Kolchin Seminar in Differential Algebra 

The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
Last updated on December 13 2014. For
Schedules,
lecture notes and additional material, see under (or click):
• Current Schedule
•
Past Lectures–Fall
2014 •
Past Years
End of Fall Semester Seminars. Happy Holidays!
Kolchin Seminar in Differential Algebra. For 2014 Fall Semester, KSDA meets most Fridays from 12:30 PM to 2:00 PM at the Graduate Center, with occasion talks also from 2:15 PM to 3:45 PM and at Hunter College, on some Saturdays. 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 2:30–4:00 pm (please check with organizers). All are welcome. Unless the contrary is indicated, all 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 northeast
corner. 
Friday, September 5, 2014, 1:00 –2:30 p.m. Room 5382
Richard Churchill, Graduate Center and Hunter College, CUNY
Kolchin's Proof that Differential Galois Groups are AlgebraicKolchin's 1948 proof that differential Galois groups are algebraic appeals to a formulation of algebraic geometry which is no longer in fashion. Modern proofs require considerable knowledge of the Grothendieck approach to that subject, which takes considerable time to digest. In this talk I hope to convince those attending that Kolchin's proof can be understood in contemporary terms with only minor appeals to algebraic geometry, i.e. the definition of an algebraic set and the Hilbert Basis Theorem.
For lecture notes, please click here.
Friday, September 12, 2014, 12:30 –2:00 p.m. Room 5382
James Freitag, University of California at Berkeley
Effective Bounds For Finite DifferentialAlgebraic Varieties (Part I)Given a differential algebraic variety over a partial differential field, can one give a bound for the degree of its Zariski closure that depends only on the order and degree of the differential polynomials (but not the parameters) which determine the variety? We will discuss the general theory of prolongations of differential algebraic varieties as developed by Moosa and Scanlon, and use this theory to reduce the problem to a combinatorial problem (which will be discussed in detail in the second part of the talk). Along the way we will give numerous examples of the usefulness of the result, some of an arithmetic flavor. We will also describe some other applications of the theory of prolongations.
This is joint work with Omar Sanchez. For a video recording of the talk, please click here.
Friday, September 12, 2014, 2:15 –3:45 p.m. Room 5382
Omar Leon Sanchez, McMaster University
Effective Bounds For Finite DifferentialAlgebraic Varieties (Part II)We will talk about the difficulties that commutativity entails when trying to find points of the form (a, d_{1}(a),...,d_{m}(a)) in algebraic subvarieties of prolongations. We discuss how to deal with these issues by passing to higher order prolongations (where the order is "uniform"). We use this to establish effective bounds for finite differentialalgebraic varieties.
This is joint work with James Freitag. For a video recording of the talk, please click here.
Friday, September 19, 2014, 12:30 –2:00 p.m. Room 5382
Ronnie Nagloo, Graduate Center of CUNY
Model Theory and the Painlevé EquationsThe Painlevé equations are nonlinear 2^{nd} order ODEs and come in six families P_{1}, …, P_{6}, where P_{1} consists of the single equation y''=6y^{2}+t, and P_{2}, …, P_{6} come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20^{th} century but have arisen in a variety of important physical applications, including for example random matrix theory and general relativity.
To view the video recording of this talk, please click here and here.
Tuesday (Friday Schedule), September 23, 2014, 12:30 –2:00 p.m. Room 5382
Raymond Hoobler, CCNY and Graduate Center of CUNY
A Differential Poincaré LemmaLet X be a smooth scheme over a field k of characteristic 0. I will show that the complex
0 → O^{Δ}_{X} → O_{X} → Ω^{1} _{X/k} → …
constructed from the de Rham complex of X is exact in the dfinite topology if Δ consists of all "derivations" of X. In particular, this means that the dfinite cohomology of a variety matches the singular cohomology of X with complex coefficients and agrees with the étale cohomology of X with torsion coefficients. Most of the talk will be an outline of the necessary steps that lead up to this result. Potential applications will be described.
To view a video recording of this talk, please click here.
Friday, September 26 and October 3, 2014, School Holidays, no seminar.
October 2, 2014, 1:00 –2:00 p.m. Room 6/113 North Academic Center, CITY COLLEGE
David Marker, University of Illinois at Chicago
Model Theory and ExponentiationMethods from mathematical logic have proved surprisingly useful in understanding the geometry and topology of sets definable in the real field with exponentiation. When looking at the complex exponential field, the definability of the integers is a seemingly insurmountable impediment, but a novel approach due to Zilber leads to a large number of interesting new questions.
This is a crosslisting from Model Theory Seminar and the CCNY Mathematics Colloquium. You are welcome to join the speaker for lunch at noon in the Faculty Dining Hall on the 3rd Floor of the North Academic Center, CCNY.
