Kolchin Seminar in Differential Algebra
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Academic year 2008–2009

Last updated on May 14, 2017.
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In Fall 2008, we welcomed Professor Michael Singer, North Carolina State University, as a visiting member of KSDA. The Fall semester was mainly devoted to studying the notes on Differential Algebraic Groups by Bernard Malgrange, beginning with prerequisites. A copy of the notes is available on request.

On April 1, 2009, Professor Jerald J.Kovacic, a long time organizer of KSDA, passed away. We all felt deeply the loss of this brilliant differential algebraist who had contributed greatly to the field. Our condolences goes to his partner Mr. Dan Kiser and to his family. We wish to thank all his friends and relatives for their generous donations to KSDA in his name. Jerry will be sorely missed.

Friday, September 5, 2008, 10:30 AM

Camilo Sanabria, Graduate Center, CUNY
Hors d'oeuvre to Malgrange ideas: Jet Bundles

Abstract: Jet Bundles are heavily used in Malgrange's later work. The purpose of the talk is to familiarize the audience with the concept of Jet Bundles and its basic properties.

For lecture notes, please click here.

Friday, September 12, 2008, 10:30 AM

Camilo Sanabria, Graduate Center, CUNY
Hors d'oeuvre to Malgrange ideas: Jet Bundles (continued)

For lecture notes, please click here.

Friday, September 19, 2008, 10:30 AM

Graduate Center, CUNY
General Discussions: Malgrange ideas: Jet Bundles

Abstract: Jet bundles are heavily used in Malgrange's later work. The purpose of these discussions is to continue to review the concept of jet bundles and their basic properties.

Friday, September 26, 2008, 10:30 AM

Varadharaj Ravi Srinivasan, University of Oklahoma
On Certain Towers of Extensions by Antiderivatives

Abstract: Let F be a characteristic zero differential field with an algebraically closed field of constants, E be a no-new-constant extension of F by antiderivatives of F and let y1, ..., yn be antiderivatives of E. The antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if yi in E satisfies certain conditions. We will discuss a new proof for the Kolchin-Ostrowski theorem and generalize this theorem for a tower of extensions by J-I-E antiderivatives and use this generalized version of the theorem to classify the finitely differentially generated subfields of this tower. In the process, we will show that the J-I-E antiderivatives are algebraically independent over the ground differential field. An example of a J-I-E tower is extensions by iterated logarithms. We will discuss the normality of extensions by iterated logarithms and produce an algorithm to compute the finitely differentially generated subfields of these extensions.

For lecture notes, please click here.

For further information please visit http://math.ou.edu/~vsrinivasan/Thesis-I.pdf.

Fridays, October 3, 10, 17, 24 2008, 10:30 AM

Camilo Sanabria, Graduate Center, CUNY
Entrée to Malgrange ideas: General Involutivity Theorem, I, II, III, IV

Abstract: I will motivate through various examples the meaning of Malgrange's Involutivity Theorem. I will expose the curvature as a particular case. The computations involved will smoothly lead to Malgrange's use of the Koszul Complex and its homology.

For lecture notes, please click here and here.

Friday, October 31, 2008, 10:30 AM

General Discussions: General Involutivity Theorem

The general discussion planned originally for October 24 was postponed to October 31. We had an open discussion on the Koszul Complex, its homology groups and their relation to the General Involutivity Theorem for partial differential equations on complex manifolds.

Friday, November 7, 2008, 10:30 AM

Richard Cohn, Rutgers University at New Brunswick
Differentially Transcendental Functions

Abstract: I shall discuss differentially transcendental functions, that is, analytic functions which do not satisfy algebraic differential equations with coefficients in the field of rational functions. Examples will be given and the Kolchin analogue of Liiouville's theorem will be discussed. Finally I shall present an old paper by Ritt and Gourin which uses Cantorian methods to prove the existence of a large class of such functions.

Friday, November 14, 2008, 10:30 AM

No Seminar. See Conference Below.

November 13-16, 2008, 10:30 AM

Rutgers University at Newark
The Third International Workshop on Differential Algebra and Related Topics

Invited speakers: Matthias Aschenbrenner: UCLA; Robert Bryant: Duke University and MSRI; Guy Casale: University of Rennes 1; Julia Hartmann: University of Heidelberg; Evelyne Hubert: INRIA Sophia Antipolis; Jerry Kovacic: City University of New York (CCNY); Anton Leykin: IMA, University of Minnesota; Snigdhayan Mahanta: University of Toronto; Sylvie Paycha: University of Blaise Pascal; Anand Pillay: University of Lees; Emma Previato: Boston University; Bin Zhang: Sichuan University. Full program and abstracts are available here.

