Kolchin Seminar in Differential Algebra 
 The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
During the Fall, 2016 semester, Professors Alice Medvedev (City College) and Alexey Ovchinnikov (Queens College and the Graduate Center) organized the 7th International Workshop on Differential Algebra and Related Topics (DART VII) from September 30th (Friday) to October 4th (Tuesday), 2016 in New York City.
For a review of the lectures at DARTVII, please click here.
Last updated on October 10, 2016.
Friday, September 30Tuesday, October 4, 2016. DART VII:
7th International Workshop on Differential Algebra and Related Topics
To review the lectures on a particular day, please click:
 Friday, September 30, 2016,
 Saturday, October 1, 2016,
 Sunday, October 2, 2016,
 Monday, October 3, 2016,
 Tuesday, October 4, 2016.
For a list of posters with titles and abstracts, please click Posters.
Friday, September 30: Graduate Center, 4th Floor, Room 4102 (Science Center).
Julia Hartmann, University of Pennsylvania
Differential Embedding Problems over Complex Function FieldsDifferential Galois theory studies differential field extensions generated by solutions of differential equations by means of their symmetry groups. The question which groups can occur in this context is known as the inverse problem. Differential embedding problems can be considered as a refinement of this question. They are used to study how PicardVessiot extensions—and their differential Galois groups—fit together in towers. In this talk, we show that over a complex function field in one variable, every embedding problem has a proper solution. The key ingredient is a patching result for differential torsors.
This is joint work with A. Bachmayr, D. Harbater, and M. Wibmer.For a review of the lecture, please click video.
Gal Binyamini, Weizmann Institute of Science
Density of Rational Points on Transcendental VarietiesI will discuss results connected to the asymptotic density of rational points on transcendental varieties, starting with the theorem of PilaWilkie on general ominimal structures and some of its diophantine applications. I will then discuss the Wilkie conjecture asserting a significant sharpening of the PW theorem for the structure Rexp, and a recent proof (joint with Novikov) of this conjecture for sets definable using only restricted exponentiation (and restricted sine). I will briefly discuss the proof, and in particular the significant role played by the algebraic differential equations for exp and sin. Finally, I will discuss some related questions of diophantine significance for structures generated by elliptic, modular and other special functions, and what role the differentialalgebraic structure of these functions may play in resolving these questions.
For a review of the lecture, please click video.
Omar LeonSanchez, McMaster University
Order Bounds and Kolchin PolynomialsRecently a new upper bound for the order of differential polynomials of a characteristic set of each prime component of a differential ideal has been found (for an arbitrary number of derivations m). This bound is computationally feasible (at least in the cases m = 1; 2; 3) and does "not" depend on the degrees of the defining equations. For instance, for the case m = 2, the bound is 2^{n}r where n is the number of variables and r is the order of the original equations. In this talk I will present some applications around: 1) Securing an upper bound for the typical differential dimension, and 2) The definability of Kolchin polynomials in families.
For a review of the lecture, please click video.
Michael SemenovTianShansky, Université de Bourgogne
Poisson Structure for Differential/Difference Lax Equations and Differential Galois Theory.We discuss the transfer of the Poisson structure in the space of differential/difference Lax operators to the space of solutions of the corresponding auxiliary linear problem (the space of wave functions). A peculiar symmetry breaking gives rise to a nontrivial Poisson structure on the associated Galois group.
For a review of the lecture, please click video.
Richard Gustavson, Graduate Center (CUNY)
Differential Kernels and Bounds for the Consistency of Differential EquationsA differential kernel is a field extension of a differential field that encodes certain possible solutions to a system of polynomial differential equations. Determining if that possible solution is an actual solution is then equivalent to asking whether a regular realization of the differential kernel exists. Differential kernels were first studied by Lando in 1970 in the case of a single derivation, in which case every differential kernel has a regular realization. With more than one derivation, however, regular realizations do not always exist. In 2014, Pierce computed an upper bound for the number of prolongations of a given differential kernel needed to ensure the existence of a regular realization. In this talk, I will present an improved upper bound for the number of prolongations needed. This effective bound leads to significantly better results in the case of small numbers of derivations, and has applications to other areas of effective differential algebra, such as improved upper bounds for the order of characteristic sets of prime differential ideals and the effective differential Nullstellensatz.
