Artinian level monomial algebras with type two in three variables have unimodal Hilbert functions. In this talk, we will look at pure O-sequences of type three in three variables and pure O-sequences of type two in four variables, showing that they are strictly unimodal   November 21, 2014 Speaker:  Hans Schoutens, NYCCT and The Graduate Center, CUNY Title: Big Cohen-Macaulay modules in mixed characteristic for complete toric rings Abstract: Over a local ring R, a module is called Cohen-Macaulay (CM) if its depth equals the dimension of R (`big' refers to the fact that there is no need to assume that they are finitely generated). Such modules are of great interest as they can be used to prove many of the homological conjectures. The current state of affairs: they always exist in equal characteristic, and up to dimension three in mixed characteristic. Consequently, many homological conjectures are still open in mixed characteristic in higher dimensions. Geometers love toric varieties (=variety containing a torus together with the action of this torus), as one can compute more easily in them (e.g., resolution of singularities). Moreover, Hochster has shown that normal toric varieties are CM. Since the normalization of a toric variety is again toric, any local ring of a toric variety admits therefore a finitely generated CM. It is well-known that toric varieties are defined by binomial equations, and in the spirit of Eisenbud-Sturmfelds, we define the following more general class: a complete toric ring is the quotient of a complete regular local ring modulo an ideal defined by binomials (in some chosen regular system of parameters) with coefficients arbitrary units; if the coefficients are all +/-1 then I call it purely toric. I do not know whether Hochster’s result is still true (are normal complete toric rings CM?), nor whether their (completed) normalizations are again toric. In positive characteristic, I can nonetheless prove in certain cases that they admit a finitely generated CM, but this talk is about mixed characteristic. Here I prove that purely toric complete local rings admit a (big) CM. The proof uses Witt vectors and the construction of big CM's in positive characteristic via absolute integral closure due to Hochster-Huneke   December 5, 2014 Speaker:  Leila Khatami, Union College Title: Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit     SPRING 2010   February 19, 2010 Speaker: Jooyoun Hong, Southern Connecticut State University Title: The equations of almost complete intersections.   February 26, 2010 Postponed due to inclement weather Speaker: Janet Striuli, Fairfield University   April 9, 2010 Speaker: Lance Bryant, Shippensburg University Title: Numerical Semigroups and One-Dimensional Rings Abstract: This talk will focus on the relationship between a numerical semigroup ring R and its corresponding semigroup. We are primarily interested in the Cohen-Macaulay and Gorenstein properties of the associated graded ring of R (more generally, of the associated graded ring of an ideal filtration in R) and how these properties are reflected in the semigroup. We will also consider a more general class of rings, namely one-dimensional analytically irreducible domains   April 23, 2010 Speaker: Timothy B.P. Clark, Northwestern University Title: Posets, CW Complexes and Free Resolutions of Monomial Ideals Abstract: Let P be a finite partially ordered set (poset) with set of atoms A and let k be a field.  Considering certain open intervals of P, we utilize a construction of Tchernev to produce a sequence of k-vector spaces and vector space maps D(P). When a poset map from P to Z^n exists, the sequence D(P) is homogenized to approximate a free resolution of R/N where N is the monomial ideal in k[x_1,\ldots,x_n] whose set of minimal generators are the images of the atoms of P under the above poset map.  When this approximation is an exact complex of multigraded modules, we call it a poset resolution of R/N. In our main results, we show that poset resolutions provide a common framework from which to view a number of (not necessarily minimal) resolutions previously constructed using distinct methods.  For various classes of monomial ideals, the posets taken under consideration include the Boolean lattice, the lcm-lattice associated to N, the Scarf complex of N, a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal N and the face poset of a regular CW complex   May 7, 2010 Speaker: Aline Hosry, University of Missouri Title: A theorem of Briançon-Skoda type for F-rational rings Abstract
 SPRING 2017 The seminar will be held on February 3, 17, March 3, 17 and April 21   February 3 , 2017 Speaker: Basanti Sharma Poudyal, University of Texas at Arlington Title:  Existence of totally reflexive modules in local graded rings with Hilbert series $1+et+(e-1)t^2$ Abstract:  Let $(A,m)$ be a Noetherian local graded ring with Hilbert series $1+et+(e-1)t^2$. It is known that the existence of exact zero divisors implies the existence of non-free totally reflexive modules. We are interested in the existence of these modules in the absence of exact zero divisors. In a recent study, Vraciu and Atkins constructed an example of a ring of codimesion 8 that does not have exact zero divisors, but has non-free totally reflexive modules. In this talk, we will give a class of rings of codimension 5 and higher admitting totally reflexive modules, but without having exact zero divisors   February 17 , 2017 Speaker: Laura Ghezzi, New York City College of Technology (CUNY) Title: Invariants of Cohen-Macaulay rings associated to their canonical ideals Abstract: In recent joint work with Shiro Goto, Jooyoun Hong and Wolmer Vasconcelos we introduce new invariants of Cohen-Macaulay local rings. Our focus is the class of Cohen-Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are integers--the type of R, the reduction number of C--that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers--the roots of R and several canonical degrees. In this talk we first give an overview of the background and then we define and focus on the (basic) canonical degree   March 3 , 2017 Speaker: Fei Ye, Queensborough Community College, CUNY Title: A characterization of big Q-divisors on surfaces and its applications Abstract: On surfaces, there are three important methods to study adjoint linear systems: Reider's method using Bogomolov instability theorem for rank 2 vector bundles; Kawamata-Viehweg vanishing theorem,  Sakai's non-vanishing theorem. In this talk, I will present a generalization of Sakai's result  for  Q-divisors and show its connections with Bogomolov's theorem and Kawamata-Viehweg vanishing theorem.  As an application, I will also explain how we use this result to study adjoint linear systems. This is a joint work with Tong Zhang and Zhixian Zhu   March 17 , 2017 Speaker: Hassan El Houari, La Guardia Community College, CUNY Title: Locally nilpotent derivations and polynomial automorphisms Abstract:  Locally nilpotent derivations over polynomial rings are objects of great importance in many ﬁelds of pure and applied mathematics. Recently, it has made remarkable progress and became an important topic in understanding affine algebraic geometry and commutative algebra. In this talk, I will start with brief introduction of the basic properties of locally nilpotent derivations and show how some classical problems in affine algebraic geometry and commutative algebra can be formulated in the language of locally nilpotent derivations.  Then, I will talk about some algorithmic applications of locally nilpotent derivations in studying polynomial automorphisms of the affine space   April 21 , 2017 Speaker: Sam van Gool, CUNY Graduate Center Title: Sheaves and duality Abstract: We look at sheaf representations through the lens of universal algebra and Stone duality theory. As our main result, we exhibit a correspondence between soft sheaf representations of universal algebras and frame homomorphisms into subframes of pairwise commuting congruences of their congruence lattices. For distributive-lattice-ordered algebras, this allows us to dualize such sheaf representations into decompositions of Stone dual spaces. This talk is on joint work with Mai Gehrke (Paris), preprint available at http://samvangool.net/papers/gehrkevangool2016.pdf