Friday, December 5, 2008

Li Guo, Rutgers University at Newark (10:30 AM)
Rota-Baxter algebra and related structures


Abstract: We discuss structures related to the Rota-Baxter algebra, such as differential Rota-Baxter algebra, integro-differential algebra, relative Rota-Baxter algebra and Rota-Baxter family.

William Sit, City College of New York (11:40 AM)
Constrained Systems


Abstract: This is an introduction to constrained systems. A constrained system is given by two differential polynomials p(y) and q(y) in the unknown y, having the property that any two elements satisfying p(y) = 0 and q(y) ≠ 0 will generate isomorphic differential fields. Examples and proofs of constrained and non-constrained systems will be given. Only basic graduate algebra is required.

Note: The second talk will be followed by an informal discussion on constrainability and computability after lunch.

Friday, December 12, 2008, 10:30 AM

Michael F. Singer, North Carolina State University
Differential Groups and the Gamma Function

Abstract: I will develop a Galois theory of linear difference equations where the Galois group is a linear differential group, that is, a group of matrices whose entries satisfy a fixed set of polynomial differential equations. These groups measure the differential dependence among solutions of linear difference equations. I will show how this theory can be used to give a new proof for Hölder's Theorem that the Gamma function satisfies no differential polynomial equation, as well as new results concerning differential dependence of solutions of higher order difference equations.

For lecture notes, please click here.

KSDA Winter break begins on Friday, December 19, 2008 (no seminar).

Friday, January 30, 2009, 10:30 AM

Richard Churchill, Graduate Center and Hunter College
A Topological Approach to Constrained Points

Abstract: Generic points and specializations are concepts from algebraic geometry, which were quite popular when the Weil approach was fashionable, but which receive scant attention in the standard contemporary texts. Moreover, when these modern texts do address the concepts, the definitions tend to be topological rather than algebraic.

In differential algebraic geometry the analogous concepts are still generally treated in the classical spirit, despite the fact that it is quite easy to bring the language up to date. Moreover, in this subject one encounters, along with generic points and specializations, the unintuitive concepts of a "constrained point," accompanied by a definition in the classical spirit. The purpose of this talk is to give a simple topological definition, and to show how the concept generalizes the notion of an element of an extension field being algebraic over the ground field.

The talk will be predominantly at the level of elementary point-set topology, and no familiarity with algebraic geometry or differential algebra will be assumed.

Friday, February 6, 2009, 10:30 AM

Richard Churchill, Graduate Center and Hunter College
Constrained Points in the Context of Differential Algebra

Abstract: In my lecture of January 30, 2009, I developed the idea of a constrained point in an arbitrary topological space, and related the idea to algebraic field extensions and algebraic closures. In this lecture I will relate the idea to differential algebraic field extensions and differential closures, with the hope to explain why differential closures are defined as they are.

Friday, February 13, 2009, 10:30 AM

Camilo Sanabria, Graduate Center, CUNY
Symmetries of meromorphic connections over Riemann Surfaces

Abstract: We give ourselves a vector bundle over a Riemann surface with a meromorphic connection. We call an automorphism of the surface a "symmetry of the connection" if it lifts to a bundle automorphism sending parallel sections to parallel sections. For example, if the automorphism of the surface is the identity the symmetries are given by the differential Galois group. I will explain how to get all the rest of symmetries using the Fano group.

The results are explained in full detail on arXiv:0805.4649.

For lecture notes, please click here

Friday, February 20, 2009, 10:30 AM

David Marker, University of Illinois at Chicago
Embedding Differential Algebraic Groups in Algebraic Groups

We will survey Pillay's proof that any differential algebraic group can be embedded into an algebraic group.

For references, please click here

For lecture notes, please click here.

Friday, February 27, 2009, 10:30 AM

Camilo Sanabria, Graduate Center, CUNY
Symmetries of meromorphic connections over Riemann Surfaces, II

Abstract: (Continuing from Feb 13). We give ourselves a vector bundle over a Riemann surface with a meromorphic connection. We call an automorphism of the surface a "symmetry of the connection" if it lifts to a bundle automorphism sending parallel sections to parallel sections. For example, if the automorphism of the surface is the identity the symmetries are given by the differential Galois group. I will explain how to get all the rest of symmetries using the Fano group.

