Kolchin Seminar in Differential Algebra
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Academic year 2010–2011

Last updated on May 14, 2017.
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In Fall, 2010, we explored new areas in the interactions between differential algebra and related fields, such as the Hopf-algebraic approach to Galois Theory in the setting where the field of constants is not necessarily algebraically closed (or differentially closed in the parametric case). We continued with topics in algebraic geometry, representation theory, computational complexity, differential geometry, model theory and number theory. Professor Camilo Sanabria of Bronx Community College gave a series of lectures on foliations and holonomy. In October, KSDA members helped organized The Fourth International Workshop on Differential Algebra and Related Topics (DART IV) October 27–30, 2010, Beijing, China. In Spring, 2011 we welcomed Professor Raymond Hoobler from The Graduate Center and The City College of CUNY, who has done an increasingly insightful review on differential schemes. He delivered a series of exploratory lectures on the subject in honor of the late Prof. Jerry Kovacic. We also welcomed Meghan Anderson, a model theorist just finishing her doctorate advised by Professor Tom Scanlon, from the University of California, Berkeley.

Friday,August 27, 2010 at 10:15 a.m. Room 5382

Richard Churchill, The Graduate Center, and Hunter College, CUNY
A Geometric Approach to Classical Galois Theory

We reformulate classical Galois theory and differential Galois theory in geometric forms, which closely parallel one another. In particular, the standard matrix representations of the associated Galois groups become strikingly similar. The talk is at an introductory level — no prior knowledge of differential Galois theory is assumed.

For lecture notes, please click here.

Friday, September 3, 2010 at 10:15 a.m. Room 5382

Noson S. Yanofsky, The Graduate Center, and Brooklyn College, CUNY
Galois Theory of Algorithms

Many different programs are the implementation of the same algorithm. This gives us a surjective map from the collection of programs to the collection of algorithms. Similarly, there are many different algorithms that implement the same computable function. This gives us a surjective map from the collection of algorithms to the collection of computable functions. Algorithms are intermediate between programs and functions:

Programs --> Algorithms --> Functions.

We investigate the many possible intermediate structures by looking at the group of automorphisms of programs that preserve functionality. The fundamental theorem of Galois theory says that the subgroup lattice of this group is isomorphic to the dual lattice of intermediate types of algorithms. Along the way, we formalize the intuition that one program can be substituted for another if they are the same algorithms.

For lecture notes, please click here.

Tuesday, September 14, 2010 at 2:00 p.m. Room 5382

Alexander Levin, The Catholic University of America
Some Invariants of Difference Field Extensions

In this talk we consider some characteristics of a difference field extension, which do not depend on the system of its difference generators. We start with the discussion of invariants carried by dimension polynomials of finitely generated difference and inversive difference field extensions (such invariants include, in particular, difference transcendental degree, difference type and typical difference transcendental degree). Then we consider transformations of the basic set of translations of an inversive difference field and show that if L/K is an inversive difference field extension with the basic set σ = {α1,..., αn} and d is the difference type of L/K (d ≤ n), then there is a "natural" transformation of σ into a set τ = {β1,..., βn} such that L is an algebraic extension of a finitely generated difference overfield of K with respect to the basic set {β1,..., βd}. The last part of the talk is devoted to algebraic (in the sense of the classical field theory) difference field extensions. We are going to concentrate on ordinary case and discuss the concept of limit degree introduced and studied by R. Cohn, as well as the notion of distant degree introduced in a recent work by Z. Chatzidakis and E. Hrushovski.

Friday, September 24, 2010, 10:00 a.m. Room 5382

Informal Morning Session with
Bernard Malgrange, Université Joseph Fourier – Grenoble.

On September 24, 2010, Anand Pillay from University of Leeds gave a 90-minute talk at the CUNY Logic Seminar, from 2:00–3:30 pm at the Graduate Center in Room 6417. The title of the talk is

Model-theoretic Approaches to Galois Theories: a Survey.

Please click here for the abstract and other information.

Friday, October 1, 2010, 11:00 a.m. Room 5382

Informal Morning Session with
Bernard Malgrange, Université Joseph Fourier – Grenoble.

Friday,October 1st, 2010 at 2:00 p.m. Room 5382

Camilo Sanabria, Bronx Community College, CUNY
Foliations and Holonomy

Foliations are ubiquitous in B. Malgrange's approach to non-linear differential equations. In this talk I will recall the definition of foliation, the concept of holonomy and some properties of foliations of co-dimension 1. The talk will have a geometric approach, so the objects studied, as well as the examples, will be modeled over the real numbers.

Friday,October 8, 2010 at 10:15 a.m.Room 5382

No meeting. See 2:00 p.m. Session.

