Kolchin Seminar in Differential Algebra 

The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
Academic
year 2011–2012 Last updated on January 18, 2018. 
Friday, August 27, 2011, No Meeting
Friday, September 2, 2011 at 10:15 a.m. Room 5382
Richard Churchill, Hunter College and Graduate Center, CUNY
An Introduction to Group CohomologyLet G be a group acting on an Rmodule N by linear automorphisms. In the talk I view N as the union of certain distinguished parameterized families of orbits of the action, and explain how group cohomology determines these families and how they are patched together.
No familiarity with group cohomology (or cohomology in general) is assumed; the necessary background will be sketched, and proofs will be detailed in lecture notes to be posted.
If time permits, an application to Galois theory will be presented.
Friday, September 9, 2011 at 10:15 a.m. Room 5382
Michael Singer, North Carolina State University
Linear Algebraic Groups as Parameterized PicardVessiot Galois GroupsAfter giving an introduction to the parameterized PicardVessiot theory (PPVtheory), I will discuss an inverse problem: which linear differential algebraic groups can occur as PPVGalois groups over k(x), where k is a differentially closed field with respect to some parametric derivations, and where x is not in k, x' = 1, and a' = 0 for all a in k. I will show that a linear algebraic group (considered as a linear differential algebraic group) is a PPVGalois group over k(x) if and only if its identity component has no one dimensional quotient as an algebraic group.
Friday, September 16, 2011 at 10:15 a.m. Room 5382
Alice Medvedev, University of California at Berkeley
Using Jet Spaces to Understand the Structure of Solutions Sets of Difference EquationsIn light of the Zilber trichotomy for minimal sets definable in differenceclosed fields, it is useful to know when a solution set of a system of difference equations admits something like a group structure. This talk will include a cursory sketch of the modeltheoretic background and a quick introduction to jet spaces, a construction in algebraic geometry. The focus of the talk is the details of using jet spaces of underlying algebraic varieties to obtain information about the structure of a solution set of a system of difference equations.
Let S be an automorphism of a field K. For a variety X with a rational dominant morphism F from X to S(X), the set of solutions of S(x) = F(x) is called the Svariety determined by X and F. When X is an algebraic group and F is a group homomorphism, this is a subgroup of X. When no proper infinite subgroup is itself an Svariety, and F is not purely inseparable, this group is minimal and falls into the second, "modular," case of the Zilber Trichotomy. Other modular nontrivial Svarieties always admit a finite cover by a modular group Svariety, and identifying these is the subject of this talk.
The main result is that any finite cover of an Svariety by a modular group Svariety must be a quotient by a (finite) group of algebraic automorphisms of the underlying algebraic group. To be more precise, let E be an algebraic group, let G be an algebraic group homomorphism from E to S(E), and let H be a morphism from the Svariety determined by E and G to the Svariety determined by X and F. If the fibers of H are finite, then there is a group N of automorphisms of E (as an algebraic group) such that the fibers of H are precisely the orbits of N.
Friday, September 23, 2011 at 10:15 a.m. Room 5382
Russell Miller, Queens College and Graduate Center, CUNY
Exploring Effectiveness in Differential FieldsFields have been well studied by computability theorists. Basic results about computable fields include Kronecker's Theorem and Rabin's Theorem. The former states that, in every computable finitely generated field F which is separable over its prime subfield, it is decidable which polynomials in F[X] are irreducible there, and it is also decidable which such polynomials have roots in F. The latter states that for every computable field E, there is a computable algebraic closure L of E, with E computably enumerable within L, but that E is a decidable subset of L if and only if irreducibility of polynomials in E[X] is decidable.
Differential fields K have not been nearly so well studied by computability theorists, and the effective versions of the corresponding theorems in the differential context have not yet been fully established. (The main existing result is Harrington's proof of the computability of differential closures of computable differential fields.) Indeed, it takes some work to determine exactly what the appropriate analogues are, let alone whether they hold. We will ask first what concept, for differential polynomials in K{Y}, best matches the notion of irreducibility for algebraic polynomials in F[X], arguing that the concept of a constrained pair is the notion we want. Then we will describe recent work, joint with Alexey Ovchinnikov, adapting part of Kronecker's Theorem to the context of differential fields and constrained pairs.
