Kolchin Seminar in Differential Algebra

Academic year 2011–2012 Last updated on January 31, 2020. Other years: 2005–2006   2006–2007   2007–2008   2008–2009   2009–2010   2010–2011   2012–2013   2013–2014   2014–2015   2015–2016   2016–2017   2017–2018   2018–2019   2019–Fall

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 Cohomology

Let G be a group acting on an R-module 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 Picard-Vessiot Galois Groups

After giving an introduction to the parameterized Picard-Vessiot theory (PPV-theory), I will discuss an inverse problem: which linear differential algebraic groups can occur as PPV-Galois 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 PPV-Galois 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 Equations

In light of the Zilber trichotomy for minimal sets definable in difference-closed 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 model-theoretic 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 S-variety 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 S-variety, and F is not purely inseparable, this group is minimal and falls into the second, "modular," case of the Zilber Trichotomy. Other modular nontrivial S-varieties always admit a finite cover by a modular group S-variety, and identifying these is the subject of this talk.

The main result is that any finite cover of an S-variety by a modular group S-variety 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 S-variety determined by E and G to the S-variety 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 Fields

Fields 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
Non-commutative Linear Almost Simple Groups Are Perfect

We will develop the notion of an almost simple differential algebraic group (due to Cassidy and Singer). Cassidy and Singer asked if non-commutative 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 non-commutative 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 Harrison-Trainor, University of Waterloo
Non-standard Methods for Bounds in Differential Polynomial Rings

We present an extension of the non-standard 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 Bernstein-Type Dimension Polynomials and\\ New Invariants of Finitely Generated D-Modules

In 1971 J. Bernstein introduced a Hilbert-type dimension polynomial associated with a finitely generated filtered D-module, which is a module over a Weyl algebra. Bernstein polynomials and their invariants play an important role in the theory of D-modules and its applications. A Bernstein polynomial of a finitely generated D-module 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 D-module 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 Bernstein-type dimension polynomials in two variables that carry more invariants than classical Bernstein polynomials. In particular, we describe some new invariants of holonomic D-modules.

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 Cases

We 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 q-shift case; and the last for shift and q-shift case. The criteria describe necessary and sufficient conditions on the existence of telescopers for hyperexponential-hypergeometric 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 vu

We 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 Δ-flat maps essentially of Δ-finite 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 Varieties

A 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 Operators

I 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 Polynomials

In 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 Xiao-Shan Gao, and Dr. Chun-Ming 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 Avatars

We will look at examples from the classical literature on non-linear 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 Cole-Hopf 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 SL₂(F ) as the stabilizer in PSL₂(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 Form

In 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 Xiao-Shan Gao and Chun-Ming 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 Groupoids

In 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 Fields

A 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 II

In 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 Groups

A 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 Jordan-Hölder 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 PDEs

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 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 I

In 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 II

In 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 Algebra

This 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 three-space, 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 Groups

Jordan-Hölder 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