This is joint work with Omar Leon Sanchez.
For a review of the lecture, please click video.
Joel Nagloo, Bronx Community College (CUNY)
The Third Painlevé Transcendents RevisitedIn this talk I will revisit the problem of existence of algebraic relations over ℂ(t) between solutions (and derivatives) of the generic third Painlevé equation. In particular, I will discuss how one can reduce that problem to studying the Riccati subvarieties.
For a review of the lecture, please click video.
Saturday, October 1: Hunter College, West Building
Lectures in Room HW 415, Refreshments and Posters, Room HW224.
Andrei Gabrielov, Purdue University
Classification of Spherical and Circular QuadrilateralsA spherical polygon (membrane) is a bordered surface homeomorphic to a closed disc, with n distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners where it has conical singularities, and such that the boundary arcs between the corners are geodesic. Spherical polygons with n = 3 and n = 4 are called spherical triangles and quadrilaterals, respectively. This is a very old problem, related to the properties of real solutions of the real Heun equation (a Fuchsian differential equation with four real singular points and real coefficients). The corresponding problem for generic spherical triangles, related to the hypergeometric equation, was solved by Felix Klein more than 100 years ago, while nongeneric cases were completely classified as late as 2011.
When all interior angles at the corners are integer multiples of π, classification of spherical quadrilaterals is equivalent to classification of rational functions with four real critical points, a special case of the B. and M. Shapiro conjecture.
Rational functions with real critical points can be characterized by their nets, combinatorial objects similar to Grothendieck's dessins d'enfants. Similarly, spherical polygons can be characterized by multi colored nets, nonalgebraic analogs of dessins d'enfants. If time permits, I'll tell about classification of circular quadrilaterals (with the sides mapped to not necessarily geodesic circles on the sphere), and about connection between isomonodromic deformation of a Fuchsian differential equation with 5 singular points and solutions of the Painlevé VI equation. This connection allows one to represent a real solution of the real Painlevé VI equation by a sequence of nets of circular quadrilaterals and special (with one angle equal to 2π) circular pentagons.
For a review of the lecture, please click video.
Taylor Dupuy, The University of Vermont
Arithmetic Deformation Classes in the Sense of BuiumIn a 1999 paper, Faltings poses a simple property that an "Arithmetic KodairaSpencer class" must satisfy. He then shows that such a class cannot exist (click here). If one modifies this definition and uses Buium's differential equations, the socalled DeligneIllusie class naturally appears and one gets a construction which satisfies a slightly modified version of Falting's axiom. We will explain all of this in our talk.
For a review of the lecture, please click video.
Amador MartinPizarro, C.N.R.S. Institut Camille Jordan Lyon
On Bounded Automorphisms of Fields with OperatorsLascar showed that the group of automorphisms of the complex field which fix the algebraic closure of the prime field is simple. For this, he first showed that there is no nontrivial bounded automorphisms. An automorphism τ is bounded if there is a finite set A such that the image τ(b) of every element b is algebraic over A∪{b}. The same result holds for a saturated differentially closed field of characteristic zero, where we replace algebraic by differentially algebraic. Together with T. Blossier and C. Hardouin, we provide a complete classification of bounded automorphisms in various fields equipped with operators, among others, for generic difference fields in all characteristics or for HasseSchmidt differential fields in positive characteristic.
For a review of the lecture, please click video (Permission from speaker required.)
Emma Previato, Boston University
An Arithmetically Integrable SystemWe construct the arithmetic analog of a completely integrable system. The motion of the Euler top is described by an algebraically completely integrable Hamiltonian system. We prove that the system is Frobenius liftable to an integral manifold, whose tangent bundle is spanned by arithmetic flows generated by commuting Hamiltonians in the arithmetic sense defined by A. Buium.
This is joint work with A. Buium.
For a review of the lecture, please click video.
