Kolchin Seminar in Differential Algebra 

The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
Academic Year
2013–2014 Last updated on January 31, 2020. 2005–2006 2006–2007 2007–2008 2008–2009 2009–2010 2010–2011 2011–2012 2012–2013 2014–2015 2015–2016 2016–2017 2017–2018 2018–2019 2019–Fall 
SAD NEWS: With our deepest sympathies and condolences for the Kolchin family, we passed on the sad news that Kate Kolchin, widow of Ellis Robert Kolchin, died on June 25, 2013, after suffering an injury from a fall. We all miss Kate, who had been a benefactor and a strong supporter of KSDA, and had welcomed many of us and our speaker guests to her home. While we no longer can enjoy her meticulously prepared feasts, we shall always remember her hospitality, generosity, loving friendship and encouragement.
An obituary appeared in the New York Times on June 30, 2013. There is an online Guest Book where you may leave a message until July 30, 2013.
Thanks to all who have left a message on the Guest Book. Peter Kolchin, son of Ellis and Kate, hosted an informal memorial for Kate on Saturday, August 17 at her Riverside Drive apartment, to celebrate her life and her 93rd birthday (August 16). Phyllis Cassidy and William Sit spoke on behalf of KSDA, and Alexander Levin and Alexey Ovchinnikov attended the memorial.
Special Announcement: Logic I by Prof. Philipp Rothmaler of Bronx Community College and the Graduate Center. This course meets Tuesdays and Thursdays, 4:155:45 pm in Room 6496 of the Graduate Center. The course is an introduction to logic and model theory, both topics are basic to following the more advanced topics in differential algebra from a model theory view point. For details, click here.
Friday, August 30, 2013 at 10:15 a.m. Room 5382
Richard Churchill, Hunter College and The Graduate Center
Transcendentals, The Goldbach Conjecture, and the Twin Prime ConjectureThe talk introduces Differential Algebra by examining elementary calculus from an algebraic pointofview and makes a few connections, both amusing and surprising, with elementary number theory. For example, we translate Euclid's proof of the infinitude of primes into a proof involving the differentiation of integers.
For lecture notes, please click here (updated 09/01/2013).
Friday, September 6 and 13, 2013 No Classes. No seminar.
Friday, September 20, 2013 at 10:15 a.m. Room 5382
Richard Gustavson, Graduate Center (CUNY)
A New Bound for the Effective Differential NullstellensatzThe differential Nullstellensatz asks when a differential equation f is a differential algebraic consequence of a system of differential equations f_{1}, ...,f_{k}. The effective differential Nullstellensatz tasks us with finding a bound on the number of derivations of the f_{i} we need when f is such a differential algebraic consequence. A first bound for the effective differential Nullstellensatz was given in terms of the Ackermann function in 2009. In this talk, I will discuss the history of this problem and present a new double exponential bound for the effective differential Nullstellensatz for ordinary differential equations over ℂ, given by D'Alfonso, Jeronimo, and Solernó.
Friday, September 27, 2013 at 10:15 a.m. Room 5382
Raymond Hoobler, CCNY and Graduate Center (CUNY)
Rethinking PicardVessiot TheoryLet A be a Δalgebra, and let (E,∇) be a finitely generated A module with a connection. We first show that the (constantbasis) functor CB from ΔAalgebras B to sets given by
CB: B ↦ { B ∣ B is an ordered basis of E⊗_{A}B consisting of constant vectors}
is representable by a ΔAalgebra; that is, there is a ΔAalgebra S_{E}^{Δ} such thatCB(B) ↔ Hom_{ΔAalg}(S_{E}^{Δ}, B)
for all ΔAalgebras B, making CB a functor into abelian groups. If A^{Δ} = C is an algebraically closed field, we then easily construct the associated PicardVessiot extension and show it is a principal homogeneous space for an affine algebraic group. We also attempt to understand the situation when C is a commutative ring with some modest results.
For lecture notes, please click here.
