Kolchin Seminar in Differential Algebra 

The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
Academic
year 2009–2010 Last updated on January 18, 2018. 
In Fall, 2009, we welcomed Professor Alexey Ovchinnikov, who joined us as an Assistant Professor at Queens College, CUNY and as a coorganizer for KSDA. We also welcomed Dr. Varadharaj Ravi Srinivasan, who joined Rutgers University at Newark as a post doctoral researcher, and Professor Lourdes Juan, who was on sabbatical leave at Penn State University from Texas Tech University. The Fall semester was
mainly devoted to studying interactions between differential algebra and related fields, such as algebraic geometry, representation theory, computational complexity, differential geometry, model theory and number theory.
In Spring, we welcomed Professor William Keigher, Rutgers University at Newark, who was on Sabbatical leave. The Spring semester continued the Fall program, and in particular, we organized a miniworkshop on New Trends in Differential Algebraic Geometry, April 1719, 2010.
Friday, August 28, 2009 at 10:30 a.m.
Richard Churchill (Graduate Center and Hunter College of CUNY)
An Algebraic Approach to Linear Ordinary Differential EquationsThis talk will be an informal introduction to an algebraic approach to linear differential equations, including the Galois theory of linear differential operators. Familiarity with differential equations, beyond what would ordinarily be encountered in an undergraduate course, is not assumed. Analogies with standard Galois theory will be stressed. In particular, the relationship between PicardVessiot extensions and differential Galois groups of linear differential equations will be introduced as the counterpart of the relationship between splitting fields and Galois groups of polynomials. Computer applications will then be discussed.
Friday, September 4, 2009 at 10:30 a.m.
Alexey Ovchinnikov, Queens College, CUNY
Introduction to Tannakian CategoriesThis is the first of three related talks and will be an elementary and detailed introduction into the subject. Examples and selected proofs will be given during the informal afternoon session. The audience should wait until the second talk to see real and very interesting applications.
Friday, September 11, 2009 at 10:30 a.m.
Alexey Ovchinnikov, Queens College, CUNY
Tannakian Categories and Algebraic GroupsThis is the second of three related talks and we will show how Tannakian categories give an intrinsic description of linear algebraic groups. In particular, we will see how such a group can be recovered from its representations. This idea leads to algorithms for computing differential Galois groups of systems of linear ODEs. Examples and selected proofs will be given during the informal afternoon session.
Friday, September 25, 2009 at 10:30 a.m.
Alexey Ovchinnikov, Queens College, CUNY
Differential Tannakian Categories and Differential Algebraic GroupsThis is the third of three related talks. Differential algebraic groups have an extra structure (differential structure) and appear as Galois groups of systems of linear ODEs with parameters. In this talk we will discuss how this differential structure can be expressed within the Tannakian framework. Examples and selected proofs will be given during the informal afternoon session. This topic will be continued later during the Fall semester, with discussions on how differential Tannakian categories are related to Atiyah classes, differential schemes, and bundles with connections (from algebraic geometry).
Friday, October 2, 2009 at 10:30 a.m.
Varadharaj Ravi Srinivasan, Rutgers University at Newark
Differential Subfields of Liouvillian ExtensionsLet F be an ordinary differential field with an algebraically closed field of constants and let E be a differential field extension of F with no new constants. We say that E is an Iterated Antiderivative Extension of F , abbreviated IAE, if E contains elements x_{1}, ..., x_{n} such that E = F (x_{1}, ..., x_{n} ) and for each i = 1, 2, ..., n , if we set F_{i} :=F_{i1} (x_{i} ) and F_{0} :=F , then the derivative x'_{i} ∈F_{i1} . In this talk, we will prove that if E is an IAE of F and if K is a differential subfield of E that contains F , then K is an IAE of F as well. We will also look at several examples of such extensions and study their differential subfields.
Friday, October 9, 2009 at 10:30 a.m.
Rahim Moosa, University of Waterloo
Differential Arcs and the InfiniteDimensional Zilber DichotomyFrom the modeltheoretic point of view, finitedimensional differential algebraic varieties are analyzable in terms of "minimal" ones. The Zilber dichotomy, proved by Hrushovski and Sokolovic, says that a minimal differential variety is either algebraic (that is, essentially the constant points of an algebraic variety) or geometrically very simple ("locally modular"). This result has far reaching consequences and is at the heart of Hrushovski's proof of the function field MordellLang in characteristic zero. Some years ago, Pillay and Ziegler found a more direct proof of this dichotomy, using "differential jet spaces", a higher order version of Kolchin's differential tangent spaces.
