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 2013–2014

All morning meetings and all afternoon meetings will be in Room 5382 of the Graduate Center. Morning sessions begin at 10:15 am and last till 11:45 am. Most (informal) afternoon sessions begin at 2:00 pm and are unannounced. Occasionally, we may schedule formal talks on Friday afternoons, or on Saturdays at Hunter College, Room E920, and these will be announced. Cancellation due to inclement weather will follow CUNY guideline and policy and will be posted accordingly.

Last updated on July 25, 2014. For Schedules, lecture notes and additional material, see under (or click):
 • Current Schedule   •  Past Lectures–Summer 2014  •  Past Years

Sad News:  On June 17, 2014, Professor Richard M. Cohn passed away at the age of 94. The New York Times published a brief obituary on June 22, 2014. Memorial Service at 11am, July 24, 2014 at Riverside Memorial Chapel, 180 W. 76th Street, NYC.

There are no more lectures until September, 2014. Have a good summer.

Kolchin Seminar in Differential Algebra. KSDA meets most Fridays from 10:15 AM to 11:45 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.

Kolchin Afternoon Seminar in Differential Algebra.This informal discussion series began during the Spring Semester of 2009 and will be continued. Occasionally, for various reasons, we may also schedule guest speakers in the afternoon. It normally goes from 2:00-5:00 pm (please check with organizers). All are welcome.

Unless the contrary is indicated, all morning and afternoon meetings will be in Room 5382. This room may be difficult to find; please read the following directions. When you exit the elevator on the 5th floor, there will be doors both to your left and to your right. Go through the doors where you see the computer monitors, then turn left and then immediately right through two glass doors. At the end of the corridor, go past another set of glass doors and continue into the short corridor directly in front of you. Room 5382 is the last room on your right.

Security. 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 (updated September 1, 2013).

Occasionally, we also meet on a Saturday at Hunter College, Room E920. Hunter College is on 68th Street and Lexington Avenue, where the No. 4,5,6 subways stop. You need to enter from the West Building (a photo ID is required), go up the escalators to the third floor, walk across the bridge over Lexington Avenue to the East Building, and take the elevator before the Library to the 9th floor. Room 920 is located in a north-east corner.

Past Lectures, Summer 2014

Monday, July 7, 2014 at 9:00–10:10 a.m. Room 5382

Markus Rosenkranz, University of Kent at Canterbury, UK
Integro-Differential Polynomials and Free Integro-Differential Algebras

Adjunction of a transcendental element to an ordinary integro-differential algebra yields an analog of differential polynomials, consisting of nested nonlinear integral operators. The resulting ring of integro-differential polynomials carries the structure of an integro-differential algebra, and it contains an isomorphic copy of the corresponding differential polynomial ring. We present effective normal forms for integro-differential polynomials and exhibit their relation to the free object in the integro-differential category. A short outlook at the case integro-differential fractions will round up the talk.

For a video viewing of the lecture, please click here.
For the revised slides of this talk, please click here.

Monday, July 7, 2014 at 10:15–11:45 a.m. Room 5382

Sonia Rueda, Universidad PolitÚcnica de Madrid
Sparse Resultant Formulas for Differential Polynomials

Differential resultant formulas are defined for a system P of ordinary Laurent differential polynomials. These are determinants of coefficient matrices of an extended system of polynomials obtained from P, through derivations and multiplications by Laurent monomials. The first construction of this type was given by G. Carrà-Ferro in 1997. One would want these determinants to have the sparse differential resultant of P (defined by W. Li, C.M. Yuan and X.S. Gao in 2012) as a factor (in the generic case). Such result is proved for linear nonhomogeneous differential polynomials, an interesting case because one can focus on the sparsity problem with respect to the order of derivation, and forget about the sparsity on the degree. The methods used in the linear case extend to the nonlinear case, to construct a system ps(P) consisting of L polynomials in L-1 algebraic variables, for which the usual algebraic theory of sparse resultants can be applied.

For a video viewing of the lecture, please click here.
For the revised slides of the talk, please click here.

Monday, July 7, 2014 at 1:30–3:20 p.m. Room 5382

Markus Rosenkranz, University of Kent at Canterbury, UK
A Differential Algebra Approach to Linear Boundary Problems

In this survey talk we present an algebraic approach to regular boundary problems for linear ordinary and partial differential equations. Expanding the structure of differential algebra by a Rota-Baxter operator, we construct an operator ring that can be used for encoding the boundary problem (differential equation + boundary conditions) as well as its resolving Green's operator (integral operator with Green's function kernel). The differential algebra setting is based on an abstract theory of linear boundary problems over infinite-dimensional vector spaces. It allows to transfer an arbitrary factorization of the differential operator of a boundary problem to an integration cascade of the latter.

For a video viewing of the lecture, please click here and here.
For the slides of this talk, please click here.

Monday, July 7, 2014 at 3:30–5:00 p.m. Room 5382

Gabriela Jeronimo, Universidad de Buenos Aires, Argentina
On the Differential Nullstellensatz: Order and Degree Bounds

The following differential version of Hilbert's Nullstellensatz was introduced by Ritt, and later extended to arbitrary differential fields: if f1,..., fs, and g are multivariate differential polynomials with coefficients in an ordinary differential field K such that every zero of the system in any extension of K is a zero of g, then some power of g is a linear combination of the fi's and a certain number of their derivatives, with polynomials as coefficients. The first known bound for orders of derivatives in the differential Nullstellensatz for both partial and ordinary differential fields was given in a paper in 2008 by Golubitsky, Kondratieva, Ovchinnikov and Szanto. We will present new order and degree bounds for the differential Nullstellensatz in the case of ordinary systems of differential algebraic equations over a field of constants K of characteristic 0. Our main result is a doubly exponential upper bound for the number of successive derivatives of f1,..., fs involved. Combining this bound with effective versions of the classical algebraic Hilbert's Nullstellensatz, we also give a bound for the power of g in the differential ideal, and for the degrees of polynomial coefficients in a linear combination of f1,..., fs and their derivatives representing this power of g.