Friday, October 4, 2013 at 10:15 a.m. Room 5382
Alice Medvedev, The City College of The City University of New York
Difference algebra in Hrushovski's Frobenius paper, IIn "Elementary Theory of the Frobenius Automorphism," Hrushovski develops some new (as far as I know) notions in difference algebra, including new difference analogs of prime and radical ideals, and several new notions of dimension. I will describe and explain these notions.
For lecture notes, please click here
Friday, October 11, 2013 at 10:15 a.m. Room 5382
Alexey Ovchinnikov, Queens College and Graduate Center (CUNY)
Computing Parameterized Differential Galois Groups: Unipotent and Reductive CasesWe will discuss the recent progress in computing the Galois group of a system of linear differential equations with parameters.
Friday, October 18, 2013 at 10:15 a.m. Room 5382
Thomas Scanlon, University of California at Berkeley
DFields as a Common Formalism for Difference and Differential AlgebraIn a series of papers with Rahim Moosa, I have developed a theory of Drings unifying and generalizing difference and differential algebra. Here we are given a ring functor D whose underlying additive group scheme is isomorphic to some power of the additive group. A Dring is a ring R given together with a homomorphism f : R → D(R). A first motivating example is when D(R) = R[ε]/(ε^{2}), so that the data of Dring is that of an endomorphism σ:R → R and a σderivation ∂:R → R (that is, ∂(rs) = ∂(r)σ(s)+σ(r)∂(s)). Another example is when D(R) = R, where a Dring structure is given by an endomorphism of R.
We develop a theory of prolongation spaces, jet spaces, and of Dalgebraic geometry. With our most recent paper, we draw out the model theoretic consequences of this work showing that in characteristic zero, the theory of Dfields has a model companion, which we call the theory of Dclosed fields, and that many of the refined model theoretic theorems (eg the Zilber trichotomy) hold at this level of generality. As a complement, we show that no such model companion exists in characteristic p under a mild hypothesis on D.
For lecture notes, please click here.
Friday, October 25, 2013 at 10:15 a.m. Room 5382
Carlos Arreche, Graduate Center (CUNY)
Computing the Unipotent Radical of a 2x2 Parameterized Differential Galois GroupSuppose that G is the parameterized PicardVessiot (PPV) group of a secondorder parameterized linear differential equation over F(x), where F is a differentially closed field of parameters. We will discuss a new method to compute the unipotent radical R_{u}(G) of G. This method is based on the procedure presented by Minchenko, Ovchinnikov, and Singer in their paper titled Unipotent differential algebraic groups as parameterized differential Galois groups, where they completely solve the problem of computing R_{u}(G) under the assumption that the maximal reductive quotient G/R_{u}(G) is differentially constant. We will show how to circumvent this assumption in order to apply their procedure to compute R_{u}(G) in all cases.
Friday, November 1, 2013 at 10:15 a.m. Room 5382
Alice Medvedev, The City College of The City University of New York
Difference algebra in Hrushovski's Frobenius paper, IIIn Elementary Theory of the Frobenius Automorphism, Hrushovski develops some new (as far as I know) notions in difference algebra, including new difference analogs of prime and radical ideals, and several new notions of dimension. I will describe and explain these notions.
For lecture notes, please click here. Please also click here for links provided by a CIRM workshop on this topic.
Friday, November 1, 2013 at 2:00 p.m.Room 5382
Alexander A. Mikhalev, Moscow State University
PBW Pairs of Varieties of Linear AlgebrasThe notion of a PBWpair of varieties of linear algebras over a field is under consideration. If (V,W) is a PBWpair of varieties and V is Schreier, then so is W. Similar results are also true for the Freiheitssatz and the Word problem. If V(X) and W(X) are free algebras of the varieties V and W, respectively, with X as the set of free generators, then V(X) is the universal enveloping algebra of W(X). In the case where V(X) is a free nonassociative algebra, this result provides a possibility to construct algorithms for symbolic computation in the algebra W(X) , such as for the recognition of automorphisms and primitive elements, and the construction of normal forms for elements and standard bases of ideals.
The talk is based on the article by A. A. Mikhalev and I. P. Shestakov (to appear in Communications in Algebra, 2014).