In the context of several commuting derivations there is a plenitude of infinitedimensional differential varieties whose structure is known to be very rich. There is an analogue of minimality here, called "regularity": infinitedimensional differential varieties are analyzable in terms of regular ones. It is still unknown whether or not the analogue of the Zilber dichotomy for regular infinitedimensional varieties is true. In this talk I will discuss the role that "differential arc spaces" play in reducing this problem to a question about differential subgroups of the additive group. I hope to explain all modeltheoretic prerequisites.
Friday, October 16, 2009 at 10:30 a.m.
Lourdes Juan, Texas Tech University
Differential Galois Extensions with Specified Galois Groups, IIn this joint work with Ted Chinburg and Andy Magid, we address the problem of recognizing a differential Galois extension E of a differential field F from weaker information than the structure of E as a differential field. Our work includes a differential counterpart of the normal basis theorem in polynomial Galois theory and the construction of an invariant that depends on the differential Galois group of the extension.
Saturday, October 17, 2009 at 10:30 a.m.
NB: This is a special Saturday meeting of the Kolchin Seminar. Please note that the meeting will be held at the North Academic Center, Room 1/511E of The City College, 160 Convent Avenue, New York, NY 10031. Please click here for directions.
Moshe Kamensky, University of Notre Dame
Differential Tensor CategoriesI will suggest an axiomatization of the categorical structure of the category Rep_{G} of representations of a linear differential algebraic group G. This is analogous to the description of Rep_{G} as a rigid abelian tensor category for a linear algebraic group G. I will present some constructions which suggest that one can do differential algebraic geometry within such a category.
In the second part of the talk, I will explain how, given a (suitably defined) fibre functor on such a category, one may reconstruct a differential algebraic group from it, similarly to the classical Tannakian formalism for algebraic groups. This result was first obtained by Alexey Ovchinnikov using algebraic methods. I will present a proof using the model theoretic notion of the binding group.
No model theory will be assumed.
Reference: Section 5 of http://arxiv.org/abs/0908.0604
The seminar will be followed by an informal discussion session after lunch, in the same room.
Friday, October 23, 2009 at 10:30 a.m.
Lourdes Juan, Texas Tech University
Differential Galois Extensions with Specified Galois Groups, IIThis is a continuation of her talk on October 16.
In this joint work with Ted Chinburg and Andy Magid, we address the problem of recognizing a differential Galois extension E of a differential field F from weaker information than the structure of E as a differential field. Our work includes a differential counterpart of the normal basis theorem in polynomial Galois theory and the construction of an invariant that depends on the differential Galois group of the extension.
Friday, October 30, 2009 at 10:30 a.m.
Benjamin Antieau, University of Illinois at Chicago
Galois Theory of Difference Equations with Difference ParametersIn this talk, I will explore an application of Dima Trushin's work on difference Nullstellensatz theorems to the creation of a Galois theory of difference equations with difference parameters. This complements the works of Cassidy, Singer, and Hardouin on the Galois theory of difference and differential equations with differential parameters.
However, serious ringtheoretical difficulties must be dealt with in the case where one has difference parameters. These are approached by building upon the initial idea of Trushin's difference closed rings (pseudofields).
Friday, November 6, 2009 at 10:30 a.m.
Leonard Scott, McConnell/Bernard Professor of Mathematics, The University of Virginia
Algebraic Group Representations and Related TopicsThis lecture will survey the theory of algebraic group representations in positive characteristic, with some attention to its historical development and its relationship to the theory of finite group representations. Other topics of a Lietheoretic nature will also be discussed in this context, including at least brief mention of characteristic 0 infinite dimensional Lie algebra representations in both the classical and affine cases, quantum groups, perverse sheaves, and rings of differential operators. Much of the focus will be on irreducible representations, but some attention will be given to other classes of indecomposable representations, and there will be some discussion of homological issues, as time permits.
This will be followed by an afternoon session of informal discussion.For lecture notes, please click here
Friday, November 13, 2009 at 10:30 a.m.