Georg Regensburger, Johannes Kepler University Linz
The Fundamental Theorem of Calculus in Differential AlgebraFor incorporating the fundamental theorem of calculus in differential algebras, we consider a linear rightinverse of the derivation. In these algebras, we automatically have an induced evaluation operation corresponding to the second part of the fundamental theorem. This setting generalizes the notion of integrodifferential algebras where in addition the evaluation has to be multiplicative. We construct the corresponding algebra of linear operators as a quotient of a tensor algebra. Based on a completion process for tensor reduction systems, we found all linear consequences (like integration by parts) of the defining relations. This, in turn, enabled us to determine normal forms for these generalized integro differential operators. Normal forms are needed for effective computations and are implemented as a Mathematica package. We also illustrate how analogs of the Taylor formula or variation of constants can be proven in this operator framework.
This is joint work with Clemens G. Raab.
For a review of the lecture, please click video.
Sunday, October 2: Hunter College, West Building
Lectures in Room HW 605, Refreshments and Posters, Room HW605.
Victor Kac, Massachusetts Institute of Technology
The Method of Lambda Brackets in the Theory of Integrable (Bi)Hamiltonian PDEI will introduce the notion of a Poisson vertex algebra (=differential algebra endowed with a lambda bracket) and explain how it is used to construct integrable evolution partial differential equations. Nonlocal and noncommutative situations will be also briefly discussed.
For a review of the lecture, please click video.
Ekaterina Shemyakova, SUNY at New Paltz
Differential Operators on the Superline, Berezinians and Darboux TransformationsWe consider nondegenerate differential operators on the superline (i.e., a supermanifold of dimension 1—1). We show how every such operator can be expressed via certain "super Wronskians" (particular Berezinians). This is applied to the classification of Darboux transformations (DT). Namely, we show that every DT on the superline corresponds to an invariant subspace of the source operator and upon a choice of a basis is given by a superWronskian formula. These formulas are nontrivial because Berezinians are fractions, not polynomials. (On the way, we obtain for Berezinians some analogs of classical row and column expansions, probably new.) We also discuss a generalization of the above to differential operators acting on the algebra of densities on the superline.
For preprints, click here and here.
For a review of the lecture, please click video.
Michael Wibmer, University of Pennsylvania
Torsors for Difference Algebraic GroupsWe introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This has applications to Parameterized PicardVessiot Theory.
For a review of the lecture, please click video.
Andrei Minchenko, University of Vienna
Simple Differential Algebraic Lie AlgebrasThe main purpose of the talk will be to describe the relation between differential algebraic Lie algebras in the sense of Cassidy and linearly compact Lie algebras. In particular, we will see how the classification of simple linearly compact Lie algebras can be used to deduce Cassidy's classification of simple linear differential algebraic groups.
For a review of the lecture, please click video.
Julien Roques, Institut Fourier, Université Grenoble Alpes, France
Hypergeometric Mirror MapsMirror maps are power series which occur in mirror symmetry as the inverse for composition of power series of the form q(z) = exp(ω_{2}(z)/ω_{1}(z)), called canonical coordinates, where ω_{1}(z) and ω_{2}(z) are particular solutions of the PicardFuchs equation associated with certain oneparameter families of CalabiYau varieties. In several cases, the mirror maps have integral coefficients. In this talk, we will give an overview of the integrality properties of mirror maps associated to the generalized hypergeometric equations.
For a review of the lecture, please click video. Permission from the speaker is required.
Monday, October 3: Hunter College, West Building
Lectures in Room HW 605, Refreshments and Posters, Room HW605.
Michael Singer, North Carolina State University
Arithmetic Finite Automata, Automatic Sets, and Difference EquationsA finite automaton is one of the simplest models of computation. Initially introduced by McCulloch and Pitts to model neural networks, they have been used to aid in software design as well as to characterize certain formal languages and numbertheoretic properties of integers. A set of integers is said to be mautomatic if there is a finite automaton that decides if an integer is in this set given its basem representation. For example powers of 2 are 2automatic but not 3automatic. This latter result follows from a theorem of Cobham describing which sets of integers are mautomatic and nautomatic for sufficiently distinct m and n. In recent work with Reinhard Schaefke, we gave a new proof of this result based on analytic results concerning normal forms of systems of difference equations. In this talk, I will describe this circle of ideas.
For a review of the lecture, please click video (an accidental technical error missed the first few minutes) and slides.