Friday, September 30, 2011 at 10:15 a.m. Room 5382
Please note there are no regular classes on September 30, 2011.
Jim Freitag, University of Illinois at Chicago
Noncommutative Linear Almost Simple Groups Are PerfectWe will develop the notion of an almost simple differential algebraic group (due to Cassidy and Singer). Cassidy and Singer asked if noncommutative almost simple groups are perfect. This question was originally answered via a definability theorem which works for more general problems. After giving a very simple instance of a result of this flavor, a different proof that noncommutative almost simple linear differential algebraic groups are perfect will be given.
Friday, September 30, 2011 at 2:00 pm a.m. Room 5382
Please note there are no regular classes on September 30, 2011.
Informal Afternoon Session with
Bernard Malgrange, Université Joseph Fourier – Grenoble.
Saturday, October 1, 2011 at 10 a.m. Room E920 Hunter College
Matthew HarrisonTrainor, University of Waterloo
Nonstandard Methods for Bounds in Differential Polynomial RingsWe present an extension of the nonstandard methods of van den Dries and Schmidt [Bounds in the Theory of Polynomial Rings over Fields: A Nonstandard Approach, Inventionnes Mathematicae, 76:77–91, 1984] to differential polynomial rings over differential fields. The motivation is the problem of the existence of bounds on degrees and orders for checking primality of radical (partial) differential ideals. We prove an equivalence of this problem with several others related to the Ritt problem, as well as giving new proofs of the existence of bounds for characteristic sets of minimal prime differential ideals and the Differential Nullstellensatz.
Joint work with Jack Klys and Rahim Moosa.
Saturday, October 1, 2011 at 2 p.m. Room E920 Hunter College
Anand Pillay, University of Leeds
The Painleve Equations: Strong Minimality and Geometric Triviality
Tuesday, October 4, 2011 at 10:15 a.m. Room 5382
This Tuesday follows a Friday schedule.
Alexander Levin, Catholic University of America
Bivariate BernsteinType Dimension Polynomials and\\ New Invariants of Finitely Generated DModulesIn 1971 J. Bernstein introduced a Hilberttype dimension polynomial associated with a finitely generated filtered Dmodule, which is a module over a Weyl algebra. Bernstein polynomials and their invariants play an important role in the theory of Dmodules and its applications. A Bernstein polynomial of a finitely generated Dmodule M is associated with a system of generators of M and carries invariants of M independent of the choice of generators. In this talk we consider a generalization of the Gröbner basis method to the case of a finitely generated free Dmodule equipped with a bifiltration associated with two natural term orderings. Then we apply properties of generalized Gröbner bases to prove the existence, determine invariants and outline methods of computation of Bernsteintype dimension polynomials in two variables that carry more invariants than classical Bernstein polynomials. In particular, we describe some new invariants of holonomic Dmodules.
Tuesday, October 4, 2011 at 2:00 p.m. Room 5382
This Tuesday follows a Friday schedule.
Informal discussions with Alexander Levin and Russell Miller
Friday, October 7, 2011. No meeting.
Friday, October 14, 2011 at 10:15 a.m. Room 5382
Shaoshi Chen, North Carolina State University
Termination Criteria for Zeilberger's Algorithm in Mixed CasesWe present three criteria on the termination of Zeilberger's algorithm in mixed cases. The first is for the differential and shift case; the second for the differential and qshift case; and the last for shift and qshift case. The criteria describe necessary and sufficient conditions on the existence of telescopers for hyperexponentialhypergeometric solutions in the above mixed cases. We will also review some results on which the criteria are based, including: a structure theorem on compatible rational functions and various generalizations of Hermite reduction in the mixed cases. This talk reports joint work with F. Chyzak, R. Feng, G. Fu, and Z. Li.
Friday, October 21, 2011 at 10:15 a.m. Room 5382
Raymond Hoobler, The City College and Graduate Center, CUNY
Spec^{Δ} déjà vuWe will review the construction of the differential spectrum and its sheaf of rings while correcting an error made at the end of the spring series of talks. We will prove almost surjectivity for differential rings and differential modules, sketch Chevalley's Theorem and conclude by showing that Δflat maps essentially of Δfinite type are open.