Friday, October 10, 2014, 12:30 –1:45 p.m. Room 5382
Ronnie Nagloo, Graduate Center of CUNY
Geometrically Trivial Strongly Minimal Sets in DCF_{0}In this talk we look at the problem of describing the `finer' structure of geometrically trivial strongly minimal sets in DCF_{0}. In particular, I will talk about the ωcategoricity conjecture, recently disproved in its general form by James Freitag and Tom Scanlon, and the unimodularity conjecture, a weakening of the above conjecture and which came to life after the work on the second Painlevé equations.
Friday, October 10, 2014, 2:00 –3:30 p.m. Room 6417
Alice Medvedev, City College, CUNY
Model Theory of Difference Fields, Part I
I'll begin by setting up the firstorder language and axioms of difference fields, and give some interesting examples, including Frobenius automorphisms of fields in positive characteristic and difference equations from analysis that give the subject its name. Differenceclosed fields, a natural analog of algebraically closed fields, have a nice model theory, starting with almostquantifier elimination. Further modeltheoretic notions—algebraic closure, elementary equivalence, forking independence—all have elementary purely algebraic characterizations that I will explain. The model theory of difference fields has been used in arithmetic geometry in several exciting ways (Hrushovski's results on the ManinMumford Conjecture; his twisted LangWeil estimates; several people's work on algebraic dynamics) that I will probably not explain in detail.
This is a crosslisting from CUNY Logic Workshop. This talk will be continued in the Model Theory Seminar the following week.
Friday, October 17, 2014, 10:45 –12:15 p.m. Room 5382
Alice Medvedev, City College, CUNY
Model Theory of Difference Fields, Part II
This is a continuation of my talk at the CUNY Logic Workshop last week. The necessary background will be summarized briefly at the beginning of this one.
ACFA, the theory of difference closed fields, is a rich source of explicit examples showing forking independence and nonorthogonality, the distinction between stable and simple theories, and the distinction between onebased and locallymodular groups. To present these examples, I will introduce difference varieties, which are the basic building blocks of definable sets in ACFA, and sigmavarieties, a more tractable special case of these.
This is a crosslisting from Model Theory Seminar.
Friday, October 17, 2014, 2:00 –3:30 p.m. Room 6417
Ronnie Nagloo, Graduate Center of CUNY
On Transformations in the Painlevé FamilyThe Painlevé equations are nonlinear 2^{nd} order ODEs and come in six families P_{1}, …, P_{6}, where P_{1} consists of the single equation y''=6y^{2}+t, and P_{2}, …, P_{6} come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20^{th} Century but have arisen in a variety of important physical applications. In this talk I will explain how one can use Model Theory to answer the question on existence of algebraic relations among solutions of different Painlevé equations from the families P_{1}, …, P_{6}.
This is a joint seminar with CUNY Logic Workshop.
Friday, October 24, 2014, 12:30 –2:00 p.m. Room 5382
Li Guo, Rutgers University at Newark
RotaBaxter Operators on Polynomial Algebras, Integration, and Averaging 0peratorsThe RotaBaxter operator is an algebraic abstraction of integration. Following this classical connection, we study the relationship between RotaBaxter operators and integration in the case of the polynomial ring k[x], where both concepts make sense. We consider two classes of RotaBaxter operators: monomial ones and injective ones. For the first class, we apply averaging operators to determine monomial RotaBaxter operators. For the second class, we make use of the double product on RotaBaxter algebras.
If time permits, we will talk more about averaging operators, which crop up naturally from this study.
This is a joint work with Markus Rosenkranz and Shanghua Zheng.
For a copy of the slides, please click here.
Friday, October 31, 2014, 12:30 –1:45 p.m. Room 5382
Anand Pillay, University of Notre Dame
Interpretations and Differential Galois ExtensionsWe prove a number of results around finding strongly normal extensions of a differential field K, sometimes with prescribed properties, when the constants of K are not necessarily algebraically closed. The general yoga of interpretations and definable groupoids is used (in place of the Tannakian formalism in the linear case).
This is joint work with M. Kamensky.
For a video recording of this talk, please click here.
Friday, October 31, 2014, 2:00 –3:30 p.m. Room 6417
Anand Pillay, University of Notre Dame
MordellLang and ManinMumford in Positive Characteristic, RevisitedWe give a reduction of function field MordellLang to function field ManinMumford, in positive characteristic. The upshot is another account of or proof of function field MordellLang in positive characteristic, avoiding the recourse to difficult results on Zariski geometries.
This work is joint with Benoist and Bouscaren.This is a cross listing from CUNY Logic Workshop.