Friday, November 8, 2013 at 10:15 a.m. Room 5382
Open Discussion
Difference Algebra in Hrushovski's Frobenius Paper, III
Friday, November 15, 2013 at 10:15 a.m. Room 5382
David Marker, University of Illinois at Chicago
LogarithmicExponential SeriesI will survey some old work of van den Dries, Macintyre and myself. We construct an algebraic nonstandard model of the theory of the real exponential field. There is a natural derivation on the LEseries which is compatible with the exponential and the archemedian valuation.
Friday, November 22, 2013 at 10:15 a.m. Room 5382
James Freitag, University of California at Berkeley
Almost Simple Differential and Superstable GroupsWe discuss a class of differential algebraic groups isolated by Cassidy and Singer, the class of almost simple differential algebraic groups. Roughly, the defining condition of this class is that every normal differential algebraic subgroup is small (but not assumed to be finite) compared to the group itself. The superstable analogue of this condition was investigated by Berline and Lascar. Following a description of some definability results for this class of groups, we will discuss a recent proof by Minchenko that noncommutative almost simple differential algebraic groups are quasi simple. We will then generalize Minchenko's technique to analyze perfect central extensions of algebraic groups, proving that if they are almost simple then they are quasi simple. We will relate this work to a theorem of Cherlin and Altinel from the finite Morley rank setting, proving that the generalization of their theorem to the general omega stable setting cannot hold without additional hypotheses.
Wednesday (Friday Schedule), November 27, 2013 : No Meeting. Happy Thanksgiving.
Friday, December 6, 2013 at 10:15 a.m. Room 5382
Omar Sanchez, McMaster University
Extending Differential Free OperatorsIn recent work, Moosa and Scanlon developed a theory of free operators on fields of characteristic zero that unifies (and generalizes) the differential and difference settings. One can develop an analogous theory where the free operators interact nicely with a fixed set of commuting derivations. In this talk we will introduce this theory of differential free operators and discuss some extensionality results. This is joint work with Rahim Moosa.
Friday, December 13, 2013 at 2:00 p.m.Room 5382
Uma Iyer, Bronx Community College, CUNY
Generic Base Algebras and Universal Comodule Algebras for Some Finite Dimensional Hopf AlgebrasGiven a Hopf algebra, E. Aljadeff and C. Kassel constructed the generic base algebra and the universal comodule algebra associated to it. We describe these algebras for the Taft algebras, the Hopf algebras E(n) and certain monomial Hopf algebras. This is joint work with C. Kassel.
For lecture slides, please click here.
Have a Merry Christmas and Happy New Year, and see you in January, 2014 .
Friday, January 31, 2014 at 10:15 a.m. Room 5382
Thomas Dreyfus, Institute of Mathematics of Jussieu
A Density Theorem for Parameterized Differential Galois theoryTo a linear differential system with coefficients that are germs of meromorphic functions, we can associate an algebraic group (the differential Galois group), which measures the algebraic relations among the solutions. The Density Theorem of Ramis gives a list of topological generators of this group with respect to the Zariski topology. More recently a Galois theory for parameterized linear differential system has been developed by Cassidy and Singer. In this theory, the Galois group, which is a differential algebraic group (with respect to the parametric derivations), measures the algebraic and the (parametric) differential algebraic relations among the solutions. We will present an analogue of the Density Theorem of Ramis for this theory.
For a copy of the lecture slides, please click here. The full preprint of the paper is available at http://www.math.jussieu.fr/~tdreyfus/ramisparam.pdf,
Friday, February 7, 2014 at 10:15 a.m. Room 5382
Alexey Ovchinnikov, Queens College and The Graduate Center, CUNY
Semigroup Actions on Tannakian CategoriesOstrowski's theorem implies that log(x), log(x +1),... are algebraically independent over ℂ(x). More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a nonzero solution y and particular transformations of y, such as derivatives of y with respect to parameters, shifts of the arguments, rescaling, etc. I will discuss a theory of Tannakian categories with semigroup actions, which could be used to attack such questions in full generality. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this talk, I will present a finite set of axioms that characterizes actions of semigroups that are finite free products of free finitely generated commutative semigroups on Tannakian categories. This is the class of semigroups that appear in many applications.