For lecture notes, please click here.
Friday, October 28, 2011 at 10:15 a.m. Room 5382
Raymond Hoobler, The City College and Graduate Center, CUNY
Projective Algebraic Varieties Are Affine Differential VarietiesA key observation of Cassidy will allow us to explain that a projective algebraic variety over constants becomes an affine differential variety over a differential field. This raises interesting questions about representations of Abelian varieties and, more specifically, the Weierstrass elliptic curve, in differential algebraic geometry.
Friday, November 4, 2011 at 10:15 a.m. Room 5382
Moshe Kamensky, University of Notre Dame
Tensor Categories for Fields with OperatorsI will describe work in progress, in which I attempt to describe the category of representations of a linear group defined in a field with operators, in the sense of the theory of Moosa and Scanlon.
Friday, November 11, 2011 at 10:15 a.m. Room 5382
Wei Li, KLMM, Chinese Academy of Sciences, Beijing
Sparse Differential Resultant for Laurent Differential PolynomialsIn this talk, we introduce the concepts of Laurent differential polynomials and Laurent differentially essential systems and give a criterion for a system to be a Laurent differentially essential one in terms of its support. We then define the sparse differential resultant for a Laurent differentially essential system and discuss its basic properties, which are similar to those of the algebraic sparse resultant. (Proofs will be given if time permits). In particular, we obtain the order and degree bounds for the sparse differential resultant. Based on these bounds, we propose an algorithm to compute the sparse differential resultant. The algorithm is single exponential in terms of the order, the number of variables, and the size of the Laurent differential system.
This is a recent joint work with my advisor XiaoShan Gao, and Dr. ChunMing Yuan. For a preprint of our paper, please click here.
For the lecture slide presentation, please click here.
Friday, November 18, 2011 at 10:15 a.m. Room 5382
Phyllis J. Cassidy, Smith College and The City College, CUNY
Examples as AvatarsWe will look at examples from the classical literature on nonlinear differential equations literature. These intriguing examples seem to embody parts of differential algebraic geometry not yet fully developed. They are are expressible in the language of differentially closed fields (both ordinary and partial), but seem to be in search of a proper context therein. Let F be a differentially closed field. Some of these examples are:
 The ColeHopf transformation of the Burgers equation (without external force) into the Heat equation, which we will interpret as an additive differential algebraic group lying over a differential algebraic variety (both varieties defined by evolution equations);
 the interpretation of a proper Zariski dense differential algebraic subgroup of SL_{2}(F ) as the stabilizer in PSL_{2}(F ) of a Riccati variety in the projective line over F;
 differential algebraic subvarieties of the Painlevé II variety that are homogeneous spaces for differential algebraic groups;
 the Painlevé I (and maybe the KdV variety), in its interpretation as a differential algebraic subvariety of the multiplicative group G_{m}(F ) whose fibers are principle homogeneous spaces for my favorite differential algebraic subgroup of G_{m}(F ) (and the link between this group and Hirota derivatives);
and, if time, stamina and preparation permit,
 a peek at conservation laws associated with the Heat equation, as interpreted in the language of differentially closed fields.
The applicable definitions and basic notions of differential algebra will be explained.
For the lecture slides, please click here.
Friday, November 18, 2011 at 2:00 p.m. Room 5382
Wei Li, KLMM, Chinese Academy of Sciences, Beijing
Differential Chow FormIn this talk, we present an intersection theory for generic differential polynomials. We show that the intersection of an irreducible differential variety of dimension d and order h with a generic differential hypersurface of order s is an irreducible variety of dimension d 1 and order h+s. We use this to prove the dimension conjecture for generic differential polynomials. We define the differential Chow form for an irreducible differential variety in this setting and establish for it most of the properties analogous to those of the Chow form in the algebraic case. Furthermore, we define a generalized differential Chow form and prove its properties. As an application of the generalized differential Chow form, we define the differential resultant of n+1 generic differential polynomials in n variables and prove properties similar to those of the Sylvester resultant for two univariate polynomials and the Macaulay resultant for multivariate polynomials.