Friday, November 7, 2014, 12:30 –2:00 p.m. Room 5382
Michael Singer, North Carolina State University
The General Solution of A First Order Differential PolynomialThis is the title of a 1976 paper by Richard Cohn in which he gives a purely algebraic proof of a theorem (proved analytically by Ritt) that gives a bound on the number of derivatives needed to find a basis for the radical ideal of the general solution of such a polynomial. I will show that the method introduced by Cohn can be used to give a modern proof of a theorem of Hamburger stating that a singular solution of such a polynomial is either an envelope of a set of solutions or embedded in an analytic family of solutions depending on whether or not it corresponds to an essential singular component of this polynomial. I will also discuss the relation between this phenomenon and the Low Power Theorem. This will be an elementary talk with all these terms and concepts defined and explained.
This is joint work with Evelyne Hubert.
For a video recording of this talk, please click here.
Friday, November 14, 2014, 12:30 –2:00 p.m. Room 5382
Rahim Moosa, University of Waterloo
Differential Varieties with Only Algebraic ImagesConsider the following condition on a finitedimensional differentialalgebraic variety X: whenever X→Y is a dominant morphism, and dim(Y) < dim(X), then Y is (a finite cover of) an algebraic variety in the constants. This property is a specialisation to differentially closed fields of a modeltheoretic condition that itself arose as an abstraction from complex analytic geometry. Nonalgebraic examples can be found among differential algebraic subgroups of simple abelian varieties. I will give a characterisation of this property that involves differential analogues of "algebraic reduction" and "descent". This is joint work with Anand Pillay.
For a partial video recording of this talk, please click here.
Friday, November 14, 2014, 2:15 –3:45 p.m. Room 5382
Uma Iyer, Bronx Community College, CUNY
Weight Modules of an Algebra of Quantum Differential OperatorsGeneralized Weyl Algebras (GWAs) were independently introduced by V. Bavula and A. Rosenberg. These algebras have been widely studied. In particular, weight modules over the GWAs have been also studied. We study weight modules over a particular algebra of quantum differential operators which contains a GWA of rank 1. This is joint work with V. Futorny.
For a video recording of this talk, please click here.
Saturday, November 15, 2014, 11:00 a.m.–12:30 p.m. Hunter College Rm HW217
Andrey Minchenko, Weizmann Institute
On a Problem of Computing Parameterized PicardVessiot GroupWe will deal with the problem of computing parameterized Galois groups of differential equations. At present, one can determine whether the group is unipotent or reductive, and compute the group algorithmically in both cases. We will consider the obstacles we face in our attempt to solve the general case (when the group is neither unipotent nor reductive) and explain how some of them may be dealt with.
Friday, November 21, 2014, 12:30 –2:00 p.m. Room 5382
Joseph Gunther, Baruch College and the Graduate Center, CUNY
Difference Algebraic GeometryWe'll examine the foundations of schemestyle difference algebraic geometry, as developed by Hrushovski to prove a generalization of the LangWeil estimates for the number of points of a variety over finite fields. This means working over not just an arbitrary ring, but over an arbitrary ring with a selfmap. We'll consider constructions and facts from standard algebraic geometry, and see how they work (or don't) in the difference algebra setting.
Friday, November 28, 2014, Thanksgiving Week, no seminar.
Friday, December 5, 2014, 12:30 –2:00 p.m. Room 5382
Carlos Arreche, North Carolina State University
On the Computation of the DifferenceDifferential Galois Group for a SecondOrder Linear Difference EquationConsider the difference field C(x) with automorphism φ: x ↦ x +1, and assume that the field of constants C is algebraically closed and of characteristic zero. The difference Galois theory of van der Put and Singer associates a linear algebraic group over C to a linear difference equation with respect to φ. This Galois group measures the algebraic relations amongst the solutions to the difference equation. There is an algorithm to compute the difference Galois group corresponding to a secondorder linear difference equation over C(x), due to Hendriks. More recently, Hardouin and Singer have developed a Galois theory for differencedifferential equations that associates a linear differential algebraic group to a linear difference equation, and this Galois group measures the differentialalgebraic relations amongst the solutions. In this talk, I will describe ongoing work towards extending Hendriks' algorithm to compute the differencedifferential Galois group associated to a secondorder linear difference equation over C(x).
Friday, December 12, 2014, 12:30 –2:00 p.m. Room 5382
Wei Li, KLMM, Chinese Academy of Sciences, and University of California at Berkeley
Sparse Difference ResultantIn this talk, we first define sparse difference resultant for a Laurent transformally essential system of difference polynomials and give a simple criterion for the existence of sparse difference resultant which requires only linear algebraic techniques. Then we discuss the basic properties of the sparse difference resultant, in particular, give its order bound in terms of the Jacobi number and degree bound. We show the projective difference space is not transformally complete. If time permits, as a special case, we introduce the difference resultant and give the precise order and degree, a determinant representation, and a Poissontype product formula for the difference resultant.
This is joint work with XianShan Gao and ChunMing Yuan.
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