Lourdes Juan, Texas Tech University
Differential Central Simple Algebras and PicardVessiot RepresentationsI will start with a brief introduction to differential Galois theory. A differential field is a field K with a derivation, that is, an additive map D:K→K satisfying D (fg )=D (f ) g +fD (g ) for f,g in K. The field of constants C of K is the kernel of D. A differential central simple algebra (DCSA) over K is a pair (A,D) where A is a central simple algebra and D is a derivation of A extending the derivation D of its center K. Any DCSA, and in particular a matrix differential algebra over K, can be trivialized by a PicardVessiot (differential Galois) extension E of K. In the matrix algebra case, there is a correspondence between Kalgebras trivialized by E and representations of the differential Galois group of E over K in PGL_{n}(C ), which can be interpreted as cocycles equivalent up to coboundaries.
Reference: Differential central simple algebras and PicardVessiot representations, with Andy Magid, Proc. Amer. Math. Soc. 136(6) (2008), 19111918.
Friday, November 20, 2009 at 10:30 a.m.
Richard Cohn, Rutgers University at New Brunswick
The Low Power and Low Weight ConditionsLet R be a differential polynomial ring. Let A, B ∈R be such that A is irreducible, B ≠ 0 , and B is annulled by the principal component V of the variety of A . Is V a component of the variety of B , or is it merely properly contained in a component? The answer is provided by the famed low power condition which is necessary and sufficient for V to be a component. For the corresponding question concerning difference polynomials, the analogous low weight condition is necessary, but sufficient only with further restrictions.
After a brief review of the differential case, I will define the low weight conditions and give examples to illustrate the claims above, and sketch the proof of necessity.
Friday, November 27, 2009 at 10:30 a.m.
Thanksgiving Week, No Meeting
Friday, December 4, 2009 at 10:30 a.m.
Raymond Hoobler, Graduate Center and The City College, CUNY
FPQC Descent and Grothendieck Topologies in a Differential SettingAbstract: I will summarize flat descent for the category of differential ring and then introduce a new Grothendieck topology: the δflat topology. Basic theorems on descent, sheaves, and nonabelian H^{1} will be carefully stated and proved using Milne: Etale Cohomology, as a reference. This talk generalizes Galois descent and nonabelian Galois cohomology to the differential ring setting. I will attempt to properly and carefully motivate the concepts involved.
For lecture notes (fourth version), please click here
Friday, December 11, 2009 at 10:30 a.m.
Raymond Hoobler, Graduate Center and The City College, CUNY
Differential Azumaya AlgebrasAbstract: I will use the descent and Grothendieck topology machinery to classify differential line bundles and differential locally free sheaves over a differential ring R. Differential Azumaya algebras will then appear as a natural generalization of differential central simple algebras as defined by Lourdes Juan and Andy Magid. Finally some short exact sequences of sheaves in the δflat topology will be constructed using PicardVessiot theory which tie together results over R and the ring R^{δ} of constants of R.
For lecture notes (revised version), please click here.
Friday, December 11, 2009 at 2:00 p.m.
Please note this is an afternoon talk that will take place in Room 8404.
Alexander Levin, The Catholic University of America
Dimension Polynomials of Intermediate Fields and Krulltype Dimension of Finitely Generated Differential Field ExtensionsLet L be a finite generated differential field extension of a differential field K of zero characteristic. We introduce the concepts of transcendence type and dimension of the extension L/K by considering chains of its intermediate differentials fields. Using the technique of differential dimension polynomials we obtain relationships between the transcendence type and dimension of L/K and differential birational invariants of this extension carried by its dimension polynomial.
Saturday, December 12, 2009 at 10:30 a.m. to 12:30 p.m.; 2:00 p.m. to 5:00 p.m.
NB: This is a special Saturday meeting of the Kolchin Seminar. Please note that the meeting will be held at the North Academic Center, Room 1/511E of The City College, 160 Convent Avenue, New York, NY 10031. Please click here for directions.
Dijana Jakelic, University of North Carolina, Wilmington
Finitedimensional Representations of Quantum Affine Algebras at Roots of UnityQuantum groups emerged from quantum statistical mechanics in the mid 1980s. Since then they have turned out to be fundamental algebraic structure behind many branches of mathematics and mathematical physics. The category of finitedimensional representations of quantum affine algebras is one of the most studied topics within representation theory of quantum groups.
The morning session of the talk will be structured as an introduction to this subject. In particular, we will start with the definition of quantum groups and their basic representation theory for generic values of the quantization parameter. We will then survey a few results of Lusztig on finitedimensional representations of quantum groups at roots of unity. Several examples illustrating the differences between the two settings (generic and root of unity) will be provided along the way.