The results are explained in full detail on arXiv:0805.4649.

Friday, March 6, 2009, 10:30 AM

Raymond Hoobler, City College and Graduate Center, CUNY
Differential Spectrum and Other Matters

Abstract: I will discuss the Tannakian approach to Fano's Theorem following Nguyen's thesis and a paper of Compoint and Weil. I will state, but not prove, a descent theorem for linear operators.

Friday, March 13 and 20, 2009, 10:30 AM

Raymond Hoobler, City College and Graduate Center, CUNY
Differential Spectrum and Other Matters

Abstract: I will approach Kovacic's Diff Spec from the standpoint of local ringed spaces and attempt to answer some of the questions raised in his talk last year.

For lecture notes, please click here

Friday, March 27 and April 3, 2009, 10:30 AM

Leon F. Pritchard, York College, CUNY
William Keigher, Rutgers University at Newark
On Partitioned Differential Quasifields, I and II

Abstract: A differential quasifield is a natural generalization of a differential field in characteristic p > 0. Elementary properties of differential quasifields are considered, and a generalized version of the theorem on the connection between linear independence over constants and the Wronskian is presented.

For lecture notes, please click here

April 8 through 17, 2009: Spring Recess (No seminar scheduled currently. Bernard Malgrange visiting April 6 to April 25)

Friday, April 24, 2009, 10:30 AM

Li Guo, Rutgers University at Newark
Classification of Rota-Baxter type operators

Abstract: Several linear operators have arisen over the years from studies in mathematics and mathematical physics, such as the differential operator, integral (Rota-Baxter) operator, averaging operator and Nijenhuis operator. Many years ago, Rota proposed to find a classification of such operators, but little progress has been made since. We formulate a framework to consider the classification problem of Rota-Baxter type operators. We then discuss our conjectural answer to this problem. Related classification problems for differential operators will also be considered. (This is joint work with W. Sit and R. Zhang.)

For lecture notes, please click here.

Friday, April 24, 2009, 2:00 PM to 5:00 PM

Bernard Malgrange met informally with KSDA in Room 8404.

Friday, April 24, 2009, 6:30 PM

There was a gathering to celebrate the life and achievement of Dr. Jerald J. Kovacic at his home on April 24, 2009, starting at 6:30 pm. R.S.V.P. ksda@sci.ccny.cuny.edu. Please click the links here for more information.
Copy of New York Times Obituary on April 9, 2009
To view the original New York Times article and to leave condolences, upload photos, etc. on legacy.com

Friday, May 1, 2009, 10:30 AM

David Marker, University of Illinois at Chicago
Embdedding Differential Algebraic Groups in Algebraic Groups, II

Abstract: In my talk on February 20, I discussed Pillay's result that every connected differential algebraic group can have a differential embedding into an algebraic group in the special case when the original group is finite dimensional. In this talk I will explain the general case.

For lecture notes, please click here.

Friday, May 8, 2009, 10:30 AM

William Sit, City College of New York
Algebraic Constraints on Initial Values of Differential Equations

Abstract: Initial value problems of differential algebraic equations are of practical importance in many applications. Three related problems have been investigated: singularities, numerical solutions, and existence and uniqueness of solutions. I will briefly discuss the difficulties posed by each. In this talk, we describe a computational approach to obtain algebraic constraints on initial values that would guarantee existence and uniqueness of solutions. These constraints may be implicitly implied by the differential equations themselves. We apply this approach to a class of non-linear systems of first order ordinary differential equations and in addition to obtaining the constraints, the algorithms will also provide equivalent systems where the first derivatives of the dependent variables (unknowns) are explicitly given in terms of the unknowns. This vector field can be integrated in a numerically stable way. Examples where singularities are exposed by the algorithm will be given. (This is joint work with F. Leon Pritchard of York College.)

Reference: F. L. Pritchard and W. Sit, "On Initial Value Problems for Ordinary Differential-Algebraic Equations," in Grobner Bases in Symbolic Analysis, M. Rosenkranz and D. Wang eds., Radon Series Comp. Appl. Math 2, 283-340, Walter de Gruyter, 2007.

For lecture notes, please click here


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