Kolchin Seminar in Differential Algebra

Friday, November 13, 2009 at 10:30 a.m.

Lourdes Juan, Texas Tech University
Differential Central Simple Algebras and Picard-Vessiot Representations

I will start with a brief introduction to differential Galois theory. A differential field is a field K  with a derivation, that is, an additive map D:K→K  satisfying D (fg )=D (f ) g  +fD (g ) for f,g  in K. The field of constants C  of K  is the kernel of D. A differential central simple algebra (DCSA) over K  is a pair (A,D) where A  is a central simple algebra and D is a derivation of A  extending the derivation D  of its center K. Any DCSA, and in particular a matrix differential algebra over K, can be trivialized by a Picard-Vessiot (differential Galois) extension E  of K. In the matrix algebra case, there is a correspondence between K-algebras trivialized by E  and representations of the differential Galois group of E  over K  in PGL_n(C ), which can be interpreted as cocycles equivalent up to coboundaries.

Reference: Differential central simple algebras and Picard-Vessiot representations, with Andy Magid, Proc. Amer. Math. Soc.  136(6) (2008), 1911-1918.

Friday, November 20, 2009 at 10:30 a.m.

Richard Cohn, Rutgers University at New Brunswick
The Low Power and Low Weight Conditions

Let R be a differential polynomial ring. Let A, B ∈R  be such that A  is irreducible, B ≠ 0 , and B  is annulled by the principal component V  of the variety of A . Is V  a component of the variety of B , or is it merely properly contained in a component? The answer is provided by the famed low power condition which is necessary and sufficient for V  to be a component. For the corresponding question concerning difference polynomials, the analogous low weight condition is necessary, but sufficient only with further restrictions.
After a brief review of the differential case, I will define the low weight conditions and give examples to illustrate the claims above, and sketch the proof of necessity.

Friday, November 27, 2009 at 10:30 a.m.

Thanksgiving Week, No Meeting

Friday, December 4, 2009 at 10:30 a.m.

Raymond Hoobler, Graduate Center and The City College, CUNY
FPQC Descent and Grothendieck Topologies in a Differential Setting

Abstract: I will summarize flat descent for the category of differential ring and then introduce the δ-flat Grothendieck topology. Basic theorems on descent, sheaves, and non-abelian H¹ will be carefully stated and proved using Milne: Etale Cohomology, as a reference. This talk generalizes Galois descent and non-abelian Galois cohomology to the differential ring setting. I will attempt to properly and carefully motivate the concepts involved.

Friday, December 11, 2009 at 10:30 a.m.

Raymond Hoobler, Graduate Center and The City College, CUNY
Differential Azumaya Algebras

Abstract: I will use the descent and Grothendieck topology machinery to classify differential line bundles and differential locally free sheaves over a differential ring R. Differential Azumaya algebras will then appear as a natural generalization of differential central simple algebras as defined by Lourdes Juan and Andy Magid. Finally some short exact sequences of sheaves in the δ-flat topology will be constructed using Picard-Vessiot theory which tie together results over R  and the ring R^δ  of constants of R.

KSDA will meet most Fridays at 10:30 AM at the Graduate Center. The purpose of these meetings is to introduce the audience to differential algebra. The lectures will be suitable for graduate students and faculty and will often include open problems. Presentations will be made by visiting scholars, local faculty, and graduate students. Unless the contrary is indicated, all morning meetings will be in Room 6421. This room may be difficult to find; please read the following directions. When you exit the elevator on the 6th floor, there will be doors both to your left and to your right. Go through the doors where you see the computer monitors, turn right and then immediately left near the drinking fountains. At the end of the corridor turn left, and then left again into a short hallway just past the drinking fountains. The room is at the end of this short corridor. All afternoon meetings will be in Room 8404, which is to the right of the cafeteria on the 8th floor. When you go to the GC you will have to sign in, and it is required that you have some photo ID with you. For directions to the Graduate Center, please click here, and for more on security requirements for entering the premise, please click here.

For past lecture notes and additional material, see below under "Other Academic Years".

Announcement: Kolchin Afternoon Seminar in Differential Algebra. This informal series began during the Spring Semester of 2009 and will be continued. We have reserved Room 8404 at the Graduate Center for discussions on Friday afternoons from 2:00-5:00 pm (please check with organizers). All are welcome.