Friday, February 14, 2014 at 10:1511:15 a.m. Room 5382
Carlos Arreche, The Graduate Center, CUNY
A PicardVessiot Topology for Differential SchemesWe present a new Grothendieck topology for differential schemes, called the PicardVessiot topology, in which every O_{X}coherent module with a connection is locally trivial (i.e., generated by horizontal sections). The main examples of differential schemes are smooth algebraic varieties and prime spectra of differential rings. We will discuss analogies (and contrasts) with the étale topology, as well as potential applications of the PicardVessiot topology to some problems in algebraic geometry and differential algebra. We will motivate the abstract theory with a brief account of the role of étale cohomology in the proof of the Weil conjectures, and we will recall the relevant definitions from (differential) algebraic geometry and the theory of Grothendieck topologies.
Friday, February 21, 2014 at 10:15 a.m. Room 5382
Moshe Kamensky, The Hebrew University, Jerusalem
PicardVessiot StructuresA PicardVessiot (PV) extension associated to a linear differential equation is the analogue, in differential Galois theory, of the splitting field of a polynomial. A classical result asserts that a PV extension exists, and is unique, so long as the base field of constants is algebraically closed. I will explain a generalisation of this result, for other fields of constants. The proof uses a generalisation of PV extensions in a general first order setting, and a geometric description of the collection of such extensions.
Alert:Video recordings of Moshe Kamensky's talk in both the morning and afternoon sessions are available for viewing Morning and Afternoon.
Friday, February 28, 2014 at 10:1511:15 a.m. Room 5382
Raymond Hoobler, City College and Graduate Center, CUNY
Carlos Arreche, The Graduate Center, CUNY
A PicardVessiot Topology for Differential Schemes, Part IIThis is a continuation of the talk given on February 14. We present a new Grothendieck topology for differential schemes, called the PicardVessiot topology, in which every O_{X}coherent module with a connection is locally trivial (i.e., generated by horizontal sections). The main examples of differential schemes are smooth algebraic varieties and prime spectra of differential rings. We will discuss analogies (and contrasts) with the étale topology, as well as potential applications of the PicardVessiot topology to some problems in algebraic geometry and differential algebra. We will motivate the abstract theory with a brief account of the role of étale cohomology in the proof of the Weil conjectures, and we will recall the relevant definitions from (differential) algebraic geometry and the theory of Grothendieck topologies.
There will be no scheduled Kolchin Seminar Talk on March 7, 2014. You are invited to attend instead:
Friday, March 7, 2014 at 12:302:00 p.m. Room
6417
Russell Miller, The Graduate Center, CUNY
Turing Degree Spectra of Differentially Closed FieldsThis is a crosslisting from CUNY Model Theory Seminar. For abstract, click here.
Friday, March 14, 2014 at 10:15 a.m. Room 5382
Andrew Parker, New York City College of Technology, I
Grothendieck Topologies—Sieves, Sites and SheavesThis talk will serve as an introduction to the categorytheoretical foundations necessary for defining Grothendieck Topologies. Starting with the Yoneda embedding, our aim will be to replace the standard notion of "open cover of a topological space X " with that of "covering sieves for a category C " in such a way that the classical formalism of sheaves and sheaf cohomology remain meaningful in a more general context.
Friday, March 21, 2014 at 10:15 a.m.–11:45 a.m., and 2:00 p.m.–5:00 p.m. Room 5382
Alexandru Buium, University of New Mexico
Arithmetic Differential Equations on GL_{n}, Parts I and IIMotivated by the search of a concept of linearity in the theory of arithmetic differential equations, we introduce an arithmetic analogue of Lie algebras and an arithmetic analogue of the MaurerCartan connections. There is a family of such arithmetic connections for each of the classical involutions of GL_{n}. Finally we discuss the Galois groups attached to the resulting arithmetic differential equations. These Galois groups (generally) appear as subgroups of GL_{n} over the "algebraic closure of the field with one element".