This is a joint work with XiaoShan Gao and ChunMing Yuan.
For the lecture slide presentation, please click here.
Friday, November 25, 2011 (Thanksgiving holiday, no seminar)
Friday, December 2, 2011 at 10:15 a.m. Room 5382
Nathan Penton, Graduate Center, CUNY
The Tannakian Theory of GroupoidsIn a 2008 thesis, Giorgio Trentinaglia proved a Lie groupoid analogue of the Tannakian correspondence between groups and categories of representations. I will describe his results, and, as time permits, speculate on how they could be adjusted to apply to the various groupoids arising in nonlinear settings, potentially leading towards a nonlinear analogue of the Tannakian approach to differential Galois theory.
Friday, December 9, 2011 at 10:15 a.m. Room 5382
David Marker, University of Illinois at Chicago
Strongly Minimal Sets in Differential FieldsA fundamental insight of modern model theory is that understanding the combinatorial geometry of strongly minimal sets in differentially closed fields is a crucial first step in differential algebraic geometry. In this lecture I will give a detailed overview and if time permits, I will go through the proofs of some of the key parts of the trichotomy theorem.
Friday, December 9, 2011 at 2:00 p.m. Room 5382
Nathan Penton, Graduate Center, CUNY
The Tannakian Theory of Groupoids, Part IIIn the second part of this talk, we will go into more detail on background material on Tannakian Categories and groupoid associated to PDEs, and finally state the main duality result. If time permits, we will begin discussing background material on stacks, leading to a modest generalization of the main result.
Saturday, December 10, 2011 at 10:00 a.m. Room E920 Hunter College
Andrey Minchenko, University of Western Ontario
Almost Simple Differential Algebraic GroupsA differential algebraic group (DAG) G is almost simple if all its proper normal differential algebraic subgroups have smaller differential type. For any DAG G, the JordanHölder Theorem says that G has a subnormal series whose factors are almost simple DAGs. This result shows the importance of almost simple DAGs, but in general, the structure of such a DAG G is still not clear. However, it was recently shown by Freitag that if G is nonabelian, then it must be perfect. We use this fact to study the structure of nonabelian almost simple linear differential algebraic groups.
December 16, 2011 through January 20, 2012: Winter Break.
Friday, January 27, 2012 at 10:15 a.m. Room 5382
Nathan Penton, Graduate Center, CUNY
Jet Bundles and PDEsThe language of jet bundles allows a formalization of the geometric aspects of nonlinear partial differential equations, in much the same way that varieties formalize the geometric aspects of polynomials. As such, they provide a natural setting for the application of differential algebra to nonlinear problems. In this talk, I will provide an introduction to PDEs and their symmetries from the jet bundle viewpoint, including details and examples, with the goal of defining some of the differential algebraic structures and transformation groupoids that arise in this study.
Although this talk consists of expansions, clarifications, and corrections of notions touched upon in my talk last month, nothing from that talk will be assumed.
Friday, February 3, 2012 at 10:15 a.m. Room 5382
Christian Dönch, Research Institute for Symbolic Computation. Johannes Kepler University, Linz, Austria
Alexander Levin, The Catholic University of America
Evaluation of the Strength of Systems of Partial Differential and Difference Equations via Differential and Difference Dimension Polynomials, Part IIn the first part of this talk the speakers will review basic facts about univariate and multivariate differential and difference dimension polynomials and their relation to the strength of systems of algebraic differential and difference equations. We will also describe invariants of such polynomials and main methods of their computation.
Friday, February 3, 2012 at 2:00 p.m. Room 5382
Christian Dönch, Research Institute for Symbolic Computation. Johannes Kepler University, Linz, Austria
Alexander Levin, The Catholic University of America
Evaluation of the Strength of Systems of Partial Differential and Difference Equations via Differential and Difference Dimension Polynomials, Part IIIn the second part of the talk we will introduce the concepts of a Gröbner basis with respect to several orderings and relative Gröbner basis and show how properties of these bases can be applied to the computation of multivariate dimension polynomials. Then we will present algorithms of computation of such polynomials and use them to determine the strength of several systems of PDEs of mathematical physics and their difference analogs obtained from different types of difference schemes.