Preliminary Announcements

Friday, December 11, 2009 at 2:00 p.m.
Please note this is an afternoon talk that will take place in Room 8404.

Alexander Levin, Catholic University of Washington
Title: TBA

Saturday, December 12, 2009 at 10:30 a.m.

NB: This is a special Saturday meeting of the Kolchin Seminar. Please note that the meeting will be held at the North Academic Center, Room 1/511E of The City College, 160 Convent Avenue, New York, NY 10031. Please click here for directions.

Dijiana Jakelic, University of North Carolina, Wilmington
Title: TBA

Friday, December 18, 2009 at 10:30 a.m.

Varadharaj Ravi Srinivasan, Rutgers University at Newark
Title: TBA

Past Talks, Fall 2009

Friday, August 28, 2009 at 10:30 a.m.

Richard Churchill (Graduate Center and Hunter College of CUNY)
An Algebraic Approach to Linear Ordinary Differential Equations

This talk will be an informal introduction to an algebraic approach to linear differential equations, including the Galois theory of linear differential operators. Familiarity with differential equations, beyond what would ordinarily be encountered in an undergraduate course, is not assumed. Analogies with standard Galois theory will be stressed. In particular, the relationship between Picard-Vessiot extensions and differential Galois groups of linear differential equations will be introduced as the counterpart of the relationship between splitting fields and Galois groups of polynomials. Computer applications will then be discussed.

Friday, September 4, 2009 at 10:30 a.m.

Alexey Ovchinnikov, Queens College, CUNY
Introduction to Tannakian Categories

This is the first of three related talks and will be an elementary and detailed introduction into the subject. Examples and selected proofs will be given during the informal afternoon session. The audience should wait until the second talk to see real and very interesting applications.

Friday, September 11, 2009 at 10:30 a.m.

Alexey Ovchinnikov, Queens College, CUNY
Tannakian Categories and Algebraic Groups

This is the second of three related talks and we will show how Tannakian categories give an intrinsic description of linear algebraic groups. In particular, we will see how such a group can be recovered from its representations. This idea leads to algorithms for computing differential Galois groups of systems of linear ODEs. Examples and selected proofs will be given during the informal afternoon session.

Friday, September 25, 2009 at 10:30 a.m.

Alexey Ovchinnikov, Queens College, CUNY
Differential Tannakian Categories and Differential Algebraic Groups

This is the third of three related talks. Differential algebraic groups have an extra structure (differential structure) and appear as Galois groups of systems of linear ODEs with parameters. In this talk we will discuss how this differential structure can be expressed within the Tannakian framework. Examples and selected proofs will be given during the informal afternoon session. This topic will be continued later during the Fall semester, with discussions on how differential Tannakian categories are related to Atiyah classes, differential schemes, and bundles with connections (from algebraic geometry).

Friday, October 2, 2009 at 10:30 a.m.

Varadharaj Ravi Srinivasan, Rutgers University at Newark
Differential Subfields of Liouvillian Extensions

Let F  be an ordinary differential field with an algebraically closed field of constants and let E  be a differential field extension of F  with no new constants. We say that E  is an Iterated Antiderivative Extension  of F , abbreviated IAE, if E  contains elements x₁, ..., x_n  such that E = F (x₁, ..., x_n ) and for each i =  1, 2, ..., n , if we set F_i :=F_i-1 (x_i ) and F₀ :=F , then the derivative x'_i ∈F_i-1 . In this talk, we will prove that if E  is an IAE of F  and if K  is a differential subfield of E  that contains F , then K  is an IAE of F  as well. We will also look at several examples of such extensions and study their differential subfields.

Friday, October 9, 2009 at 10:30 a.m.

Rahim Moosa, University of Waterloo
Differential Arcs and the Infinite-Dimensional Zilber Dichotomy

From the model-theoretic point of view, finite-dimensional differential algebraic varieties are analyzable in terms of "minimal" ones. The Zilber dichotomy, proved by Hrushovski and Sokolovic, says that a minimal differential variety is either algebraic (that is, essentially the constant points of an algebraic variety) or geometrically very simple ("locally modular"). This result has far reaching consequences and is at the heart of Hrushovski's proof of the function field Mordell-Lang in characteristic zero. Some years ago, Pillay and Ziegler found a more direct proof of this dichotomy, using "differential jet spaces", a higher order version of Kolchin's differential tangent spaces.