Alert:Video recordings of Alexandru Buium's talk in both the morning and afternoon sessions are available for viewing Part Ivideo1 and PartIvideo2; PartIIvideo1 and PartIIvideo2.
Friday, March 28, 2014 at 10:15 a.m. Room 5382
Andrew Parker, New York City College of Technology
Grothendieck Topologies—Sieves, Sites and Sheaves, IIThis talk will serve as an introduction to the categorytheoretical foundations necessary for defining Grothendieck Topologies. Starting with the Yoneda embedding, our aim will be to replace the standard notion of "open cover of a topological space X " with that of "covering sieves for a category C " in such a way that the classical formalism of sheaves and sheaf cohomology remain meaningful in a more general context.
Friday, April 4, 2014
There will be no scheduled meeting. Those interested in an informal discussion on Grothendieck Topology are welcome to meet at 10:15 a.m. in Room 5382.The following is a crosslisting from the Commutative Algebra & Algebraic Geometry Seminar.
Friday, April 4, 2014 at 4:00 p.m. Room 6417
Andrew Parker, New York City College of Technology, CUNY
A^{1}Homotopy Theory.In this talk, we will lay the mathematical foundations for replacing the standard unit interval with the affine line, for the purposes of applying topological arguments in an algebrogeometrical context.
Friday, April 11, 2014 at 10:15 a.m.–11:45 a.m. Room 5382
Omar Sanchez, McMaster University
On Differential Algebraic, But Not Constrained, FamiliesWe will give a negative answer to the question: Is every finitely generated differential algebraic extension, with no new constants, of ℂ, a constrained extension? We then use this to answer a question on Poisson prime ideals of Poisson algebras. This is joint work with J. Bell, S. Launois, and R. Moosa.
A video recording of this talk is available for viewing here.
Friday, April 11, 2014 at 12:30 p.m. –2:00 p.m. Room 5382
Andrey Minchenko, Weizmann Institute of Science
Central Extensions of Simple Linear Differential Algebraic GroupsWe will classify central extensions of simple linear differential algebraic groups. In particular, we will see that every simple linear differential algebraic group over a nonordinary differential field has a perfect linear differential central extension with an infinite center. An important step in our classification is to show that H^{2}(SL_{2}, G_{a}) is a free Hom(G_{a}, G_{a})=k[Δ]module generated by m(m1)/2 independent elements, where k stands for the ground Δfield and m= Δ.
A video recording of this talk is available for viewing here.
Friday, April 11, 2014 at 3:00 p.m. –6:00 p.m.
Saturday, April 12, 2014 at 10:00–11:50 a.m. and 3:30–5:50 p.m.
Special Session on Differential Algebra and Galois Theory, at the AMSMeeting at Lubbock, TX
A video recording of the Special Session I (Friday afternoon) is available here .
A video recording of the Special Session II (Saturday morning) is available here .
A video recording of the Special Session III (Saturday afternoon) is available here .
Slides of the talks will be available here as they are made available from the authors.
Friday, April 18, 2014, No Seminar due to Spring Break.
Friday, April 25, 2014 at 10:15 a.m. Room 5382
Li Guo, Rutgers University at Newark
GröbnerShirshov Bases of Free Integral Differential AlgebrasIntegrodifferential algebras have been introduced recently in the study of boundary problems of differential equations. We will discuss a recent construction of free commutative integrodifferential algebras using the method of GröbnerShirshov basis based on the construction of free commutative RotaBaxter algebras via mixable shuffles.
For a copy of the lecture slides, please click here.
Friday, May 2, 2014 at 10:15 a.m. Room 5382
Benjamin Steinberg, The City College, CUNY
Affine Monoids, Toric Varieties and Rational Polyhedral ConesAn affine monoid is a finitely generated submonoid of ℤ^{n}. The category of affine monoids is dually equivalent to the category of affine toric varieties and also to the category of Zariski closed, irreducible (multiplicative) submonoids of ℂ^{n} for some n. The classification of normal affine monoids, or equivalently normal affine toric varieties, is equivalent to the classification of rational polyhedral cones. Thus the classification of affine monoids is much more complicated a problem than the classification of finitely generated abelian groups and is closely connected to algebraic geometry and discrete geometry. Nonetheless, the isomorphism problem for finitely generated commutative monoids is decidable. So things are not completely hopeless.