Friday, October 8, 2010 at 2:00 p.m. Room 5382

Camilo Sanabria, Bronx Community College, CUNY
Foliations and Groupoids

This is a continuation of the talk from last week. I will recall the concepts of groupoid and Lie groupoid and I will explain how these concepts are used in the study of foliations. If time allows I will also talk about orbifolds and their relation to foliations with compact leaves. The talk will have a geometric approach, so the objects studied, as well as the examples, will be modeled over the real numbers.

Friday, October 15, 2010 at 10:15 a.m. Room 5382

Alexey Ovchinnikov, Queens College, CUNY
Tannakian Categories and Algebraic Groupoids—Preliminaries

We will discuss Hopf algebroids in the framework of Tannakian categories and look at basic examples.

Friday, October 15, 2010 at 2:00 p.m. Room 5382

Camilo Sanabria, Bronx Community College, CUNY
Foliations and Groupoids

This is a continuation of the talk from last week.

Saturday, October 16, 2010 at 10:15 a.m.
Room 920, Hunter College, East Building

Direction: Please be advised that Hunter College has a ``card-swipe'' security system. Attendees coming to the seminar on 10/16 will have to enter the campus via the West Building on the southwest corner of 68th and Lexington Ave,(probably) show some kind of id and/or sign in, go up to the third floor, take the bridge over Lexington Ave to get into the East Building, and then take the elevator to the ninth floor to get to the room.

Moshe Kamensky, University of Notre Dame
Tannakian Categories

I will give a survey on Deligne's paper Catégories Tannakiennes. Among the main results of the paper are the statement that any two fibre functors on a Tannakian category are locally isomorphic; the construction of the fundamental group of a Tannakian category; the existence of fibre functors in characteristic zero; and an alternative construction of Picard-Vessiot extensions and the Galois group of a linear differential equation.
I plan to explain in some detail the statement of the results, and then go into some of the proofs.

Friday, October 22, 2010 at 10:15 a.m. Room 5382

Anton Leykin, University of Illinois at Chicago
Multiplier ideals via computational D-modules theory

After an introduction to computational methods in D-modules theory, we will provide an overview of the new algorithms for generalized Bernstein-Sato polynomials for an arbitrary variety. These lead to algorithms for singularity theory invariants: log canonical thresholds, jumping coefficients, and multiplier ideals. (Based on joint work with Christine Berkesch.)

Friday, October 29, 2010

No seminar.
Fourth International Workshop on Differential Algebra and Related Topics (DART IV) October 27–30, 2010, Beijing, China.

Friday, November 5, 2010 at 10:15 a.m. Room 5382

James Freitag, University of Illinois at Chicago
Definability of Rank for Differential Varieties

We will work over an ordinary differentially closed field of characteristic zero. Given a family of differential algebraic varieties parameterized by points in affine space, we will consider the subfamily with a common coefficient and leading term in their Kolchin polynomials. We will prove that this is a constructable condition in the Kolchin topology using geometry and model theory. We will show that this essentially comes from the Zariski topology (or definability of Morley rank in strongly minimal theories). Then we will discuss the barriers to a similar theorem for partial differential fields.

Friday, November 12, 2010 at 10:15 a.m. Room 5382

Alexey Ovchinnikov, Queens College, CUNY
Title: Differential representations of SL(2)

W. Sit's characterization of differential algebraic subgroups of SL(2) will be presented next week. In this talk, we will be discussing the representation theory of SL(2) including some unexpected examples.

Friday, November 19, 2010 at 10:15 a.m. Room 5382

William Sit, City College of New York, CUNY
Differential Algebraic Subgroups of SL(2), Part I

A differential algebraic subgroup of SL(2) is a subgroup whose elements, when viewed as a quadruple in affine space, satisfy and are defined by a system of partial differential equations. A classification of all such subgroups up to conjugation over a ground field, which is a partial differential field, was completed in 1972. In this two-part talk, we review this classification and outline the method used to obtain it. Part I provides the motivation for the problem, a classification of the algebraic subgroups of SL(2), some useful results on linear partial differential equations, and examples of differential algebraic subgroups.

Reference: William Sit, Differential algebraic subgroups of SL(2) and strong normality in simple extensions, Amer. J. Math., 97 (3) (1975), pp. 627–698.

For lecture slides, please click here

Friday, November 26, 2010

No meeting. Thanksgiving.

Friday, December 3, 2010 at 10:15 a.m. Room 5382

Ravi Srinivasan, Rutgers University
Hopf Algebraic approach to Picard-Vessiot Theory

This talk is based on a paper by M. Takeuchi. We will use Hopf algebras to formalize the notion of a Picard-Vessiot extension and to characterize PV extensions as a minimal splitting field of a linear differential equation. We will also establish a Galois correspondence between Hopf ideals and intermediate differential subfields.

