Kolchin Seminar in Differential Algebra
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The Graduate Center
365 Fifth Avenue, New York, NY 10016-4309
General Telephone: 1-212-817-7000

Academic Year 2016–2017

Last updated on May 14, 2017.
Other years:   2005–2006   2006–2007   2007–2008   2008–2009   2009–2010   2010–2011   2011–2012   2012–2013   2013–2014   2014–2015   2015–2016


Highlights of 2016–2017 activities and quick links:

Friday, August 26, 2016, 10:15–11:45 a.m. Room 5382

Richard Churchill, Hunter College and the Graduate Center, CUNY
Model Theory as Ordinary Mathematics

Contemporary formulations of group theory, topology, algebraic geometry, differential algebra, etc. are generally done in terms of set theory/category theory, but formulations of Model Theory seem far more dependent on formal logic. I will sketch how that subject can be presented in a set-theoretic/category-theoretic framework.

Friday, September 2, 2016, 10:15–11:45 a.m. Room 5382

Informal Session. No scheduled seminar talk.

Friday, September 2, 2016, 2:00–3:30 p.m. Room 6417

Hans Schoutens, The City University of New York
A Model Theory of Affine n-Space via Differential Algebra

For abstract, click here.

This is a cross-listing from CUNY Logic Workshop.

Friday, September 9, 2016, 10:15–11:45 a.m. Room 5382

Informal Session. No scheduled seminar talk.

Friday, September 16, 2016, 10:15–11:45 a.m. Room 5382

Reid Dale, University of California at Berkeley
An Introduction to Pillay's Differential Galois Theory (Part 1)

In a series of papers from the 1990s and early 2000s, Pillay used the machinery of model-theoretic binding groups to give a slick geometric account and generalization of Kolchin's theory of strongly normal extensions and constrained cohomology. This series of two talks is intended to be expository, with its main goal being to introduce and frame the relevant model-theoretic notions of internality and binding groups within the context of differential algebra, as well as to go through Pillay's argument that his generalized strongly normal extensions arise from logarithmic differential equations defined over algebraic D-groups.

For a review of Part 1, please click video.

Friday, September 16, 2016, 12:30–1:45 p.m. Room 6417

Reid Dale, University of California at Berkeley
An Introduction to Pillay’s Differential Galois Theory (Part 2)

Abstract: See Part 1.

This series of two lectures is a joint seminar with Model Theory Seminar.

For a review of Part 2, please click video.

Friday, September 23, 2016, 10:15–11:45 a.m. Room 5382

Emma Previato, Boston University
Integrable Systems as Strongly Normal Differential Extensions

The notion of algebraically completely integrable Hamiltonian system was introduced in the 1970s. In particular, the flows of motion can be completed to complex algebraic tori. We show that they can be realized by Picard-Vessiot extension, generalizing—in a sense—Kolchin's example where the group of the extension is an elliptic curve. It is valuable to note that the base field can be transcendental, and to give a suitable interpretation of the Casimir functions that may occur in the original phase space.

This is joint work with A. Buium, who first (1986) observed this fact for the Euler system.

For a review of the lecture, please click video.

Friday, September 30--Tuesday, October 4, 2016. DART VII:
7th International Workshop on Differential Algebra and Related Topics

To review the lectures on a particular day, please click:

For a list of posters with titles and abstracts, please click Posters.

Friday, October 7, 2016, 10:15–11:45 a.m. Room 5382

No Seminar Scheduled.

Friday, October 21, 2016, 10:15–11:45 a.m. Room 5382

Silvain Rideau, University of California at Berkeley
Imaginaries in Valued Differential Fields I: Finding Prolongations

In 2000, Scanlon described a theory of existentially closed differential fields where the derivation is contractive: v(d(x)) ≥ v(x), for all x. He also proved a quantifier elimination result for this theory. Around the same time, Haskell, Hrushovski and Macpherson classified all the quotients of definable sets by definable equivalence relations in an algebraically closed valued field by proving elimination of imaginaries (relative to certain quotients of the linear group). In analogy with the pure field situation where elimination of imaginaries for differentially closed fields can be derived from elimination of imaginaries in the underlying algebraically closed field, it was conjectured that Scanlon's theory of existentially closed contractive valued differential fields also eliminated imaginaries relatively to those same quotients of the linear group.