(Joint work with Lisi D'Alfonso and Pablo Solernˇ).
For a video viewing of the lecture, please click here.

Tuesday, July 8, 2014 at 9:00–10:30 a.m. Room 4419

Moulay Barkatou, XLIM Research Institute, Limoges, France
Algorithms for Finding Rational Solutions of Linear Differential and Difference Equations and Their Complexity

In this talk we will give a general survey on the algorithms for finding (efficiently) rational solutions of systems of linear differential and difference equations and investigate their complexity.
Problems of finding more general solutions or solving other classes of equations will be discussed as well.

For a video viewing of the lecture, please click here.

Tuesday, July 8, 2014 at 10:45 a.m.–12:15 p.m. Room 4419

Markus Rosenkranz, University of Kent at Canterbury, UK
A Noncommutative Mikusinski Calculus for Linear Boundary Problems

While the classical Mikusinski calculus views a derivation as the reciprocal of a fixed integral operator via a commutative localization of a convolution algebra, the incorporation of boundary conditions leads to a noncommutative localization that employs Green's operators as reciprocals. Unlike the classical approach, this construction is applicable to a large class of abstract integro-differential algebras. We will also discuss the relation between localized boundary problems and the integro-differential analog of the (first) Weyl algebra, which exhibits a certain discrepancy between two-sided inverses of the derivation and natural actions on the underlying polynomial ring.

For a video viewing of the lecture, please click here.
For the revised slides of this talk, please click here.

Tuesday, July 8, 2014 at 2:00–5:00 p.m. Room 4419

Markus Rosenkranz, University of Kent at Canterbury, UK
I: Software for Symbolic Boundary Problems and Applications in Actuarial Mathematics
II: Singular Boundary Problems and Generalized Green's Operators

Part I: We give a short overview of available software for the symbolic treatment of linear boundary problems. Applications in actuarial mathematics are presented, where the description of renewal risk models leads to boundary problems for high-order linear ordinary differential equations. A factorization approach is employed for deriving an explicit form of their general solution parameterized by the order of the equation and the given actuarial parameters.

For a video viewing of the lecture, please click here.
For the revised slides of this talk, please click here.

Part II. In this talk we consider boundary problems for linear ordinary differential equations that are singular in the sense that their boundary conditions do not admit a solution for every forcing function. Specifically, we will be dealing with overdetermined boundary conditions, where additional constraints on the forcing function are to be determined to ensure the existence of solutions. If these constraints are satisfied, a generalized Green's operator can be determined for solving the corresponding boundary problem. We use an abstract differential algebra setting, thus generalizing the classical theory of Moore-Penrose inverses for boundary problems in Hilbert space.
(This part of the talk is based on A. Korporal's PhD thesis, Johannes Kepler University, Linz, Austria, December 2012.)

For a video viewing of the lecture, please click here.
For the slides of this talk, please click here.

Wednesday Through Saturday, July 9‐12, 2014

Applications of Computer Algebra (ACA) 2014 will be held at Fordham University, New York, from July 9 through 12, 2014. The conference is organized by Robert H. Lewis (General and Local Chair), Tony Shaska and Illias Kotsireas (Program Co-chairs) with an Advisory Committee composed of Eugenio Roanes-Lozano, Stanly Steinberg,and Michael Wester. There will be a Special Session on Computational Differential and Difference Algebra, organized by Alexey Ovchinnikov of Queens College and the Graduate Center, in addition to the satellite lectures listed above and below.

Monday, July 14, 2014 at 10:00–11:15 a.m. Room 4419

Julien Roques, Institut Fourier, UniversitÚ Grenoble 1
Regular Singular q-Difference Equations and Birkhoff Matrices

We will first recall the classification of regular singular q-difference equations by means of Birkhoff matrices. Then, we will study a notion of rigidity based on the residues of the Birkhoff matrices.

Monday, July 14, 2014 at 11:30–12:45 a.m. Room 4419

Suzy S. Maddah, XLIM Research Institute, Limoges, France.
Moser-based algorithms over Univariate and Bivariate (Differential) Fields

Moser-based algorithms are algorithms based on the regularity notion introduced by Moser in 1960 for linear singular differential systems. They have proved their utility in the symbolic resolution of systems of linear functional equations (see, e.g., Barkatou'1997, Barkatou-Broughton-Pfluegel'2007, Barkatou-Pfluegel'2009) and the perturbed algebraic eigenvalue-eigenvector problem (Jeannerod-Pfluegel'1999). This gave rise to the package ISOLDE (Barkatou-Pfluegel) which is written in the computer algebra system Maple and dedicated to the symbolic resolution of linear functional matrix equations. However, such algorithms have not been considered yet over bivariate fields. This will be the interest of this talk. We give a generalization of the Moser-based algorithm given by Barkatou'1995 to simplify the symbolic resolutions of singularly-perturbed linear differential systems and completely integrable Pfaffian systems with normal crossings in two variables. This unified treatment of these two well-known differential systems, which exhibit dissimilar kinds of difficulties, illustrates the flexibility of such algorithms in the bivariate case and paves the way for further applications. Our algorithms are implemented and examples are illustrated in Maple.

For an extended abstract with additional references, please click here.

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