Kolchin Seminar in Differential Algebra

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

Thanks to Franz Winkler, a doctoral student of his, Thieu Vo Ngoc, from Research Institute for Symbolic Computation at Johannes Kepler University is visiting the GC for a few months during the Spring Semester and early summer.

The KSDA website moved to https:// ksda.ccny.cuny.edu.

During the Spring, 2016 semester, Professors Alice Medvedev (City College) and Alexey Ovchinnikov (Queens College and the Graduate Center) organized a thematic workshop on differential algebra and related topics on April 8–10 and a second one on May 13–15.

Friday, September 11, 2015, 10:15–11:45 a.m. Room 5382

Xing Gao (Lanzhou University, China and Rutgers University at Newark)
Integro-Differential Algebra of Combinatorial Species

The concept of structure is fundamental and recurring in all branches of mathematics, as well as in computer science. Informally, a combinatorial species is a class of finite structures on arbitrary finite sets which is closed under arbitrary "relabellings" along bijections. Various combinatorial operations can be defined on species of structures, such as addition, multiplication, substitution, differentiation and integration, giving rise to combinatorial algebras. Roughly speaking, an integro-differential algebra (R,d, P) is an algebraic abstraction of the familiar setting of derivatives and integrals in analysis, where one employs a notion of differentiation d together with a notion of integration P on some (real or complex) algebra of functions. In this talk, we give an integro-differential algebra structure on combinatorial species of structures.
The lecture is available: including slides, and video at CUNY, or on Youtube.

Friday, September 11, 2015, 12:30–1:45 p.m. Room 6417
This is a cross-listing from Model Theory Seminar

Russell Miller (Queens College and Graduate Center, CUNY)
Degree Spectra of Real Closed Fields

The degree spectrum of a countable structure is the set of all Turing degrees of isomorphic copies of that structure. This topic has been widely studied in computable model theory. Here we examine the possible degree spectra of real closed fields, finding them to offer far more complexity than the related theory of algebraically closed fields. The co-author of this project, Victor Ocasio Gonzalez, showed in his dissertation that, for every linear order L, there exists a real closed field whose spectrum is the pre-image under jump of the spectrum of L. We add further results, distinguishing the cases of archimedean and non-archimedean real closed fields, and splitting the latter into two subcases based on the existence of a least multiplicative class of positive nonstandard elements. If such a class exists, then finiteness in the field is always decidable, but the case with no such class proves more interesting.

Friday, September 18, 2015, 10:15–11:45 a.m. Room 5382

Michael Wibmer, University of Pennsylvania
Strongly Étale Difference Algebras and Babbitt's Decomposition

We introduce a certain class of difference algebras whose role in the study of difference equations is analogous to the role of étale algebras in the study of algebraic equations. We use these difference algebras to deduce an improved version of Babbitt's decomposition theorem. We also present applications to difference algebraic groups and the compatibility problem. This is joint work with Ivan Tomasic.
For a review of the talk, please click video-1 and video-2.

Friday, September 25, 2015, NO SEMINAR (Tuesday schedule for CUNY)

Friday, October 2, 2015, 10:15–11:45 a.m. Room 5382

Xiao-Shan Gao, Academy of Mathematics and Systems Science, Beijing, China
Differential and Difference Chow Form, Sparse Resultant, and Toric Variety

In this talk, I will give a survey on the recent work on differential and difference Chow forms, sparse resultants, and toric varieties. Chow forms are used as canonical representations as well as coordination for algebraic cycles. Sparse resultants are powerful tools for elimination of sparse polynomial systems. Chow forms and sparse results are connected through toric varieties. More precisely, for a given set A of monomial supports, the Chow form of the toric variety defined by A is the A-sparse resultant. We will show how these results are extended to differential algebra and difference algebra.
For a review of the lecture, please click video and slides.

Friday, October 2, 2015, 12:30–1:45 p.m. Room 6417
This is a cross-listing from Model Theory Seminar

Richard Gustavson, The Graduate Center (CUNY)
Effective bounds for the Existence of Differential Field Extensions

We present a new upper bound for the existence of a differential field extension of a differential field (K; D) that is compatible with a given field extension of K. In 2014, Pierce provided an upper bound in terms of lengths of certain antichain sequences of ℕ ^m equipped with the product order. Pierce’s theory has interesting applications to the model theory of fields with m commuting derivations, and his results have been used when studying effective methods in differential algebra, such as the effective differential Nullstellensatz problem. We use a new approach involving Macaulay’s theorem on the Hilbert function to produce an improved upper bound. In particular, we see markedly improved results in the case of two and three derivations.
This is joint work with Omar Leon Sanchez.