Friday, December 10, 2010 at 10:15 a.m. Room 5382

Andrey Minchenko, University of Western Ontario
Differential representations of SL(2)

In order to describe the linear representations of a group, it is sufficient to find all of its indecomposable representations. It is known that indecomposable algebraic representations of G=SL(2) correspond to irreducible subrepresentations of G in the ring R of polynomials in two variables x and y. Given a derivation ' on the ground field, R extends to a G-representation R' by adding variables x',y',x'',y'',etc. We will investigate indecomposable subrepresentations of R' and discuss their relation to description of all differential representations of G.

Friday, December 17, 2010 at 10:15 a.m. Room 5382

William Sit, City College of New York, CUNY
Differential Algebraic Subgroups of SL(2), Part II

The Zariski closures in SL(2) of differential algebraic subgroups of SL(2) are algebraic subgroups of SL(2). In Part II, we discuss "lifting" the classification of the algebraic subgroups to obtain a classification for the differential case. If time permits, we will discuss some applications with examples of strongly normal extensions and their differential Galois groups.

For lecture notes, please click here.

Friday, January 28, 2011 at 10:15 a.m. Room 5382

Ravi Srinivasan, Rutgers University, Newark
Hopf algebraic approach to Picard-Vessiot Theory

This talk is a continuation of my talk from December 3rd 2010. We will formalize the notion of a Picard-Vessiot extension using Hopf algebras. I will give several examples and discuss briefly the Galois correspondence between Hopf ideals and intermediate differential subfields. I will also give a quick overview on the materials from my last lecture.

Friday, February 4, 2011 at 10:15 a.m. Room 5382

Carlos Arreche, The Graduate Center, CUNY
Differential Galois theory in arbitrary characteristic for modules with iterative connection

About a decade ago, Matzat and van der Put described a Picard-Vessiot theory for iterative differential fields in arbitrary characteristic generalizing the classical theory in characteristic zero, but their Galois correspondence was shown to be incomplete. Recently, Maurischat (arXiv:0712.3748) described a Galois theory for modules with iterative connection which generalizes that of Matzat and van der Put and gives a complete Galois correspondence which is equivalent to Takeuchi's in this setting. I will motivate and describe Maurischat's work and relate it to the approaches of Matzat-van der Put and Takeuchi.

Friday, February 11, 2011

No meeting. President Lincoln’s Birthday.

Friday, February 18, 2011 at 10:15 a.m. Room 5382

Raymond Hoobler, Graduate Center and The City College, CUNY
A Grothendieck approach to differential Azumaya algebras

Gothendieck introduced connections on a sheaf on a scheme $X$ over $S$ by considering the first order neighborhood of the diagonal map $X\rightarrow X\times_{S}X$. I will explain this definition and connect it to the usual definition. Then I will show that an Azumaya algebra $\Lambda$ on an affine scheme $Spec(A)$ satisfies Grothendieck's definition by calculating the Hochschild cohomology of $\Lambda$. Time permitting, I will then connect it to my talk last spring using the $\delta$-flat topology to interpret the differential Brauer group of a differential ring $A$.

For lecture notes, please click here.

Friday, February 25, 2011 at 10:15 a.m. Room 5382

Raymond Hoobler, Graduate Center and The City College, CUNY
Differential Schemes

I will begin by summarizing Kovacic's work and the proper sheafification procedure. I will then explain when faithfully flat descent holds for differential schemes and interpret this result in terms of differential principal homogeneous spaces for a differential group. This provides the connection between Kolchin's constrained cohomology and the $\Delta$-flat cohomology. Given sufficient time, I will also discuss varying the partial differential structure using adjoint functors.

Friday, March 4, 2011 at 10:15 a.m. Room 5382

Raymond Hoobler, Graduate Center and The City College, CUNY
Differential Cohomology

I will begin by discussing constrained extensions and show that differential principal homogeneous spaces always have points in differentially closed fields. Using this I will show that $\Delta$-flat cohomology extends Kolchin's constrained cohomology to differential schemes and show that if the coefficients are algebraic groups, then $\Delta$-flat (= constrained) cohomology classes split by passing to the algebraic closure of a differential field are precisely those coming from the (non differential) Galois cohomology.

Friday, March 11, 2011 at 10:15 a.m. Room 5382

Dmitry Trushin, Moscow State University, Moscow
A non-standard geometric approach to differential and difference equations

I will present a non-standard geometric approach to differential and difference equations. I will show that there are four natural classes of rings playing the role of universal domains containing all necessary solutions. These rings are: differentially closed fields of characteristic zero and quasifields of prime characteristic in the differential case, and difference closed fields and pseudofields in the difference case.