In the afternoon session, we will touch upon representation theory of affine algebras and discuss a joint work with A. Moura on finitedimensional representations of quantum affine algebras at roots of unity focusing on tensor products and characters of irreducible modules. We shall also describe the block decomposition of the underlying abelian category. We will then take a closer look at some examples in an informal way.
Friday, December 18, 2009 at 10:30 a.m.
Varadharaj Ravi Srinivasan, Rutgers University at Newark
PicardVessiot Extensions for Certain Unipotent Algebraic GroupsLet F be a characteristic zero differential field with an algebraically closed field of constants C. A PicardVessiot extension of F, whose differential Galois group is isomorphic to a unipotent algebraic group, is an iterated antiderivative extension of F. In this talk, I will sketch a construction of a (PicardVessiot) iterated antiderivative extension for the group U(n ;C ) of all upper triangular matrices in GL(n ;C ) with 1's in the main diagonal. If F = C(x), then for distinct complex numbers c_{0} := 1; c_{1}, c_{2}, ... ,c_{n}, our method produces a linear homogeneous differential equation with regular singularities at these c_{i} and the solutions of this differential equation can be expressed in terms of polylogarithms.
Friday, January 29, 2010 at 10:30 a.m.
William Keigher, Rutgers University at Newark
Automorphisms of Hurwitz seriesThe ring of Hurwitz series has been shown to be a useful tool for differential algebra. We will provide some results which show that it is even more useful than had been thought. For example, we will examine the relationship between derivations on the ring and automorphisms of the ring of Hurwitz series. We will also show that there are natural exp and log maps associated with the ring of Hurwitz series.
Friday, February 5, 2010 at 10:30 a.m.
Camilo Sanabria, Graduate Center, CUNY
Generalizing an Early Result of F. Klein on Linear Differential EquationsWe concentrate our attention on linear differential equations over compact Riemann surfaces and we address the problem of descent. An early result in this setting is Klein's Theorem, which states that any second order linear differential equation with algebraic solutions is the pullback of a standard hypergeometric equation. M. Berkenbosch, M. van Hoeij and J.A. Weil introduced the concept of standard equation, leading to Berkenbosch's generalization of Klein's theorem to the third order. In this talk I will explain how to broaden the scope of Klein's Theorem, via differential Galois theory, to equations with reductive Galois group of arbitrary order. All the concepts involved will be defined and I will motivate the result with some examples.
Friday, February 12, 2010 at 10:30 a.m.
Lincoln's Birthday, School closed and no seminar
Friday, February 19, 2010 at 10:30 a.m.
Uma Iyer, Bronx Community College of CUNY
Quantum Differential OperatorsThe universal enveloping algebra of a Lie algebra has a Hopf algebra structure. When the universal enveloping algebra acts on a ring via its Hopf structure, the action is given by differential operators. Lunts and Rosenberg, in 1997, provided a definition of quantum differential operators which allow for the action of quantum groups via its Hopf structure to be through these quantum differential operators. This talk will be an introduction to the quantum differential operators, and a construction of the same over the polynomial ring in one variable, and the quantum plane.
Friday, February 26, 2010 at 10:30 a.m.
Seminar cancelled due to snow storm.
Friday, March 5, 2010 at 10:00 a.m. Room 5382 Note Time Change.
Andrei Minchenko, Cornell University
Zariski Closures of Reductive Differential Algebraic GroupsIntroduced by Phyllis Cassidy, linear differential algebraic groups appear as Galois groups of systems of linear differential equations with parameters. Various properties of such groups determine the corresponding properties of the solutions. One can study linear differential algebraic groups using their Zariski closures, which are algebraic groups. We will discuss several important results obtained this way. In particular, we will establish an isomorphism of the Zariski closures that correspond to representations of minimal dimension of reductive differential algebraic groups.
Friday, March 12, 2010 at 10:00 a.m. Room 5382 Note room and time.
(Rescheduled after cancelation on February 26 due to snow.)
Kerry Ojakian, Queens College, CUNY
An Introduction to Computation over the Reals: Computable Analysis, Analog Computation, and Computing with Polynomial Differential EquationsWhile my talk is preparatory for this afternoon's CUNY Logic Workshop by Campagnolo [1], the talk should make sense on its own. I will only assume very mild familiarity with the theory of computation (basically, what a Turing machine is).