Friday, October 9, 2015, 10:15–11:45 a.m. Room 5382

Eugenia Cheng, University of Sheffield and School of the Art Institute of Chicago
Operads: from loop spaces to n-categories

This talk was cancelled by the speaker due to a severe mosquito attack. We thank Prof. Li Guo of Rutgers University for presenting a brief introduction to operads.

It is known that n-groupoids are too strict to model homotopy n-types, and instead we must use "weak n-groupoids", whose axioms are only satisfied up to equivalence. The problem is to make this definition in a coherent way; these equivalences must satisfy some other axioms of their own, possibly only up to equivalence. In this talk we will discuss how to approach this problem using operads. Operads were originally introduced as a powerful tool for handling the "weak monoid" structure of loop spaces, and we will show how to harness this same power to handle the structure of weak n-categories according to the work of Trimble and Batanin, with further developments by Leinster and Cheng. The talk will be introductory; in particular no knowledge of n-categories will be assumed.

Friday, October 16, 2015, 10:15–11:45 a.m. Room 5382

Alice Medvedev (session chair), The City College of New York
L. diVizio and C. Hardouin: Intrinsic Approach to Galois Theory—q-Difference Equations

As an experimental project for the Fall semester, we will designate several Fridays (mornings and occasionally afternoons) discussing (mainly Part IV of) the above paper.
From the Introduction: "The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of q-difference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of q-difference equations and we use it to establish the relations among the different theories in the literature."
Graduate students and faculties with an interest in Hopf algebras, Tannakian categories, algebraic groups, difference equations, or differential algebra are welcome to join us and participate in the discussions!
The first meeting will be chaired by Alice Medvedev (CCNY). We shall go through the scope of the paper and choose the topics of common interest as well as their prerequisites. We shall then agree on a plan, which is to include the dates and session chairs of future meetings.

Friday, October 23, 2015, 10:15–11:45 a.m. Room 5382

Mengxiao Sun, Graduate Center
L. diVizio and C. Hardouin: Intrinsic Approach to Galois Theory—q-Difference Equations

This is the first study/discussion of selected topics from the paper. This Friday's topic will be the basics of q-difference equations based on Chapter I.
From the Introduction: "The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of q-difference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of q-difference equations and we use it to establish the relations among the different theories in the literature."

Friday, October 30, 2015, 10:15–11:45 a.m. Room 5382

Mengxiao Sun, Graduate Center
L. diVizio and C. Hardouin: Intrinsic Approach to Galois Theory—q-Difference Equations

This is the second study/discussion of selected topics from the paper. This Friday, the speaker will continue on the basics of q-difference equations based on Chapter I.
From the Introduction: "The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of q-difference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of q-difference equations and we use it to establish the relations among the different theories in the literature."

Friday, October 30, 2015, 12:30–1:45 p.m. Room 6417
This is a cross-listing from Model Theory Seminar

Joel Nagloo, The Graduate Center (CUNY)
On the Existence of Parameterized Strongly Normal Extensions

In this talk we look at the problem of existence of differential Galois extensions for parameterized logarithmic equations. More precisely, if E and D are two distinguished sets of derivations and K is an (E∪D)-field of characteristic zero, we look at conditions on (K^E, D), the set of E-constants of K as a D-field, that guarantee that every (parameterized) E-logarithmic equation over K has a parameterized strongly normal extension.
This is joint work with Omar Leon Sanchez.

Friday, November 6, 2015, 10:15–11:45 a.m. Room 5382

Richard Gustavson and Eli Amzallag, Graduate Center (CUNY)
Effective Difference Nullstellensatz

The difference Nullstellensatz says that in a difference closed pseudofield R (which is a difference ring with some extra properties) a system of polynomial difference equations F = 0 has a common solution in R if and only if the difference ideal generated by F does not contain 1.  The effective difference Nullstellensatz tasks us with finding an upper bound on the number of times we must apply the automorphism to the elements of the system in order to obtain this difference-algebraic relation.  It is currently unknown if such a bound exists.  In this talk, we will present an introduction to the problem along with our approach to solving it, modeled after recent breakthroughs in the differential case.

Friday, November 13, 2015, 10:15–11:45 a.m. Room 5382

Joel Nagloo and Peter Thompson, Graduate Center (CUNY)
On Degree Bounds for Integrability of Differential Equations

In this talk we look at the question of existence of degree bounds for integrability of differential equations. We discuss the failure of the "ultraproduct" methods for bounds in this context and look at well known non-integrable ODEs such as the Painlevé  equations.