Saturday, March 12, 2011 at 1:00 a.m. Room R6/113, CCNY

Dmitry Trushin, Moscow State University, Moscow
A non-standard geometric approach to differential and difference equations

This is an informal continuation of Friday's talk, concentrating on difference equations.

Location: City College of New York, North Academic Center, 6th Floor, Room 113 (green side).

Friday, March 18, 2011 at 10:15 a.m. Room 5382

Michael Wibmer, RWTH, Aachen
A Chevalley theorem for difference equations

By a theorem of Chevalley the image of a morphism of varieties is a constructible set. The algebraic version of this fact is usually stated as a result on ``extension of specializations'' or ``lifting of prime ideals''. We present a difference analog of this theorem. The approach is based on the philosophy that occasionally one needs to pass to higher powers of $\sigma$, where $\sigma$ is the endomorphism defining the difference structure. In other words, we consider difference pseudo fields (which are finite direct products of fields) rather than difference fields. We also prove a result on compatibility of pseudo fields and present some applications of the main theorem, e.g. constrained extension and uniqueness of $\sigma$-Picard-Vessiot rings for linear differential equations with a difference parameter.

Saturday, March 19, 2011 at 1:00 a.m. Room 920, East Building, Hunter College

Michael Wibmer, RWTH, Aachen
A Chevalley theorem for difference equations

This is an informal continuation of Friday's talk.

Location: Room 920, Hunter College, East Building.

Friday, March 25, 2010, 10:15 a.m. Room 5382

Dmitry Trushin, Moscow State University, Moscow
A non-standard geometric approach to differential and difference equations

A continuation of the talk from March 11.

Friday, April 1, 2011 at 10:15 a.m. Room 5382

Brainstorming session.

Friday, April 8, 2011 at 10:15 a.m. Room 5382

Meghan Anderson, University of California, Berkeley
Solutions to Linear Equations in Valued D-fields

A model complete theory of valued D-fields was developed by Scanlon in his 1997 thesis. In this theory, valued fields are endowed with linear operator D which specializes to a derivative in the residue field, but which in the valued field obeys a twisted Leibniz rule and is interdefinable with a valuation preserving automorphism. The theory has good model theoretic properties, notably quantifier elimination, which should allow for some analysis of the upstairs difference field in terms of the downstairs differential structure. However, it also presents its own challenges, even in the relatively simple setting of solution spaces to linear equations, some of which I will discuss.

Friday, April 15, 2011 at 10:15 a.m. Room 5382

Meghan Anderson, University of California, Berkeley
Solutions to Linear Equations in Valued D-fields

This is a continuation of the talk from last week. A model complete theory of valued D-fields was developed by Scanlon in his 1997 thesis. In this theory, valued fields are endowed with linear operator D which specializes to a derivative in the residue field, but which in the valued field obeys a twisted Leibniz rule and is interdefinable with a valuation preserving automorphism. The theory has good model theoretic properties, notably quantifier elimination, which should allow for some analysis of the upstairs difference field in terms of the downstairs differential structure. However, it also presents its own challenges, even in the relatively simple setting of solution spaces to linear equations, some of which I will discuss.

Friday, April 29, 2011 at 10:15 a.m. Room 5382

Raymond Hoobler, Graduate Center and The City College, CUNY
DiffSpec Redux

They say that three times is a charm and so it appears. A Max $\Delta$ ring is a $\Delta$ ring in which all maximal ideals are $\Delta$ ideals. I will give a straightforward definition of the structure sheaf of a Max $\Delta$ ring using the usual definition with inverting $f$ to get sections over $D(f)$ to define the $\Delta$ structure sheaf. Any $\Delta$ ring can be made into a Max $\Delta$ ring by inverting all differential units.
I will show that there are no non-zero differential zeros in a Max $\Delta$ ring and, even better, that the tensor product of two Max $\Delta$ rings over a third Max $\Delta$ ring is a Max $\Delta$ ring. This makes many of the standard tools from algebraic geometry available for differential algebraic geometry. If there is enough time, I will suggest an application to differential group scheme extensions.

For lecture notes, please click here.

Friday, May 6, 2011 at 10:15 a.m. Room 5382

Raymond Hoobler, Graduate Center and The City College, CUNY
Projective Delta schemes

I will outline the procedure for defining projective differential schemes in a form similar to affine delta schemes. An effort will be made to describe Kolchin's results in this case. A number of interesting questions for future work will be posed.


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