Kolchin Seminar in Differential Algebra 

The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
Academic year 2012–2013 Last updated on January 18, 2018. 
Friday, August 31, 2012, NO SEMINAR
Friday, September 7, 2012 at 10:15 a.m. and 2:00 p.m. Room 5382
Alexander R. Its, Indiana UniversityPurdue University Indianapolis
Isomonodromy Deformations and the RiemannHilbert MethodOne of the interesting developments in mathematical physics for the last three to two decades is the transformation of the classical monodromy theory of systems of linear ODEs into a new analytical tool — the RiemannHilbert method. This method has allowed researchers to solve a number of longstanding problems in many different areas of mathematics and physics. As key examples we consider the theory of integrable nonlinear equations of the KdV and Painlevé types, statistical mechanics, orthogonal polynomials, and random matrices.
This survey talk is based on the works done by many people and spanned over more than two decades. It will be delivered in two parts; the second part is from 2:00 pm to 3:30 pm.
Friday, September 7, 2012 at 3:30 p.m. Room 5382
Informal session with
Bernard Malgrange, Université Joseph Fourier–Grenoble
Friday, September 14, 2012 at 10:15 a.m. Room 5382
Carlos Arreche, Graduate Center, CUNY
An Algorithm to Compute the PPV Group of a Second Order Linear Differential Equation with One Differential ParameterWe develop algorithms to compute the Parameterized PicardVessiot (PPV) group corresponding to a second order linear differential equation with one differential parameter. We work in the setting of the PPV theory as developed by Cassidy and Singer in an analogous manner to the construction in classical PicardVessiot theory of Kolchin, where the Galois group of such an equation is now a linear differential algebraic group. Our algorithms compute the polynomial differential equations that define the PPV group as a differential algebraic subgroup of GL(2).
We first perform a standard change of variables to obtain an associated unimodular equation. We apply Kovacic's algorithm to compute the Liouvillian solutions for this associated equation, if they exist, and we compute the unimodular PPV group of the associated equation from these data. We then describe an algorithm to compute the PPV group of the original equation, as a quotient by a zerodimensional kernel of the direct product of this unimodular PPV group and the changeofvariables group. A more detailed version of these results is available in my preprint.
This talk may be extended to the afternoon session, starting at 2:00 pm if necessary.
Friday, September 14, 2012 at 3:15 p.m. Room 5382
Informal session with
Bernard Malgrange, Université Joseph Fourier–Grenoble
Friday, September 21, 2012 at 10:15 a.m. Room 5382
Andy Magid, University of Oklahoma
Rationally Differential Commutative AlgebrasA differential commutative ring R is a differentially rational algebra over the differential field F (which we assume to be of characteristic zero with an algebraically closed field of constants) if every element of R satisfies a linear homogeneous differential equation over F. A differential R module M is differentially rational if every element of M satisfies a linear homogeneous differential equation over R. We consider the category of differentially rational R modules in general, and in the case R is differentially simple. In the latter case we show that all differential R monomorphisms of rationally differential R modules are R split. We use this to prove that all square zero differential extensions (in the simple case) with differentially rational kernel are F algebra split, and develop a criterion for when extensions are differentially F algebra split. When R is the PicardVessiot ring of a PicardVessiot extension of F, this criterion implies that all square zero differential extensions with differentially rational kernel are split as differential F algebras.
Friday, September 21, 2012 at 2:00 p.m. Room 5382
Carlos Arreche, Graduate Center, CUNY
An Algorithm to Compute the PPV Group of a Second Order Linear Differential Equation with One Differential Parameter, PART IIThis is a continuation of the talk on September 14, 2012. We develop algorithms to compute the Parameterized PicardVessiot (PPV) group corresponding to a second order linear differential equation with one differential parameter. We work in the setting of the PPV theory as developed by Cassidy and Singer in an analogous manner to the construction in classical PicardVessiot theory of Kolchin, where the Galois group of such an equation is now a linear differential algebraic group. Our algorithms compute the polynomial differential equations that define the PPV group as a differential algebraic subgroup of GL(2).