Carlos Arreche, North Carolina State University
Projectively Integrable Linear Difference Equations and Their Galois GroupsA differencedifferential field (of characteristic 0) is a field k equipped with an automorphism σ and a derivation δ that commute with each other. A linear difference equation is integrable if its solutions also satisfy a linear differential system of the same size. The difference equation is projectively integrable if it becomes integrable "after moding out by scalars." Based on recent results of R. Schaefke and M. Singer, we show that when k = C(x) and σ is either a shift, q, or Mahler operator, the differencedifferential Galois group G attached to a projectively integrable difference equation has a very special form. These results have applications for the direct problem of computing G for a given linear difference equation, as well as for the inverse problem of deciding which linear differential algebraic groups occur as differencedifferential Galois groups for such equations.
This is joint work with Michael Singer.
For a review of the lecture, please click video.
Charlotte Hardouin, Institut de Mathématiques de Toulouse
Parametric Genericity of Special FunctionsAmong the functions of a complex variable, we distinguish the rational and algebraic functions from the transcendental functions. In the latter class, we consider a new hierarchy depending on the differential dependence of the function. Starting from the holonomic or Dfinite functions that satisfy a linear differential equation, this hierarchy defines a new level of complexity, called hypertranscendence, that corresponds to functions that do not satisfy a polynomial relation with their derivatives. A celebrated example is the Gamma function, which is hypertranscendental by a result of Hölder. Recently parametrized Galois theories have been developed (by HardouinSinger, DiVizioHardouinWibmer, and OvchinnikovWibmer) in order to understand theses questions from a geometric point of view. In this talk, I will try to expose a strategy developed in two collaborations with Dreyfus and Roques that allows us to show that if a function is transcendental then it is also hypertranscendental. We mainly focus on solutions of qdifference equations and consider discrete and continuous hypertranscendence. This strategy heavily relies on a beautiful classification result for differential algebraic groups by Cassidy, which allows reducing the question of hypertranscendence to a question of holonomy.
For a review of the lecture, please click video.
James Freitag, University of Illinois at Chicago
Ranks in Differential FieldsWe will introduce various model theoretic ranks on differential varieties, and talk about what is known about their relationship to Kolchin polynomials. After giving some examples and open problems, we will explain how recent work (joint with Moosa) on a Jouanoloutype theorem together with recent work (joint with Nagloo and Thieu Vo) on vector fields lead to answering a question of Hrushovski and Scanlon on ranks in differential fields.
For a review of the lecture, please click video.
Wei Li, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Partial Differential Chow Forms and a Type of Partial Differential Chow VarietiesIn this talk, we first discuss an intersection theory for quasigeneric partial differential polynomials in partial differential algebraic geometry. Then based on the intersection theory, we define the partial differential Chow form for an irreducible differential variety whose Kolchin differential polynomial is of the form
(d + 1) (
t+m m ) − (
t+m−s m )
and we show that most of the basic properties of the differential Chow form in the ordinary case are established for its partial differential counterpart. Furthermore, we prove that a special type of partial differential Chow varieties exist.For a review of the lecture, please click video.
Anand Pillay, Notre Dame University
PicardVessiot Closed Differential Fields and Constrained CohomologyWe prove that a differential field K is both algebraically closed and PicardVessiot closed if and only if H_{c}^{1}(K,G) = 1 for any linear differential algebraic group over K.
For a review of the lecture, please click video.
Tuesday, October 4: Graduate Center, 4th Floor, Room 4102 (Science Center).
Tom Scanlon, University of California at Berkeley
Algebraic Independence of Functions Satisfying Nonlinear Polynomial Mahler EquationsI will report on some work in program with Alice Medvedev and Khoa Dang Nguyen. We say for k ∈ ℚ_{+}, a positive rational number not equal to 1, that an analytic function f(z) is a kMahler function of polynomial type if there is a polynomial P ∈ ℂ(z)[Y] of degree at least two for which f(z^{k}) = P(z, f(z)). Using methods from the model theory of difference fields, we study the following question: is it possible to have a pair of transcendental functions f and g with f a kMahler function of polynomial type and g an ℓMahler function of polynomial type, with g and f algebraically dependent over ℂ(z) and k and ℓ multiplicatively independent?
For a review of the lecture, please click video.