The goal of the theory of computation is to formalize exactly what computers can and cannot do. Traditionally, this has been attained using a model such as the Turing machine, where the inputs and outputs to the theoretical machine are finitely representable objects, such as natural numbers. Various models extend the theory of computation to allow for nonfinitely representable objects, such as real numbers. Computable Analysis is one standard way to extend computation to allow real number inputs. The idea, which originated with Turing himself, is to in fact use Turing machines as follows: To compute a real function f (x ), the machine takes a rational number approximation of x, and outputs a rational number approximation of f (x ). In this approach, time proceeds in discrete steps, as usual with Turing machine computation.
A different variety of approach to computation over the reals is to use tools from analysis to define computation, leading to a number of models referred to as "analog," since time is not seen as proceeding in discrete time steps, but rather it proceeds in a continuous fashion. One major analog model is Shannon's General Purpose Analog Computer (GPAC); it is a circuit model that has a number of simple kinds of gates, its most significant one being a gate that takes a function as input and outputs its integral. It was discovered (by a series of authors) that the GPAC is in fact equivalent to roughly the following model of computation: Take the solution of a system of differential equations defined using polynomials.
I will focus my talk on the GPAC and polynomial differential equations, discussing their properties and the connection between the models.
It turns out that the GPAC is weaker than Computable Analysis. However, in the work of Campagnolo and others (to be discussed during the afternoon CUNY Logic Workshop [1]), the GPAC is enhanced so that it captures exactly the functions of Computable Analysis.
[1] Manuel Lameiras Campagnolo, Instituto Superior de Agronomia (Lisbon) and University of Maryland (College Park). For the slides from his March 12 talk, please click here.
Friday, March 19, 2010 at 10:00 a.m. Room 5382 Note room and time.
Earl Taft, Rutgers University at New Brunswick
Recursive Sequences and Combinatorial IdentitiesLinearly recursive sequences have a bialgebra structure. Polynomially recursive (or Dfinite) sequences have a topological bialgebra structure. If such a sequence is of a combinatorial nature, a formula for its coproduct can often be interpreted as a combinatorial identity. We illustrate this for the sequences whose nth term is ((n / i)(n!)) for a fixed nonnegative i (where (n / i) is the binomial coefficient). The resulting combinatorial identity is of an iterated Vandermonde type.
Friday, March 26, 2010 at 10:00 a.m.
William Keigher, Rutgers University at Newwark
Module Structures on the Ring of Hurwitz SeriesLet k be a field of characteristic p > 0 . We consider monic linear homogeneous differential equations (LHDE) over the ring of Hurwitz series Hk of k . We obtain explicit recursive expressions for solutions of such equations and show that Hk admits a full a set of solutions as well. We then consider the notion of intertwining of Hurwitz series to reduce the study of solutions of an n^{th} order equation to a system of n first order equations in a particularly simple form. For every LDHE over Hk we will associate a module (over a suitable quasifield extension of k ), which is closed under the shift derivation of Hk and discuss the structure of the group of module automorphisms that commutes with the shift derivation.
Friday, April 2, 2010
March 29 through April 5, Spring Recess, no seminar
Friday, April 9, 2010 at 10:00 a.m. Room 5382 Note Time Change.
Camilo Sanabria, Graduate Center, CUNY
Jet Bundles and Higher Order DifferentialsWe will study the abstract construction of jet bundles and higher order differentials as introduced by Moosa and Scanlon in their joint paper, "Jet and Prolongation Spaces." In particular, there is a unified approach to differential and difference "Prolongation" Spaces over algebraic varieties using Weil's restriction of scalars. We will work out a couple of examples of interest during the exposition.
FridaySunday, April 1618, 2010
MiniWorkshop: Differential Algebraic GeometryFor abstracts of this MiniWorkshop, please click here.
Tuesday, April 20, 2010, 4:30 pm
(Columbia University: Kolchin Memorial Lecture)
Lehman Auditorium, 202 Altschul Hall, Barnard College (Enter at 117th Street & Broadway)
Tea served at 4:00 pm in 508 Mathematics Hall.