We first perform a standard change of variables to obtain an associated unimodular equation. We apply Kovacic's algorithm to compute the Liouvillian solutions for this associated equation, if they exist, and we compute the unimodular PPV group of the associated equation from these data. We then describe an algorithm to compute the PPV group of the original equation, as a quotient by a zerodimensional kernel of the direct product of this unimodular PPV group and the changeofvariables group. A more detailed version of these results is available in my preprint.
Friday, September 28, 2012 at 10:15 a.m. Room 5382
Raymond Hoobler, CCNY and Graduate Center of CUNY
Differential Deformation Theory — Picard Vessiot CaseWe will build on Andy Magid's work to describe a deformation theory in the Picard Vessiot Case.
Friday, October 5, 2012 at 10:15 a.m. Room 5382
Alexandru Buium, University of New Mexico
Arithmetic PDEsIt is possible to develop two arithmetic PDE theories as follows. The first theory considers two directions, one arithmetic direction (in which derivatives are replaced by Fermat quotients with respect to one fixed prime) and one geometric direction (in which one considers a genuine derivative operator). The second theory considers, again, two directions but now the two directions are both arithmetic (and they correspond to Fermat quotient operators with respect to two fixed primes). One can then attach to various "group related" varieties (such as abelian varieties or Shimura varieties) arithmetic analogues of hyperbolic, parabolic, and elliptic PDEs. The study of solutions of these arithmetic PDEs exhibits interesting analogies with the Fourier analytic study of solutions of classical linear PDEs.
Prof. Buium will be also giving a talk on Differential Calculus with Integers at the Number Theory Seminar, on Thursday, October 4, 2012, Room C198, at 5:30pm to 7:00pm.
Friday, October 12, 2012 Informal discussion session (with Carlos Arreche) at 10:15 a.m. Room 5382
Friday, October 12, 2012 at 2:00 p.m. Room 5382
John Nahay, Bioenergia America, LLC, Princeton, NJ
The Wall of Complexity: Summations Over Partitions of IntegersWhat do the following problems have in common?
 Solving nonlinear ODEs/PDEs by recursively differentiating the PDEs to get their Taylor series
 Using the Lagrange Inversion Formula to invert a hypergeometric function
 Finding the local extrema of a transcendental function
 Finding a smooth function which goes through a given finite set of monotonically increasing points such that the function is monotonically increasing on the entire interval containing the points. (This problem arose in a cell culture course in biotechnology.)
 Computing all infinitely many roots of a transcendental function, such as the pseudopolynomial z^{a}  z + 2 = 0 where a is an arbitrary complex number (My latest published paper in the Journal: Mathematics in Computer Science)
 Computing the Kostka numbers in the transition matrix from one basis of symmetric functions to another
 Computing terms in the Faa di Bruno formula
 Proving certain desired identities (like those in the book of 500 Combinatorial Identities by Henry Gould)
Answer: They all involve trying to express some large formula involving summations over partitions of some indexing integer n, usually in some kind of factored form, such as the familiar infinite product formula for the gamma function.
I will show with examples how far I could get before my hopes were dashed by the mysterious "wall of complexity".
For the lecture notes (Revised 10/26/2012) , please click here.
Friday, October 19, 2012 at 10:15 a.m. Room 5382
Shaoshi Chen, North Carolina State University
Rooks, Recurrences and ResiduesThere are 3,968,310 ways for a Rook on one corner square of an 8 × 8 chessboard to reach the opposite corner square. Techniques originated by Wilf and Zeilberger using Dmodule theory allow one to write down a three term recurrence relation for R(n), the number of ways a rook can move from one corner square to the opposite corner square on an n × n chessboard. When one tries to generalize these results to three dimensional chessboards, the WilfZeilberger techniques become much too computationally complex to effectively yield a recurrence.