In this talk, I will describe the first part of the proof that this result indeed holds. Our main goal will be to explain a construction that can be interpreted as finding "generic" prolongations for valued differential constructible sets.

For a review of this lecture, please click video.

Friday, October 14, 2016, Tuesday Schedule, No Seminar.

Friday, October 21, 2016, 12:30–1:45 p.m. Room 6417

Silvain Rideau, University of California at Berkeley
Imaginaries in Valued Differential Fields II: Computing Canonical Bases

In 2000, Scanlon described a theory of existentially closed differential fields where the derivation is contractive: v(d(x)) ≥ v(x), for all x. He also proved a quantifier elimination result for this theory. Around the same time, Haskell, Hrushovski and Macpherson classified all the quotients of definable sets by definable equivalence relations in an algebraically closed valued field by proving elimination of imaginaries (relative to certain quotients of the linear group). In analogy with the pure field situation where elimination of imaginaries for differentially closed fields can be derived from elimination of imaginaries in the underlying algebraically closed field, it was conjectured that Scanlon's theory of existentially closed contractive valued differential fields also eliminated imaginaries relatively to those same quotients of the linear group.

In this talk, I will describe the second part of the proof that this result indeed holds. Our goal will be to explain a result, joint with Pierre Simon, on definable types in enrichments of NIP theories, which is crucial to prove elimination of imaginaries. We show that under certain hypothesis if a type in some NIP theory T is definable in an enrichment of T, then it is already be definable in T.

These two talks are jointly organized with Model Theory Seminar.

Friday, October 28, 2016, 10:15–11:45 a.m. Room 5382

Informal Session. No scheduled seminar talk.

Friday, November 4, 2016, 10:15–11:45 a.m. Room 5382

Informal Session
No Seminar.

Due to a conflict of schedule, the previously announced talk by Joel Nagloo has been canceled and rescheduled to November 11.

Friday, November 11, 2016, 10:15–11:45 a.m. Room 5382

Joel Nagloo, Bronx Community College (CUNY)
On the Algebraic Independence Conjecture for the Generic Painlevé Equations

In this talk we explain how one can show that the solutions (and derivatives) of the generic sixth Painlevé equation are algebraically independent over ℂ(t). This extends recent progress made on the third Painlevé equations and hence fully proves the algebraic independence conjecture for the generic Painlevé equations.

For a review of the lecture, please click video.

Friday, November 18, 2016, 10:15–11:45 a.m. Room 5382

Gleb Pogudin, Johannes Kepler University
Jet Ideals and Products of Ideals in Differential Rings

We will start with the following question emerged recently in the algorithmic studies of algebraic differential equation: assume that you have several ideals (not necessarily differential) in a differential algebra, how are the product of their derivatives and derivatives of their product related? This question turned out to be closely related to the classical membership problem for the differential ideal [x1···xn] in the algebra of differential polynomials in x1, …, xn.
Both these questions can be approached using the jet schemes of the ideal (x1···xn) in the polynomial algebra in x1, …, xn. In the talk we will describe the structure of these jet schemes and use the obtained results to solve initial problems in differential algebra.

For a review of the lecture, please click video-1 and video-2.

Friday, November 25, 2016, Thanksgiving Holiday. No Seminar.

Friday, December 2 and 9, 2016, 10:15–11:45 a.m. Room 5382

William Sit, The City College (CUNY)
Rota-Baxter Type Operators, Rewriting Systems, and Gröbner-Shirshov Bases, Part I and II

G.-C. Rota asked for a complete list of all possible algebraic identities that can be satisfied by a linear operator on an associative algebra over a field. Known as Rota's Problem, or Rota's Classification Problem on linear operators, it remains unsolved for decades and only a few were known to Rota. In this talk, which will be in two parts, we present 14 Rota-Baxter type identities and their common properties. These identities, first obtained by symbolic computation, can be uniformly characterized as a class in several ways. For example, an operator identity is of Rota-Baxter type if and only if for all free operator algebra on a well-ordered set, a certain (uniformly defined) rewriting system is convergent, if and only the identity is compatible with a certain (uniformly constructed) monomial order, and if and only if the identity determines (uniformly) a Gröbner-Shirshov basis with respect to the monomial order. We obtain (uniformly) a construction of the free operator algebra satisfying any given Rota-Baxter type identity on any given well-ordered set.