For the lecture slides, please click here.
Friday, February 10, 2012 at 10:15 a.m. Room 5382
William Sit, City College of New York
Anomaly of Intersection in Differential AlgebraThis is a tutorial talk which should be of interest to algebraic geometers interested in differential algebraic geometry. Ritt gave an example of two differential algebraic varieties each of differential dimension two in differential affine threespace, whose intersection contains only one single (singular) point. This is in stark contrast to the well behaved intersection result in algebraic geometry. In this talk, I will review briefly the statements of a few deep theorems in differential algebra (the Component Theorems, the Preparation Equation and Congruence, Levi's Lemma and the Low Power Theorem) and work out the details of Ritt's example as an illustration to these theorems.
For lecture notes, please click here.
Friday, February 17, 2012 at 10:15 a.m. and 2:00 p.m. Room 5382
Nathan Penton, Graduate Center, CUNY
Jet Bundles and PDEs
Parts II (10:15 am) and III (2:00 pm)
Both talks canceled due to speaker falling sick. Rescheduled for March 16.
Friday, February 24, 2012 at 10:15 a.m. Room 5382
Andrey Minchenko, University of Western Ontario
Almost Simple Differential Algebraic GroupsJordanHölder theorem for differential algebraic group reveals the importance of the problem of classification of almost simple differential algebraic groups. After a review of basic definitions and theorems, we will discuss a new approach for solving this problem.
Andrey Minchenko will also be giving a talk on Linear conjugacy in simple algebraic groups at the Representation Theory Seminar, at 1:00 pm, Room 3209
Friday, March 2, 2012 at 10:15 a.m. Room 5382
Informal discussion on Differential Dimension
Friday, March 2, 2012 at 2:00 p.m. Room 6417
This talk is part of the CUNY Logic Workshop.
Zoé Chatzidakis, Université Paris
Algebraic dynamics, Difference Fields and Model Theory
Friday, March 9, 2012 at 10:15 a.m. Room 5382
William Simmons, University of Illinois, Chicago
Identifying Complete Differential VarietiesIn classical algebraic geometry, the geometric role of compactness is played by the property of completeness: An algebraic variety V is complete if for every variety W, the projection V × W → W is a closed map with respect to the Zariski topology. The fundamental theorem of elimination theory asserts that projective varieties are complete. What happens with differential varieties, i.e., solution sets of differential polynomial equations over differential fields? We discuss several approaches to the problem, with our main focus being a positive quantifier elimination test of van den Dries that was adapted to a differential valuative criterion by Pong.
For lecture notes, please click here.
Friday, March 16, 2012 at 10:15 a.m. and 2:00 p.m. Room 5382
Nathan Penton, Graduate Center, CUNY
Jet Bundles and PDEs
Parts II (10:15 am) and III (2:00 pm)The language of jet bundles allows a formalization of the geometric aspects of nonlinear partial differential equations, in much the same way that varieties formalize the geometric aspects of polynomials. As such, they provide a natural setting for the application of differential algebra to nonlinear problems. In these two talks, I will begin with a quick review of the introduction to PDEs started in the first talk on January 27 and continue to explore their symmetries from the jet bundle viewpoint, including details and examples, with the goal of defining some of the differential algebraic structures and transformation groupoids that arise in this study.
Friday, March 23, 2012 at 10:15 a.m. Room 5382
Carlos Arreche, Graduate Center, CUNY
Differential Galois Theory for Modules with (Nonintegrable) Connection over Differential RingsIn arXiv:math/0203274v1, Y. André develops a general Galois theory of bimodules with connection, which contains as special cases the classical PicardVessiot theory, as well as the more recent Galois theories of difference and differentialdifference equations. This is accomplished essentially by showing that, under suitable conditions, the category of such objects is Tannakian. We will restrict our attention to the (classical) case of modules with connection over a differential ring. After briefly recalling the Tannakian formalism as it applies to PicardVessiot theory, we will develop the relevant notions from André's paper and prove that if a finitely generated module over a simple differential ring admits a (not necessarily integrable) connection, then it must be projective. This is the main ingredient in the proof of the Tannakian theorem. We will develop an explicit example of a module with nonintegrable connection to illustrate the theory.