Recently, we have developed a technique, based on simple fact concerning residues of differentials on compact Riemann surfaces, that overcomes this bottleneck. This talk will be an elementary exposition of this technique and its application to other combinatorial problems. This is a joint work with Manuel Kauers (RISC) and Michael F. Singer (NCSU).For lecture slides, please click here.
Friday, October 26, 2012 No morning session.
Friday, October 26, 2012 at 2:00 p.m. Room 5382
John Nahay, Bioenergia America, LLC, Princeton, NJ
The Wall of Complexity: Summations Over Partitions of Integers, Part IIThis is a continuation of Dr. Nahay's October 12 lecture.
For lecture notes (Revised 10/26/2012) , please click here.
Friday, November 2 and 9, 2012 at 10:15 a.m. No Seminar due to postSandy problems.
Friday, November 16, 2012 at 10:15 a.m. Room 5382
James Freitag, University of California at Berkeley
Indecomposability for Differential Algebraic GroupsWe will talk about two techniques for proving subgroups of differential algebraic groups are closed in the Kolchin topology. The first is a modeltheoretic technique from stability theory, which suffices to provide one of the necessary ingredients of Andrei Minchenko's recent proof that noncommutative almost simple differential algebraic groups are actually quasisimple. The second technique is designed to work for more general strongly connected differential algebraic groups, where the model theoretic technique does not work.
For slides from the lecture, please click here
For additional notes related to the lecture, please click here
Friday, November 23, 2012: NO SEMINAR (Thanksgiving Weekend).
Friday, November 30, 2012 at 10:15 a.m. Room 5382
Alexander Levin, Catholic University of America
On the Maximality Condition for Certain Types of Difference IdealsWe will discuss several types of difference ideals and present an analog of R. Cohn's process of "shuffling" for the construction of the mixed difference ideal generated by a set. Then we will show that the ring of difference polynomials over a difference field does not satisfy the maximality condition for mixed difference ideals. This result solves an open problem of difference algebra stated by E. Hrushovski in connection with the development of difference algebraic geometry.
Friday, December 7, 2012 at 10:15 a.m. Room 5382
Carlos Arreche, Graduate Center, CUNY
Computing the PPV Group for a Parameterized Second Order Equation in the NonUnimodular CaseGiven a secondorder ordinary linear differential equation, there is a classical changeofvariables procedure which allows one to express its solutions in terms of the solutions for an associated unimodular equation (i.e., whose differential Galois group is a subgroup of SL(2)). The explicit nature of the procedure makes it relatively simple to compute the classical PicardVessiot (PV) group corresponding to the original equation in terms of the PV group of the associated unimodular equation and a changeofvariables group (an algebraic subgroup of the multiplicative group G_{m}, which can be computed explicitly). The problem of computing the PV group of an arbitrary secondorder equation is thereby reduced to the computation of this group for a unimodular equation only.
We will discuss how to carry out an analogous reduction in the parameterized setting. Although the situation is conceptually similar to the classical one, it is complicated in a nontrivial way by the richer differentialalgebraic structure of the parameterized PicardVessiot (PPV) groups. Our method relies on the parameterized Galois correspondence, the classification of the differential algebraic subgroups of G_{m}, and the classical theorem of KolchinOstrowski.
Friday, December 14, 2012 at 2:00 p.m. Room 5382
This talk was originally scheduled on November 9 and had to be canceled and rescheduled due to unforeseen circumstances.John Nahay, Bioenergia America, LLC & Broadway Performance Systems
Entropy: Why Doing Nothing at All is Better than Doing AnythingHow much energy goes into growing a human being, or any other animal, or a tree? How much energy gets wasted in the process, harming sentient beings along the way? What is the entropy differential among our choices, each of which includes, by how we vote, all the laws we support and oppose? Whom/what/how much we consume, and how much we reproduce? While I had contemplated such questions since the early 1980s and have been working on the problem intermittently, intense work did not begin on actually computing the utility values of these choices until I was asked to give a talk on behalf of Bioenergia America, LLC for a course in renewable energy at Burlington County College in Pemberton, New Jersey.