David Eisenbud (University of California, Berkeley),
Vector Bundles and Free Resolutions
Friday, April 23, 2010 at 10:00 a.m. Rm 5382
Miodrag Cristian Iovanov, University of Southern California and Rutgers University at New Brunswick
Why Coalgebras?  Frobenius Algebras, Hopf Algebras and Compact Groups via CoRepresentation Theory, with ApplicationsWe explore the question whether the proofs of some fundamental results in Hopf algebras can be done in a way that parallels what's happening in compact groups. For example, in compact groups, the bijectivity of the antipode (for the Hopf algebra of representative functions) is a given (by the map g → g ^{1}), and the uniqueness of the integrals is then not difficult to obtain. For general Hopf algebras, the main obstacle at the start was maybe because the bijectivity is not yet known: the uniqueness of the integrals is proved first, and then the bijectivity is obtained as a consequence. We wrote some short proofs that show the bijectivity of the antipode (for coFrobenius coalgebras) without using the uniqueness of the integrals, and then we derived the uniqueness in a mutatismutandis way from compact groups, or from multiplier Hopf algebras (as done by van Daele under the assumption that the antipode is bijective).
Our recently finished work reveals an even closer parallel between Hopf algebras and compact groups. This work, and in fact, some other work in which the above idea originated, is related to the treatment of a general theory of abstract integrals in algebra, which in turn is related to the notion of a generalized Frobenius algebra (which has some most important features of a Frobenius algebra but is not finite dimensional). We first noted that for compact groups and Hopf algebras, there is a connection between the existence of integrals and (co)representation theoretic properties of the Hopf algebra (i.e. homological: injectives=projectives). It turns out that only one of the structures of the Hopf algebra is needed to develop such a general theory, and the classical results of Hopf algebras follow as corollaries once they are generalized in the context of coalgebras. Moreover, characterizations of these generalized Frobenius algebras (and even of "generalized" quasiFrobenius algebras) can be obtained in a way that parallels what is known in the finite dimensional case, although the proofs are highly nontrivial. Connections to compact groups can be revealed as well. As application, we recently obtained the bijectivity of the antipode for (co)quasiHopf algebras, which could not be obtained by the classical means.
For lecture notes, please click here.
Friday, April 30, 2010 at 10:00 a.m. Rm 5382
Miodrag Cristian Iovanov, University of Southern California and Rutgers University at New Brunswick
Tensor Categories and a Certain Class of Frobenius AlgebrasWe give an application of the representation theory of tensor categories that answers the question of when weak Hopf algebras (and weak quasiHopf algebras) are Frobenius. Finite dimensional Hopf algebras, and later quasiHopf algebras were shown to be Frobenius algebras. These algebras can be seen as the support for the general situation of finite tensor categories. A certain integral theory can be developed for weak Hopf algebras, but it remained unclear and open whether or when they are Frobenius (a proof in the affirmative in a paper in J.Algebra around 2004 was found to be in error by G. Bohm). In a recent work, we completely solve this problem, and show precisely when weak Hopf algebras (or the more general weak quasiHopf algebras) are Frobenius.
Iovanov will give a talk at the CUNY Representation Theory Seminar at 1:00 pm, Rm 6493.
"Finite Tensor Categories and a certain class of Frobenius algebras."For lecture notes, please click here.
Friday, May 7, 2010 at 10:00 a.m.
Jim Freitag, University of Illinois at Chicago
Generic Points Are Not Necessarily GenericWe work over a differential field. For points on differentia varieties, there is a topological notion of genericity and a model theoretic notion of genericity. We will talk about cases where these notions agree (groups, low transcendence degree) and an example where the collections of points satisfying the two different notions are actually disjoint.
Friday, May 14, 2010 at 10:00 a.m.
Varadharaj Ravi Srinivasan, Rutgers University at Newark
An Algorithm for Solving the Inverse Problem in Differential Galois Theory for Certain Unipotent GroupsLet F be a differential field of characteristic zero with field of constants C. A PicardVessiot extension of F, whose differential Galois group is isomorphic to a unipotent algebraic group, is an iterated antiderivative extension of F . In this talk, I will sketch a construction of a (PicardVessiot) iterated antiderivative extension for U(n, C), which is the group of all upper triangular matrices in GL(n, C) with 1's in the main diagonal. The construction will not require that the field C be algebraically closed. The construction enables us to generate differential equations for these extensions. I will provide differential equations for U(n, C) for 2 < n < 11 when F = C(x), with the derivation d/dx (for larger n my computer crashes). I will also sketch how to generalize the construction to other unipotent algebraic groups.
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