In the context of several commuting derivations there is a plenitude of infinite-dimensional differential varieties whose structure is known to be very rich. There is an analogue of minimality here, called "regularity": infinite-dimensional differential varieties are analyzable in terms of regular ones. It is still unknown whether or not the analogue of the Zilber dichotomy for regular infinite-dimensional varieties is true. In this talk I will discuss the role that "differential arc spaces" play in reducing this problem to a question about differential subgroups of the additive group. I hope to explain all model-theoretic prerequisites.

Friday, October 16, 2009 at 10:30 a.m.

Lourdes Juan, Texas Tech University
Differential Galois Extensions with Specified Galois Groups, I

In this joint work with Ted Chinburg and Andy Magid, we address the problem of recognizing a differential Galois extension E  of a differential field F  from weaker information than the structure of E  as a differential field. Our work includes a differential counterpart of the normal basis theorem in polynomial Galois theory and the construction of an invariant that depends on the differential Galois group of the extension.

Saturday, October 17, 2009 at 10:30 a.m.

NB: This is a special Saturday meeting of the Kolchin Seminar. Please note that the meeting will be held at the North Academic Center, Room 1/511E of The City College, 160 Convent Avenue, New York, NY 10031. Please click here for directions.

Moshe Kamensky, University of Notre Dame
Differential Tensor Categories

I will suggest an axiomatization of the categorical structure of the category Rep_G  of representations of a linear differential algebraic group G. This is analogous to the description of Rep_G  as a rigid abelian tensor category for a linear algebraic group G. I will present some constructions which suggest that one can do differential algebraic geometry within such a category.

In the second part of the talk, I will explain how, given a (suitably defined) fibre functor on such a category, one may reconstruct a differential algebraic group from it, similarly to the classical Tannakian formalism for algebraic groups. This result was first obtained by Alexey Ovchinnikov using algebraic methods. I will present a proof using the model theoretic notion of the binding group.

No model theory will be assumed.

Reference: Section 5 of http://arxiv.org/abs/0908.0604

The seminar will be followed by an informal discussion session after lunch, in the same room.

Friday, October 23, 2009 at 10:30 a.m.

Lourdes Juan, Texas Tech University
Differential Galois Extensions with Specified Galois Groups, II

This is a continuation of her talk on October 16.
In this joint work with Ted Chinburg and Andy Magid, we address the problem of recognizing a differential Galois extension E  of a differential field F  from weaker information than the structure of E  as a differential field. Our work includes a differential counterpart of the normal basis theorem in polynomial Galois theory and the construction of an invariant that depends on the differential Galois group of the extension.

Friday, October 30, 2009 at 10:30 a.m.

Benjamin Antieau, University of Illinois at Chicago
Galois Theory of Difference Equations with Difference Parameters

In this talk, I will explore an application of Dima Trushin's work on difference Nullstellensatz theorems to the creation of a Galois theory of difference equations with difference parameters. This complements the works of Cassidy, Singer, and Hardouin on the Galois theory of difference and differential equations with differential parameters.

However, serious ring-theoretical difficulties must be dealt with in the case where one has difference parameters. These are approached by building upon the initial idea of Trushin's difference closed rings (pseudo-fields).

Friday, November 6, 2009 at 10:30 a.m.

Leonard Scott, McConnell/Bernard Professor of Mathematics, The University of Virginia
Algebraic Group Representations and Related Topics

This lecture will survey the theory of algebraic group representations in positive characteristic, with some attention to its historical development and its relationship to the theory of finite group representations. Other topics of a Lie-theoretic nature will also be discussed in this context, including at least brief mention of characteristic 0 infinite dimensional Lie algebra representations in both the classical and affine cases, quantum groups, perverse sheaves, and rings of differential operators. Much of the focus will be on irreducible representations, but some attention will be given to other classes of indecomposable representations, and there will be some discussion of homological issues, as time permits.
This will be followed by an afternoon session of informal discussion.

For lecture notes, please click here