We will begin the long complicated process of answering these questions and computing these values with a completely naive model—one that counts the number of distinct arrangements of atoms in chemical bonds and makes use of only the following physical laws: one on the existence of common chemical bonds and another relating their energies of formation. These laws may be used to compute the probabilities of certain bond formations. The computation of entropy at this level of modeling is helped by combinatorial algebra.
This is an exploratory talk. Audience comments are welcome. An introductory article is available here: Last revised March 1, 2013.
Friday, February 1, 2013 at 10:15 a.m. Room 5382
Organization Meeting, no seminar.Friday, February 8, 2013 at 10:15 a.m. Room 5382
Richard Churchill, Hunter College and Graduate Center, CUNY
A SetTheoretic Approach to Model TheoryAlthough one always employs logic in proofs, the foundations of many branches of mathematics appear to be predominantly settheoretic: one defines a topological space to be a pair (X, τ) consisting of a set X and a collection τ of subsets satisfying certain wellknown properties; one defines a group to be a pair (G, μ) consisting of a set G and a subset μ ⊂ G × G × G as the binary operation satisfying certain wellknown properties (of course, for a group one needs a bit more to handle the identity); etc. There are advantages to this commonality, particularly if one is wellversed in category theory: one can move from one area to the other and still have a fairly good idea of what the major problems are and the sort of techniques one might expect to see. In contrast, in Model Theory, the foundation appears to be heavily based on logic, and as a result the language and terminology can seem foreign to those who work in more widely publicized areas of mathematics. Rather than "sets of groups", one hears about "sets of formulas"; rather than products (Cartesian, fibered, direct, or semidirect), one hears of "ultraproducts"; rather than "reducing to a simpler case", one is told about "eliminating quantifiers".
In this talk I will indicate how some of the basic ideas of Model Theory can be formulated settheoretically, that is, in the topological and algebraic spirit indicated above.
For lecture notes and slides, please click Revised as of April 4, 2013.
Friday, February 15, 2013 at 10:15 a.m. Room 5382
Roman Kossak*, Graduate Center, CUNY
On the Existence of SetsI will review the axioms of ZFC. I will focus on the axiom schema of replacement. I will say a bit about its history and discuss some of its consequences.
*Due to our scheduled speaker falling sick, this talk will be given by David Marker of University of Illinois at Chicago instead.
Friday, February 22, 2013 at 10:15 a.m. Room 5382
William Sit, City College of New York, CUNY
Basics of Dimension in Differential AlgebraThis will be a review of dimension concepts in differential algebra, with a goal to relate them to those in Model Theory. I will discuss differential transcendence degree, differential type, typical differential dimension, and the differential dimension polynomial (Kolchin polynomial) of a finitely generated differential field extension and state certain properties. I will sketch a proof of the wellordering theorem on numerical polynomials that included the Kolchin polynomials. Time permitting, examples will be given to suggest connections between some concepts as introduced by Kontrateva et al., Berline and Lasker, Aschenbrenner and Pong, and others.
For lecture slides, please click here.
Friday, March 1, 2013 at 10:15 a.m. Room 5382
Igor Krichever, Columbia University
Analytic Theory of Difference Equations with Rational CoefficientsFor a copy of the lecture slides, please click here.
For a paper by the author related to this topic, please visit author's website or click here.
Friday, March 8, 2013 at 10:15 a.m. Room 5382
Omar Leon Sanchez, University of Waterloo
Differential Dgroups and Galois TheoryWe present differential Dgroups as an extension (to infinitedimension) of algebraic Dgroups. Then we develop the Galois theory of logarithmic equations on differential Dgroups. This theory generalizes both the parameterized PicardVessiot theory and the differential Galois theory of algebraic Dgroups.
This talk will continue at 2:00 pm in the afternoon seminar.