Friday, March 30, 2012 at 10:15 a.m. Room 5382
David Marker, University of Illinois, Chicago
Internality in Differentially Closed FieldsI will discuss the PillayZiegler internality theorem which gives a relatively straightforward approach to the HrushovskiSokolovic result that nonmodular strongly minimal sets are nonorthogonal to the constants.
Friday, March 30, 2012 at 2:00 p.m. Room 5382
Carlos Arreche, Graduate Center, CUNY
Kovacic's Algorithm for Second Order Homogeneous Linear Differential Equations with ParametersIn 1986, using the classification of the algebraic subgroups of SL(2,C), where C is an algebraically closed field of characteristic zero, J. Kovacic developed an algorithm for solving second order linear homogeneous differential equations with coefficients in C(x). The key fact is that the differential Galois group associated to such an equation is realizable as an algebraic subgroup of SL(2, C) after a possible change of variables. In this talk we will discuss how to generalize Kovacic's algorithm to compute the differential Galois group associated to a second order linear homogeneous differential equation with one differential parameter. In this case the differential Galois group is realizable as a differential algebraic subgroup of SL(2,K), where K is some differential field, again possibly after a change of variables.
Friday, April 6 and 13, 2012, School Holidays, No Seminar.
Friday, April 20, 2012 at 10:15 a.m. Room 5382
Earl Taft (Rutgers University)
Lie Bialgebra Structures on Witt and Virasoro Algebra and their Continuous DualsWe survey the known Lie bialgebra structures on the Witt algebras W_{1} = Der k[x] and W = Der k[x, x^{1}], and the Virasoro algebra V, the central extension of W, focusing on remaining open problems. The continuous duals of these Lie algebras can be identified with the space of linearly recursive sequences, and we discuss their structures as Lie bialgebras. In particular, we give a recursive relation for the Lie bracket of two linearly recursive sequences in terms of the recursive relations satisfied by the two sequences.
Friday, April 27, 2012 at 10:15 a.m. Room 5382
Li Guo, Rutgers University at Newark
Differential Type Operators, Rewriting Systems and GroebnerShirshov BasesRota asked the question of classifying all linear operators that can be defined on associative algebras. There are quite a few such operators, such as endomorphisms, derivations and RotaBaxter operators. But their classification was out of reach until recently when we put the question in the framework of operated algebras, and related this problem to rewriting systems and GroebnerShirshov bases. We illustrate how good operators are equivalent to good systems and good bases in the special case of differential type operators. This is a joint work with William Sit and Ronghua Zhang.
For lecture slides, please click here.
Friday, April 27, 2012 at 2:15 p.m. Room 5382
Uma Iyer, Bronx Community College, CUNY
On the Quantum TorusDerivations and skewderivations on the quantum torus have been wellstudied. We present our work on the algebra of quantum differential operators on the quantum torus. In particular, this algebra is a simple, left and right Noetherian, domain of GKdimension 3n where n is the number of variables. We also present this algebra as a skew group algebra over a simple domain. This is joint work with D. A. Jordan and T. C. McCune.
For lecture slides, please click here.
Friday, May 4, 2012 at 10:30 a.m. Room 3209
This is a talk at the New York Algebra Colloquium; and there is no talk at the Kolchin Seminar on May 4.
Alexey Ovchinnikov, Queens College, CUNY
Integrability Conditions for Systems of Ordinary Linear Differential Equations with Parameters.It turns out that one can substantially reduce the number of integrability conditions to be checked for a system of parameterized ordinary linear differential equations. We will discuss how to do this using the Galois theory of linear differential equations with parameters.
Friday, May 11, 2012 at 10:15 a.m. Room 5382
Andrey Minchenko, University of Western Ontario
Almost Simple Differential Algebraic GroupsWe will continue the discussion of the relation between almost simple and quasisimple linear differential algebraic groups (LDAGs). By a theorem of Altinel and Cherlin, every almost simple LDAG of finite Morley rank is quasisimple. We will look at the proof of this theorem and, using general theory of central extensions, discuss possible ways one could avoid the requirement for the Morley rank to be finite.
Have a nice summer.
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