Kolchin Seminar in Differential Algebra 
 The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
During the Spring, 2016 semester, Professors Alice Medvedev (City College) and Alexey Ovchinnikov (Queens College and the Graduate Center) organized a thematic workshop on differential algebra and related topics on April 8–10 and a second one on May 13–15.
Friday, April 8, 2016, Workshop at The Graduate Center
Julien Roques, Université Joseph Fourier, Grenoble
On the Intermediate Singularities of the qDifference EquationsIn this talk, I will mainly review some questions and works related to the socalled intermediate singularities of the algebraic linear qdifference equations over the complex projective line. No prerequisite required.
For a review of this talk, please click video1 and video2.
Michael Singer, North Carolina State University
Consistent Systems of Linear Differential and Difference EquationsI will discuss systems:
d/dx Y(x) = A(x) Y(x), A(x) ∈ gl_{n}(ℂ(x))
Y(σ(x)) = B(x) Y(x), B(x) ∈ GL_{n} (ℂ(x))
satisfying an integrability condition, where
σ(x) = x+1,
or σ(x) = qx, (q ≠ 0,1),
or σ(x) = x^{q}, (q an integer ≥ 2).
I will explain how these systems can be reduced to systems of a very simple form and how this allows one to characterize functions that satisfy both a differential and difference equation with respect to these operators, generalizing results of Ramis and Bézivin. I will show how these results also place a restriction on those groups that can occur as Galois groups of certain integrable linear difference equations. This is joint work with Reinhard Schäfke.For a review of this talk, please click video
Anand Pillay, Notre Dame University
Classification of Strongly Normal Extensions of a Differential Field, and Related Issues; Part IThe material is taken from a joint paper with M. Kamensky, Interpretations and differential Galois extensions. Given a differential field K with field of contants k, and a logarithmic differential equation over K, the strongly normal extensions of K for the equation correspond (up to isomorphism over K) with the connected components of G(k) where G is the Galois groupoid of the equation. This generalizes to other contexts (parameterized theory, ...), and is also the main tool in existence theorems for strongly normal extensions with prescribed properties.
This is a crosslisting from Logic Workshop.
For a review of this talk, please click video.
Askold Khovanskii, University of Toronto
Topological Galois TheoryIn Topological Galois Theory, we consider functions representable by quadratures as multivalued analytical functions of one complex variable. It turns out that there are some necessary topological restrictions in order for the Riemann surface of a function to be representable by quadratures can be positioned over the complex plane.
This approach has the following advantage besides its geometrical appeal. The topological obstructions are related to the character of a multivalued function. They hold not only for functions representable by quadratures, but also for a wider class. This class is obtained by adding to the functions representable by quadratures all meromorphic functions and allowing the presence of such functions in all formulae. Hence the topological results on the non representability by quadratures are stronger that those of algebraic nature. The reason for this is that the composition of two functions is not an algebraic operation. In differential algebra, instead of the composition of two functions, one considers the differential equation that they satisfy. But, for instance, the Euler function Γ does not satisfy any algebraic differential equation; therefore it is pointless to look for an equation satisfied, say, by the function Γ(exp x). The only known results on the non representability of functions by quadratures and, for instance, by the Euler functions Γ are those obtained by our method.
On the other hand, this method cannot be used to prove that a particular singlevalued meromorphic function is the not representable by quadratures.
Using the differential Galois theory (more precisely, its linearalgebraic part, dealing with the linear algebraic groups and their differential invariants), one can prove that the only reasons for the unsolvability by quadratures of linear differential equations of Fuchs type are topological. In other words, if there are no topological obstructions to solvability by quadratures for a differential equation of Fuchs type, then that equation is solvable by quadratures.
There are the following topological obstructions to the representability of functions by quadratures.
Firstly, functions representable by quadratures can have no more than countably many singular points in the complex plane. (However even for the simplest functions representable by quadratures, the set of singular points can be everywhere dense).
Secondly, the monodromy group of a function representable by quadratures is necessarily solvable. (However even for the simplest functions representable by quadratures, the monodromy group can have the cardinality of the continuum).
This is a crosslisting from the Commutative Algebra / Algebraic Geometry Seminar.
For a review of this talk, please click video.
Saturday, April 9, 2016, Workshop at Hunter College, North Building (HN)
Please note that the building and room have changed for the entire day.