Friday, May 9, 2014 at 10:15 a.m. Room 5382
Taylor Dupuy, University of California at Los Angeles and MSRI
Jet Spaces and Diophantine ProblemsWe will review the construction of various jet spaces of a variety. These are generalizations of the tangent space in usual algebraic geometry. I will also talk about some applications of jet spaces techniques to problems in diophantine geometry.
For a video recording of the talk, please click here.
Friday, May 9, 2014 at 2:00 p.m. Room 5382
Informal Afternoon Session with Bernard Malgrange, Université Joseph Fourier–Grenoble.
For the video recording of the session, please click here.
Friday, May 16, 2014 at 10:15 a.m. Room 5382
Victor Kac, Massachusetts Institute of Technology
Algebraic Theory of Integrable Systems, II will explain an approach to integrable systems of PDE via Poisson vertex algebras and Lie conformal algebras. As a digression, I will explain how the Lie conformal algebras are related to differential Lie algebras. No knowledge of integrable systems, Poisson vertex algebras or Lie conformal algebras will be assumed.
For the lecture slides, please click here.
To view the video recordings of these lectures, please click morning session and afternoon session.
Friday, May 16, 2014 at 2:00 p.m. Room 5382
Victor Kac, Massachusetts Institute of Technology
Algebraic Theory of Integrable Systems, IIThis is a continuation of the morning talk.
Monday, July 7, 2014 at 9:00–10:10 a.m. Room 5382
Markus Rosenkranz, University of Kent at Canterbury, UK
IntegroDifferential Polynomials and Free IntegroDifferential AlgebrasAdjunction of a transcendental element to an ordinary integrodifferential algebra yields an analog of differential polynomials, consisting of nested nonlinear integral operators. The resulting ring of integrodifferential polynomials carries the structure of an integrodifferential algebra, and it contains an isomorphic copy of the corresponding differential polynomial ring. We present effective normal forms for integrodifferential polynomials and exhibit their relation to the free object in the integrodifferential category. A short outlook at the case integrodifferential 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 Politécnica de Madrid
Sparse Resultant Formulas for Differential PolynomialsDifferential 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. CarràFerro 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 L1 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 ProblemsIn 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 RotaBaxter 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 infinitedimensional 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 BoundsThe following differential version of Hilbert's Nullstellensatz was introduced by Ritt, and later extended to arbitrary differential fields: if f_{1},..., f_{s}, 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 f_{i}'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 f_{1},..., f_{s} 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 f_{1},..., f_{s} and their derivatives representing this power of g.
(Joint work with Lisi D'Alfonso and Pablo Solernó).
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 ComplexityIn 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 video1 and video2.
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 ProblemsWhile 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 integrodifferential algebras. We will also discuss the relation between localized boundary problems and the integrodifferential analog of the (first) Weyl algebra, which exhibits a certain discrepancy between twosided 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 OperatorsPart 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 highorder 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 MoorePenrose 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 Cochairs) with an Advisory Committee composed of Eugenio RoanesLozano, 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, Université Grenoble 1
Regular Singular qDifference Equations and Birkhoff MatricesWe will first recall the classification of regular singular qdifference 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.
Moserbased algorithms over Univariate and Bivariate (Differential) FieldsMoserbased 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, BarkatouBroughtonPfluegel'2007, BarkatouPfluegel'2009) and the perturbed algebraic eigenvalueeigenvector problem (JeannerodPfluegel'1999). This gave rise to the package ISOLDE (BarkatouPfluegel) 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 Moserbased algorithm given by Barkatou'1995 to simplify the symbolic resolutions of singularlyperturbed linear differential systems and completely integrable Pfaffian systems with normal crossings in two variables. This unified treatment of these two wellknown 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.
For a video viewing of the lecture, please click video1 and video2.
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