Friday, March 15, 2013 at 2:00 p.m. Room 5382
Please note that this talk starts at 2:00 pm.
Richard Gustavson, Graduate Center (CUNY)
Some Open Problems in Differential Galois TheoryIn this talk we look at some open problems in differential Galois theory and their partial solutions. The two main problems we examine are the direct problem and the inverse problem. The direct problem asks whether the differential Galois group for a given system of differential equations exists, and if it does, to construct it. The inverse problem asks if a given group is the Galois group of some system of differential equations. If time permits, we will examine some open problems unrelated to the direct and inverse problems, including questions concerning monodromy groups of differential equations and factorization in the ring of linear differential operators.
Friday, March 22, 2013 at 10:15 a.m. Room 5382
Raymond Hoobler, Graduate Center (CUNY)
Deligne Revisited I: The RiemannHilbert CorrespondenceLet (E, ∇ ) be a bundle with an integrable connection on a smooth variety U over ℂ. Deligne showed that the analytic de Rham cohomology, H^{i}_{DR}(U_{an} (E, ∇ )), agreed with the algebraic de Rham cohomology, H^{i}_{DR}(U_{alg}, (E, ∇ )) if the connection was regular using Hironaka's resolution of singularities. André and Baldassari have more recently given an entirely algebraic proof of this result by using Artin neighborhoods to reduce it to a one dimensional case. In Part I, I will define Artin neighborhoods and outline the strategy for proving the result which rests on establishing several properties of (E, ∇ ). Part II will be devoted to outlining the proof of one of these properties to be selected at the end of Part I.
Friday, March 29, 2013 Spring Break. No seminar.
Friday, April 5, 2013 at 10:15 a.m. Room 5382
Raymond Hoobler, Graduate Center (CUNY)
Deligne Revisited II: GaussManin Connections and Regular Singular PointsLet (E, ∇ ) be a bundle with an integrable connection on a smooth, not necessarily complete variety X over ℂ. The notion of regular singular points for ∇ is a kind of finiteness condition that readily provides local solutions for the system of differential equation defined by ∇. We will define this condition as well as show, for a given a smooth map between smooth varieties f : X → Y, that R^{i}f_{*}((E, ∇ )) has a natural integrable connection
ℵ : R^{i}f_{*}((E, ∇ )) → R^{i}f_{*}((E, ∇ )) ⊗ Ω^{1}_{X/Y},
which is known as the GaussManin connection. We will sketch a proof that if (E, ∇ ) has only regular singular points, then so does (R^{i}f_{*}((E, ∇ )), ℵ ).
Friday, April 12, 2013 at 10:15 a.m. Room 5382
David Marker, University of Illinois at Chicago
Canonical Definitions in Differentially Closed FieldsCanonical definitions play an important role in modern model theory. In most structures one must add "imaginary elements" to give canonical definitions. Poizat showed that for differentially closed fields this is unnecessary. This is related to the the existence of definable quotients. I will survey the basic definitions, applications, and Poizat's proof.
Friday, April 19, 2013 at 10:15 a.m. Room 5382
Alexander Levin, Catholic University of America
Dimension Quasipolynomials in Differential and Difference AlgebraIn this talk we consider Hilberttype functions associated with differential and difference field extensions and systems of algebraic differential and difference equations in the case when the basic derivations or translations are assigned some rational weights. We will show that such functions are quasipolynomials, which can be obtained as linear combinations of Ehrhart quasipolynomials. In particular, we will obtain generalizations of the theorems on differential and difference dimension polynomials.
Friday, April 26, 2013 at 10:15 a.m. Room 5382
Philipp Rothmaler, Bronx Community College (CUNY)
What is a Type?I will first remind the audience of the concept of formula in its various disguises: syntactic object, definable set, sentence (i.e., formula with no free variable) in an expansion by constants, subfunctor of the forgetful functor and, element of the socalled LindenbaumTarski algebra. This latter structure is a Boolean algebra whose Stone space (under Stone duality) is the space of (complete) types. Its importance, in turn, lies in the fact that it is a compact space, which is the content of the celebrated Compactness Theorem and which I will explain if time permits.