Andrey Minchenko, Weizmann Institute
Differential Algebraic Groups and Their ApplicationsAt the most basic level, differential algebraic geometry studies solution spaces of systems of differential polynomial equations. If a matrix group is defined by a set of such equations, one arrives at the notion of a linear differential algebraic group, first introduced by P. Cassidy. These groups naturally appear as Galois groups of linear differential equations with parameters. Studying linear differential algebraic groups and their representations is important for applications to finding dependencies among solutions of differential and difference equations (e.g. transcendence properties of special functions). This study makes extensive use of the representation theory of Lie algebras. Remarkably, via their Lie algebras, differential algebraic groups are related to Lie conformal algebras, defined by V. Kac. We will discuss these and other aspects of differential algebraic groups, as well as related open problems.
Michael Wibmer, University of Pennsylvania
Groups Defined by Difference EquationsGroups defined by algebraic difference equations occur as the Galois groups of linear differential equations depending on a discrete parameter.
I will explain some structure results for these groups and I will show how these results can be used in the study of linear differential equations.
Taylor Dupuy, University of Vermont
Families of Rings With Extra Operations and an Application Toward Mazur's Conjecture (Uniform ManinMumford)We will talk about how to parameterize families of rings with extra operations and give an application.This talk will feature joint work with Joe Rabinoff, Eric Katz and David ZureickBrown, in which we study a certain case of effective ManinMumford for certain types of bad reduction padic curves. In this work one needs to deform between pderivations and derivations of the Frobenius. This algebraic family of operations we call total derivations.
Omar Sanchez, McMaster University
On Order Bounds for Differential PolynomialsThere are several foundational questions in (computational) differential algebra that still remain open (such as the Ritt problem). For instance, it is not known if a uniform bound exists to determine if a differential ideal is prime, or to determine the equations defining the Kolchinclosure of a Kolchinconstructible set. In this talk, I will present recent order bounds for characteristic sets, and what they can say on the equations defining the "Zariskiclosure" of a Kolchinconstructible set.
Parts of this talk appear in joint work with R. Gustavson and other parts in joint work with J. Freitag.
Informal Discussions and Collaborations (Pizza was served at 6:00 p.m.)
Sunday, April 10, 2016 Workshop at Hunter College, West Building (HW)
Anand Pillay, Notre Dame University
Classification of Strongly Normal Extensions of a Differential Field, and Related Issues; Part IIThe material is taken from a joint paper with M. Kamensky, Interpretations and differential Galois extensions. Given a differential field K with field of contants k, and a logarithmic differential equation over K, the strongly normal extensions of K for the equation correspond (up to isomorphism over K) with the connected components of G(k) where G is the Galois groupoid of the equation. This generalizes to other contexts (parameterized theory, ...), and is also the main tool in existence theorems for strongly normal extensions with prescribed properties.
This is a crosslisting from Logic Workshop.
For a review of Part II, please click video.
The topic of this talk has changed to:
Richard Gustavson, Graduate Center (CUNY)
Order bounds for the RosenfeldGröbner AlgorithmThe RosenfeldGröbner algorithm computes a regular decomposition of a radical differential ideal in a differential polynomial ring. This algorithm allows us to give a membership test for such a radical differential ideal, as well as test the consistency of a system of polynomial partial differential equation. In this talk I will describe an upper bound for the orders of the polynomials that appear in the output of the RosenfeldGröbner algorithm, with this upper bound depending only on the orders of the generators of the given radical differential ideal, the number of derivations, and the number of differential indeterminates. The upper bound is arrived at by studying lengths of certain antichain sequences in ℕ^{m}×{1, ..., n}. This is joint work with Alexey Ovchinnikov and Gleb Pogudin.
For a review of the talk, please click video.
Ngoc Thieu Vo, Research Institute for Symbolic Computation, Johannes Kepler University (visiting Graduate Center)
Computing All Rational Solutions of First Order Algebraic ODEsIn this talk, we consider the class of first order algebraic ordinary differential equations (AODEs) and study their rational solutions from some computational points of view. Three approaches are presented. A combinatorial approach gives a degree bound for rational solutions of a class of first order AODEs that do not have movable poles. A view from the theory of algebraic function fields yields an algebraic consideration for the class of first order first degree AODEs. Based on parametrization of algebraic curves, an algebraic geometric approach is presented. Combining these results, we propose an algorithm for computing all rational solutions of a big class of first order AODEs, which covers all first order AODEs from Kamke's collection.
For a review of the talk, please click video.
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