We explain the background, definitions, basics, and main results, sketch our approach and the methods used to prove the results, and discuss open problems.

This is joint work with Shanghua Zheng, Xing Gao, and Li Guo.

For a copy of the slides, please click Part I and Part II

December 16, 2016 to January 31, 2017 Winter Break, no seminars.


February 3, 2017, 10:15–11:30 a.m. Room 5382

Informal Session
Rota-Baxter (Type) Algebras

Friday, February 10, 2017, 10:15–11:30 a.m. Room 5382

Jim Freitag, University of Illinois at Chicago
Revisiting the Model Theory of Painlevé Equations

The Painlevé equations are six families of nonlinear order two ODEs with complex parameters. Around the start of the last century, the equations were isolated for foundational reasons in the analysis of ODEs. Since the 1970s, interest in the equations has steadily increased due in part to their connections with various areas of mathematics (e.g. monodromy of linear differential equations, mathematical physics, and diophantine geometry). In a recent series of works, Nagloo and Pillay established the algebraic independence of solutions of a Painlevé equation, at least when the coefficients are assumed to be transcendental, algebraically independent complex numbers. Later, Nagloo established results of a similar nature for algebraic relations between solutions of equations from different families. In this talk, we will build on the theme of Nagloo and Pillay, answering several questions left open by their work. One of the surprising aspects of the work of Nagloo and Pillay, as well as the present work, is the application of deep structural classification results from model theory to concrete problems on transcendence.

For a review of the lecture, please click video-1 and video-2.

Friday, February 17, 2017, 10:15–11:30 a.m. Room 5382

Peter Thompson, Graduate Center, CUNY
Non-Existence of Independent Commuting Derivations

Let K be a field. A derivation d1 on K[x1,…,xn] is said to be integrable if there exist derivations d2, …, dn such that all di commute pairwise and the set of di is linearly independent over K[x1,…,xn]. Let K be a field of characteristic 0. We present a class of derivations on K[x, y] that is not integrable.
This is joint work with Joel Nagloo and Alexey Ovchinnikov.

Friday, February 24, 2017, 10:15–11:30 a.m. Room 5382

Informal Session

Informal sessions at the Kolchin Seminar are open to all and attendees may bring short presentations and questions for discussion. In this informal session we expect to explore the algebraic approach to integral equations along the recent development in integral differential algebras of Rosenkranz et al.

Friday, March 3, 2017, 10:15–11:45 a.m. Room 5382

Peter Thompson, Graduate Center, CUNY
Non-Existence of Independent Commuting Derivations, Part II

Let K be a field. A derivation d1 on K[x1,…,xn] is said to be integrable if there exist derivations d2, …, dn such that all di commute pairwise and the set of di is linearly independent over K[x1,…,xn]. Let K be a field of characteristic 0. We present a class of derivations on K[x, y] that is not integrable.
This is joint work with Joel Nagloo and Alexey Ovchinnikov.

Friday, March 3, 2017, 2:00–4:00 p.m. Room 5382

Richard Gustavson (Graduate Center, CUNY)*
Elimination for Systems of Algebraic Differential Equations

We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, whether the given system of differential equations has a solution. We first look solely at the "algebraic data" of the system of differential equations through the theory of differential kernels to provide a new upper bound for proving the consistency of the system. We then prove a new upper bound for the effective differential Nullstellensatz, which determines a sufficient number of times to differentiate the original system in order to prove its inconsistency. Finally, we study the Rosenfeld-Gröbner algorithm, which approaches differential elimination by decomposing the given system of differential equations into simpler systems. We analyze the complexity of the Rosenfeld-Gröbner algorithm by computing an upper bound for the orders of the derivatives in all intermediate steps and in the output of the algorithm.
*This talk will be the doctoral dissertation defense of the speaker in the Mathematics Department at the Graduate Center. All are welcome.