Friday, November 20, 2015, 9:45–10:30 a.m. Room 5382

Eli Amzallag and Richard Gustavson, Graduate Center (CUNY)
Model Theory of Difference Fields

In this talk, we will give a brief introduction to the model theory of difference closed fields, called ACFA.  This theory comes from the seminal paper by Z. Chatzidakis and E. Hrushovski (Trans. Amer. Math. Soc. 351 (1999), 2997-3071).  We will describe the axioms of ACFA and discuss some of the elementary properties of this theory.

Friday, November 20, 2015, 10:30–12:00 noon. Room 5382

Richard Churchill (session chair), Graduate Center and Hunter College (CUNY)
L. diVizio and C. Hardouin: Intrinsic Approach to Galois Theory—q-Difference Equations

This Friday will be the third study/discussion session on the paper.
From the Introduction: "The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of q-difference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of q-difference equations and we use it to establish the relations among the different theories in the literature."

Friday, November 27, 2015, Thankgiving Weekend, NO SEMINAR

Friday, December 4, 2015, 10:15–11:45 a.m. Room 5382

William Keigher, Rutgers University at Newark
Interlacing of Hurwitz Series

We begin by reviewing some definitions and properties of the ring of Hurwitz series. Given a commutative ring A with identity, the ring of Hurwitz series over A can be thought of as consisting of all sequences (a_n) with entries a_n in A. This set HA of all sequences in A can be given a natural addition, multiplication, and derivative operator to make HA into a differential ring. In the case where ℚ ⊆ A, HA = A[[t]], the ring of formal power series over A. After describing some important properties of HA, we will define the notion of interlacing and show some important properties of interlacing, including a result on the solutions of n^th order linear differential equations over A.
This talk should be accessible to all graduate students.

For a copy of the slides, please click here.

Friday, December 11, 2015, 10:15–11:45 a.m. Room 5382

Matthew Harrison-Trainor, University of California at Berkeley
Differential-Algebraic Jet Spaces and Internality

We will show that the generic type of a differential-algebraic jet space satisfies a strengthening of almost internality to the constants, called preserving internality to the constants. This is a notion which was first introduced by Moosa and Pillay as a model-theoretic abstraction of a phenomenon in complex-analytic geometry. In contrast with the complex-analytic case, only a generic analogue holds in the differential-algebraic case: there is a finite-dimensional differential-algebraic variety X with a subvariety Z that is internal to the constants, such that the restriction of the differential-algebraic tangent bundle of X to Z is not almost internal to the constants.

For a review of this talk, please click video.

December 12, 2015--January 28, 2016 Winter Break, no seminar. Happy Holidays!

Friday, February 5, 2016, 10:15–11:45 a.m. Room 5382

Richard Gustavson, Graduate Center (CUNY)
Order Bounds for the Rosenfeld-Gröbner Algorithm

In this talk I will describe an upper bound for the orders of derivatives that appear in the Rosenfeld-Gröbner algorithm. The Rosenfeld-Gröbner algorithm computes a regular decomposition of a radical differential ideal in a differential polynomial ring. This algorithm can be used to test for membership in a radical differential ideal and to test the consistency of a system of polynomial differential equations. The upper bound depends only on the orders of the generators of the radical differential ideal, the number of derivations, and the number of differential indeterminates. This is joint work with Alexey Ovchinnikov and Gleb Pogudin.

For a copy of the slides, please click here.

Tuesday, February 9, 2016 (Friday schedule), No Seminar.

Friday, February 12, 2016 (Lincoln's Birthday, college closed): NO SEMINAR

Friday, February 19, 2016, 10:15–11:45 a.m. Room 5382

Henry Towsner, University of Pennsylvania
Constructive Bounds from Ultraproducts and Noetherianity

Techniques like ultraproducts are often used to give short, clean, but non-explicit proofs of constructive theorems. Abstract proofs of constructive results tell us that explicit bounds must be hidden in these proofs. Focusing on examples from differential algebra, we discuss how these results can be used to "mine" proofs to extract explicit (though usually sub-optimal) bounds, and in particular how explicit bounds can be extracted from proofs which use Noetherianity of polynomial ring extensions (and the differential analog).
For a review of the talk, please click video. For a copy of the slides, please click slides

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

Peter Thompson, Graduate Center (CUNY)
An Upper Bound for Lie Integrability of Certain Algebraic Derivations

A theorem of Lie says that if a derivation has enough symmetries, then it is integrable by quadratures. Such a derivation is said to be Lie integrable. Given a polynomially defined derivation on k[x,y] satisfying certain conditions, we present an upper bound on the degree of a polynomially defined derivation that commutes with it.

This is joint work with Joel Nagloo.