My foremost goal though is to define and describe types in much plainer terms, namely as intersections of definable sets and, most importantly, orbits under the action of certain automorphism groups. (The disguises corresponding to the above would be: ultrafilter of formulas, typedefinable set (=intersection of definable sets), complete theory in an expansion by constants, subfunctor of the forgetful functor (namely, intersections of the aforementioned subfunctors), and element of the Stone space of the LT algebra.)
I will indicate, in some simple examples, what forking extensions of types are (i.e., how the various Stone spaces associated with models of a complete theory interact). This may lead us to Lascar rank and a topological definition of stability due to Herzog and myself.
Disclaimer: there will be more pictures on the board than rigorous technical statements and—I apologize—differential algebra.
Please note that Philipp Rothmaler will speak also on MittagLeffler objects in definable categories of modules at the CUNY Logic Workshop at 2:00 p.m. (Room 6417, Graduate Center). For details, please visit Logic Workshop.
Friday, May 3, 2013 at 10:15 a.m. Room 5382
James Freitag, University of California at Berkeley
An Application of Differential CompletenessWe will discuss generalizations of Kolchin's work on complete differential algebraic varieties. In particular, we will examine notions of generalized Wronskians and linear dependence over general differential algebraic varieties.
Friday, May 3, 2013 at 2:00–3:30 p.m. Room 5382
Li Guo, Rutgers University at NewarkCurrent Schedule
On IntegroDifferential AlgebrasIntegrodifferential algebras have been introduced recently in the study of boundary problems of differential equations. We generalize these to integrodifferential algebras with a weight, in analogy to differential and RotaBaxter algebras. We construct free commutative integrodifferential algebras with weight generated by a differential algebra. This gives in particular an explicit construction of the free integrodifferential algebra on one generator. Properties of the free objects are studied.
If time permits, we will discuss another construction of free integrodifferential algebras. This alternative uses the method of GroebnerShirshov bases and is given in a recent joint work with X. Gao and S. Zheng.
The present work is a joint work with G. Regensburger and M. Rosenkranz.
For a copy of the slides presented, please click here.
Friday, May 10, 2013 at 10:15 a.m. Room 5382
Raymond Hoobler, Graduate Center (CUNY)
Deligne Revisited III: GaussManin Connections and Regular Singular PointsLet (E, ∇ ) be a bundle with an integrable connection on a smooth, not necessarily complete variety X over ℂ. The notion of regular singular points for ∇ is a kind of finiteness condition that readily provides local solutions for the system of differential equation defined by ∇. In previous lectures, I have defined this condition. In this talk, I will show, for a given a smooth map between smooth varieties f : X → Y, that R^{i}f_{*}((E, ∇ )) has a natural integrable connection
ℵ : R^{i}f_{*}((E, ∇ )) → R^{i}f_{*}((E, ∇ )) ⊗ Ω^{1}_{X/Y},
which is known as the GaussManin connection. I will sketch a proof that if (E, ∇ ) has only regular singular points, then so does (R^{i}f_{*}((E, ∇ )), ℵ ).
Friday, May 10, 2013 at 2:003:30 p.m. Room 5382
Carlos Arreche, Graduate Center, CUNY
PPV Groups and Differential TranscendenceWe will apply parameterized PicardVessiot theory to give simple necessary and sufficient criteria for the ∂/∂ttranscendence of the solutions to a parameterized secondorder linear differential equation of the form ∂^{2}Y/∂ x^{2}= p ∂Y/∂x, where p∈C(t,x) and C is a field of characteristic zero. These criteria imply, in particular, the differential transcendence of the incomplete Gamma function Γ(t,x) over ℂ(t,x), generalizing a result of Johnson, Reinhart, and Rubel (1995).
Have a good summer vacation, and see you in September or at one of the summer conferences.
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