Friday, March 10, 2017, CUNY Math Fest Day, NO SEMINAR

Friday, March 17, 2017, 10:15–11:30 a.m. Room 5382

Informal Session

Instead of the previously announced talk (see below), Peter Thompson will instead participate in an informal session, which is open to all and attendees may bring short presentations and questions for discussion.

Peter Thompson, Graduate Center, CUNY
Non-Existence of Independent Commuting Derivations, Part III

Let K be a field. A derivation d1 on K[x1,…,xn] is said to be integrable if there exist derivations d2, …, dn such that all di commute pairwise and the set of di is linearly independent over K[x1,…,xn]. Let K be a field of characteristic 0. We present a class of derivations on K[x, y] that is not integrable.
This is joint work with Joel Nagloo and Alexey Ovchinnikov.

Friday, March 24, 2017, 10:15–11:30 a.m. Room 5382

Due to a family emergency, the talk by William Keigher has been CANCELED.

Until further notice, we will have instead an
Informal Session.

Informal sessions at the Kolchin Seminar are open to all and attendees may bring short presentations and questions for discussion.

Friday, March 31 2017, 10:15–11:30 a.m. Room 5382

Informal Session.

Informal sessions at the Kolchin Seminar are open to all and attendees may bring short presentations and questions for discussion. In this session, the topic is expected to be an introduction to ultra-filters.

Kolchin Research and Training Workshop I, April 7–9, 2017.

For the three daily programs, please click April 7, April 8, April 9.

Friday, April 7, 2017, 10:00–17:20. All lectures in Room 5382, Graduate Center

Saturday, April 8, 2017, 10:00–17:45, All meetings in Room 6-215 at Baruch College

Sunday, April 9, 2017, 9:15–12:30, All meetings in Room 6-215 at Baruch College

Friday, April 14, 2017, Spring Recess, No Seminar

Kolchin Research and Training Workshop II, April 21–23, 2017.

For the three daily programs, please click April 21, April 22, April 23.

Friday, April 21, 2017, 10:00–17:30, Graduate Center

Saturday, April 22, 2017, 10:00–18:00, All meetings in Room 6-215 at Baruch College

Sunday, April 23, 2017, 9:15–13:15, All meetings in Room 6-215 at Baruch College

Friday, April 28, 2017, 10:15–11:30 a.m. Room 5382

Informal Session.

Informal sessions at the Kolchin Seminar are open to all and attendees may bring short presentations and questions for discussion.

Kolchin Research and Training Workshop III, May 5 and 7, 2017.
Also, Special Sessions on Differential and Difference Algebra at the AMS Sectional Meeting at Hunter College on May 6 and May 7.

For the three combined daily programs, please click May 5, May 6, May 7.

Friday, May 5, 2017, 10:00–17:40, Room 5382 at the Graduate Center  (Kolchin Workshop III)

Saturday, May 6 and 7, 2017
AMS Spring Eastern Sectional Meeting #1129

Differential and Difference Algebra: Recent Developments, Applications, and Interactions
Special Sessions I–IV

Sunday, May 7, 2017, 15:30–TBD, Room HN504, Hunter College North Building (Kolchin Research and Training Workshop III)

  • 15:30–TBD 
    Panel Discussion and One-on-One Sessions

Friday, May 12, 2017, 10:15–11:30 a.m. Room 5382

Alice Medvedev, City College (CUNY)
Sparse Difference Equations with High Transcendence Degree but Difference Krull Dimension One

For fixed integers r and m0, … mr, the difference equation

ri=0ni(x))mi = 1

defines a subgroup Gn of the multiplicative group of transcendence degree nr.

We show that whenever no zero of the polynomial χ(z):= ∑ri=0 mi zi is a root of unity, the difference Krull dimension of Gn is bounded, independently of n. Indeed, the difference Krull dimension of Gn is 1 whenever χ(z) is hereditarily irreducible, and it usually is.


Other Academic Years

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