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

Alice Medvedev, City College of New York
Difference-Algebraic Groups

A difference field is a field with a distinguished automorphism; difference equations are polynomial equations involving variables and their images under this automorphism. For example, the field of rational functions in one variable x with the automorphism taking  f (x) to f (x+1) is a difference field, and  f (x+1) = f (x)²  is a difference equation in the unknown function  f(x).

Difference equations can define infinite proper subgroups of simple algebraic groups. How can you tell when such a difference-algebraic group is simple as a difference-algebraic group? How can you tell when a difference equation is at all related to some difference-algebraic group? Come find out.

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

George Labahn, University of Waterloo, Canada
Rational Invariants of Finite Abelian Groups and Their Applications

We give a constructive procedure for determining a generating set of rational invariants of the linear action of a finite abelian group in the non-modular case and investigate its use in the symmetry reductions of polynomial and dynamical systems. Finite abelian subgroups of GL(n,K) can be diagonalized, allowing the group action to be accurately described by an integer matrix of exponents. We can make use of integer linear algebra to construct both a minimal generating set of invariants and the substitution to rewrite any invariant in terms of this generating set. The set of invariants provides a symmetry reduction scheme for polynomial systems whose solution set is invariant under a finite abelian group action.

This is joint work with Evelyne Hubert ( INRIA Méditerranée).

For a review of the lecture, please click video. For a copy of the slides, please click slides.

Friday, March 11, 2016, 1:00–2:30 p.m. Room 5382

Michael Wibmer, University of Pennsylvania
Linear Differential Equations and Groups Defined by Difference Equations

I will explain some details of the proof that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated. Moreover, I will try to give some ideas on how groups defined by difference equations can be used in the study of linear differential equations.
For a review of the lecture, please click video.

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

Free Discussion Day

Friday, March 25, 2016 (no class and no seminar)

Friday, April 1, 2016, 10:15–11:45 a.m. Room 5382

Ngoc Thieu Vo, Research Institute for Symbolic Computation, Johannes Kepler University (visiting Graduate Center)
Computing Algebraic and Rational General Solutions of First Order Algebraic ODEs

In this talk, we consider the class of first order algebraic ordinary differential equations (AODEs) and study their algebraic and rational general solutions. The method is based on a consideration of the AODE from a geometric point of view. In particular, parametrization of algebraic curves and surfaces play an important role for a transformation of the AODE to the other ODEs which are heavily studied. We give a decision algorithm for computing rational general solutions whose constant appears rationally. For the problem of computing algebraic general solutions, the current status of the author's work is presented together with an analysis of the difficulties of the problem.

For a copy of the lecture slides, please click here.

A workshop on Differential Algebra was held April 8–10, 2016.
For the three daily programs, please click April 8, April 9, or April 10.

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

Mengxiao Sun, Graduate Center (CUNY)
A Complexity Bound of Hrushovski’s Algorithm

Hrushovski presented an algorithm to compute the differential Galois group of a linear differential equation. An explicit bound that only depends on the order of the linear differential equation was given by Feng. In this talk, I will give an improved complexity bound of Hrushovski’s algorithm.

April 22 through April 30, 2016 is Spring Recess, no seminar.

Friday, May 6, 2016, 10:15–11:45 a.m. Room 5382

Sergey Khashin, Ivanovo State University
Runge-Kutta Methods From an Algebraic Point of View

The talk is devoted to the study of Runge-Kutta (RK) methods for numerical solutions of ordinary differential equations, to the finding of new methods of high order, both analytical and numerical, and to the selection of the most effective methods.

1. We propose a new algebraic approach to the solution of the order conditions (known as the Butcher equations). To an arbitrary matrix of coefficients, we introduce the upper and lower Butcher algebras, which are finite-dimensional commutative associative algebras. The resulting system of equations can be written entirely in the algebraic form, which are equivalent to some restrictions on the dimension of the lower component of the Butcher algebra of the matrix. From a computational point of view, this leads to the fact that the system of Butcher equations can be replaced by an equivalent one with far fewer equations (the original system of equations being highly redundant with many more equations than variables).
2. The proposed method allows us to find some new families of RK methods, including
   • a three-dimensional family of methods RK(5,6);
   • two three-dimensional families of 7-stage methods of order 6; and
   • analytical expressions for the 4-dimensional family 9-stage methods of order 7.
3. We developed methods for the numerical solution of the Butcher equations, giving not only individual solutions but also the local dimension of the variety of solutions in a neighborhood of the solution found. We also found numerically a large class of methods of orders 6, 7, and 8, which are not included in well-known families.

For the slides and a recording of the lecture, please click slides and video.

A Workshop on Differential Algebra was held May 13–15, 2016.
For the three daily programs, please click May 13, May 14, or May 15.

Have a great summer!

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