Ruyong Feng, Academy of Mathematics and Systems Science, CAS
Parallel TelescopingCreative telescoping is a powerful technique for treating symbolic sums and integrals in an algorithmic way. This technique now has been applied to many problems from various areas of mathematics and physics. As a kind of this technique, differential creative telescoping produces from an integrand a linear differential operator satisfied by the integral. Motivated by the direct problem in parameterized differential Galois theory, we introduce the notion of parallel telescoping, which is a natural generalization of differential creative telescoping. Precisely, let
K = k(t, x_{1}, …,x_{m}) be the field of rational functions over k and
be a differential pform, where the f_{i1⋯ip} are elements in some differential extension field of K. Parallel telescoping produces a nonzero operator L in k[t, ∂_{t}], called a parallel telescoper, such that
ω =
∑ 1 ≤ i_{1}<i_{2}⋯<i_{p}≤m f_{i1⋯ip} dx_{1}dx_{2}⋯dx_{p}
where η is a differential (p−1)form and d is the exterior derivation with respect to x_{1}, …, x_{m}. In this talk, we present a necessary and sufficient condition guaranteeing the existence of parallel telescopers for differential forms ω with f_{i1⋯ip} in the set of differentially finite functions over K. When restricting f_{i1⋯ip} to hyperexponential functions or algebraic functions over K, we present algorithms for deciding whether L exists or not and algorithms for computing L if it exists.
L(ω) =
∑ 1 ≤ i_{1}<i_{2}⋯<i_{p}≤m L(f_{i1⋯ip}) dx_{1}dx_{2}⋯dx_{p} = d(η) This is joint work with Shaoshi Chen, Ziming Li, Michael F. Singer and Stephen M. Watt.
For a review of the lecture, please click video.
Thomas Dreyfus, Université Lyon 1
Computing the Differential Galois Group Using Reduced FormsTo a linear differential system one may associate a group, the differential Galois group, that measures the algebraic relations between the solutions of the system. The latter may be seen as an algebraic group and its computation is in general a difficult task. In this talk we explain how to transform the system so that the new system belongs to the Lie algebra of the differential Galois group (such transformation always exists and the new system will be said to be in the KolchinKovacic reduced form). Furthermore, as we will see, the latter reduction will be helpful for computing the differential Galois group.
This is a joint work with JacquesArthur Weil.
For a review of the lecture, please click video.
Gleb Pogudin, Johannes Kepler University, Linz
New Bounds for Effective Differential NullstellensatzDifferential Nullstellensatz is a fundamental theorem in the algebraic theory of differential equations. It states that, if a system of algebraic differential equations does not have a solution in any extension of the coefficient field, then 1 can be expressed as a polynomial linear combination of the derivatives of the equations of the original system up to some order. Thus, finding bounds for this order is a key ingredient of the algorithmic treatment for the solvability of differential equations and related questions. We will present a new and improved bound for the effective version of the differential Nulstellensatz and discuss some differentialalgebraic problems arising in the proof.
This is joint work with Alexey Ovchinnikov and Thieu Vo Ngoc.
For a review of the lecture, please click video.
Zoe Chatzidakis, CNRS—ENS
Definable Galois Theory in Difference FieldsI will speak about the group of automorphisms of solutions (in a sufficiently rich difference field) of difference equations which are internal to the fixed field. (Typical examples of such equations are the linear difference equations.) In particular, we show that if the field K over which the equations are defined is algebraically closed, with fixed subfield pseudofinite, then the set of solutions of our system is strongly related to the set of solutions of an equation of the form f(x)=gx and x in X, where g is an element of an algebraic group G acting on an algebraic set X, and g is in G(K). I will also discuss what happens in the general case. If time permits, I will mention some applications.
(Joint work with Ehud Hrushovski)
For a review of the lecture, please click video.
Uma Iyer, Bronx Community College (CUNY)
Representations of an Algebra of Quantum Differential OperatorsThe algebra of quantum differential operators on graded algebras was introduced by V. Lunts and A. Rosenberg. D. Jordan, T. McCune and Iyer have identified this algebra of quantum differential operators on the polynomial algebra with coefficients in an algebraically closed field of characteristic zero. It contains the first Weyl algebra and the quantum Weyl algebra as its subalgebras. In this talk we present the classification of irreducible weight modules over the algebra of quantum differential operators over the polynomial algebra in one variable.
This is joint work with V. Futorny.
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