Kolchin Seminar in Differential Algebra 

The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
These Special Sessions are organized by:
Alexander Levin (The Catholic University of America, and
Omar León Sánchez (University of Manchester, UK)
The presenter is marked by an asterisk when there are multiple authors. Each title is preceded by an AMS code in the form of AABBB, where AA is MSC 2digit classification and BBB is a sequence number with a variable number of digits. Abstracts are from AMS official site for abstracts as of April 24, 2017, reproduced below in html format for convenience. For slides of talks, please click on the links provided.
Saturday, May 6, 2017, 08:00–17:00, Sessions I and II
 08:00–08:45, Andy Magid*, University of Oklahoma and Lourdes Juan, Texas Tech University
13329: Differential Projective Modules and Azumaya Algebras over Differential RingsDifferential modules over a commutative differential ring which are finitely generated and projective as ring modules, with differential homomorphisms, form an additive category. All such are shown to be direct summands of objects which are free as ring modules; those which are differential direct summands of differential direct sums of the ring are shown to be induced from the subring of constants. And any object has this form after a suitable extension of the base. Thus the Ktheory of the differential category reduces to that of ordinary Ktheory and kernels. Differential Azumaya algebras over the ring whose underlying modules are finitely generated and projective form a multiplicative category, and similar results to the above are obtained. The Ktheory of this multiplicative category can accordingly be analyzed in a similar way.
 09:00–09:20, William D. Simmons*, University of Pennsylvania and Henry Towsner, University of Pennsylvania
12187:Mining Effective Information from Nonconstructive Proofs in Differential AlgebraUltraproducts and other nonconstructive tools often yield existence results without explicit values. We examine the interplay of such arguments with ”proof mining” techniques that systematically extract effective information even when it is not apparent. Our main result is a uniform bound related to the detection of prime differential ideals.
For a review of the slides, please click slides.
For a copy of a related paper, Nonstandard methods for bounds in differential polynomial rings, (J. Alg. 2012), please click paper.
 09:30–09:50, Reid Dale, University of California at Berkeley
03450: Generalized Differential Galois ExtensionsIn a recent paper, Kamensky and Pillay give sufficient conditions for the existence of a differential Galois extension for a logarithmic differential equation defined over an algebraic group G over the constants. In this talk we extend this result to arbitrary algebraic Dgroups and find sufficient conditions for the existence and uniqueness of such extensions. This is joint work with J. Nagloo.
 10:00–10:20, Carlos E. Arreche, NC State University
13373: Differential SquareZero Extensions and PicardVessiot TheoryIn algebraic geometry, understanding squarezero extensions of commutative rings is the first step in the cohomological classification of infinitesimal deformations of schemes. Following recent work of Magid in the case of one derivation, we have developed analogous results for differential squarezero extensions of simple differential rings with several commuting derivations. We prove that such extensions become differentially split in a PicardVessiot extension. This is joint work with Raymond Hoobler.
For a review of the presentation, please click slides.
 10:30–10:50, Alice Medvedev, The City College of New York, CUNY
12507: Sparse Difference Equations with High Transcendence Degree but Difference Krull Dimension OneFor fixed integers r and m_{0}, … m_{r}, the difference equation
∏^{r}_{i=0} (σ^{ni}(x))^{mi} = 1
defines a subgroup G_{n} of the multiplicative group of transcendence degree nr.We show that whenever no zero of the polynomial χ(z):= ∑^{r}_{i=0} m_{i} z^{i} is a root of unity, the difference Krull dimension of G_{n} is bounded, independently of n. Indeed, the difference Krull dimension of G_{n} is 1 whenever χ(z) is hereditarily irreducible, and it usually is.
 15:00–15:20, Raymond T. Hoobler, City College of New York and Graduate Center, CUNY
14:315:Differential Brauer GroupLet X be a quasiprojective, compact scheme over a field of characteristic 0. Recent work shows that given a torsion element x ∈ H^{2}(X_{et}, G_{m}), there is an Azumayua algebra Λ on X admitting an integrable biconnection, i.e. a connection such that ∇(ab) = a∇(b) + ∇(a)b, whose cohomology class is X. We use this to define the differential Brauer group Br_{∇}(X) on such a scheme. We use the δflat topology to give a cohomological interpretation of Br_{∇}(X) and show its relation to the usual Brauer group. If X is smooth and projective, we illustrate this relationship with respect to Hodge theory.
For a review of the presentation, please click slides.
 15:30–15:50, Eli Amzallag*, CUNY Graduate Center; Gleb Pogudin, JKU Linz Institute for Algebra; and Andrei Minchenko, University of Vienna
34276: On the Complexity of Hrushovski's AlgorithmWe analyze the complexity of Hrushovski’s algorithm to compute the Galois group of a linear differential equation of order n over C(t), where C is an algebraically closed field of characteristic zero. Hrushovski presented his algorithm in a 2002 paper, using modeltheoretic language in his explanation of the algorithm’s various steps. In a 2015 paper, Feng described the steps using differentialalgebraic notions in place of modeltheoretic ones. He also turned to complexity considerations in that paper, his analysis beginning with the algorithm’s computation of a group that contains the Galois group of the given differential equation. His estimate of a bound for the degrees of defining polynomials of this group was sextuply exponential in n. In this talk, we will present an improved bound and discuss our approaches to analyzing the complexity of the rest of the algorithm.
 16:00–16:20, Richard Gustavson*, CUNY Graduate Center; Alexey Ovchinnikov, CUNY Queens College; and Gleb Pogudin, Johannes Kepler University
35274: New Upper Bounds for Differential Elimination AlgorithmsDifferential elimination is the process of eliminating a fixed set of differential unknowns from a system of differential equations in order to obtain consequences of the system that do not depend on that fixed set of unknowns. Decomposition algorithms approach this problem by decomposing a system of differential equations into a collection of simpler systems that can be more easily studied. In this talk, we will discuss the RosenfeldGröbner algorithm for systems of partial differential equations, one of the most common decomposition algorithms, which has been implemented in computer algebra systems such as Maple. Specifically, we will address the complexity of the RosenfeldGröbner algorithm by computing an upper bound for the orders of the derivatives that appear in all intermediate steps and in the output of the algorithm.
 16:30–16:50, David Marker, University of Illinois at Chicago
03102: The Logical Complexity of Schanuel's Conjecture and Exponential DerivationsSchanuel’s Conjecture is naturally a ∏^{1}_{1}statement. We show that it is equivalent to a ∏^{0}_{3}statement in arithmetic by showing that if there are counterexamples, then there are computable counterexamples. The main ideas in the proof come from the work of Johnathan Kirby on exponential algebraic closure and exponential derivations. I will survey Kirby’s work and explain the application.
For a review of the presentation, please click slides.
Sunday, May 7, 2017, 08:00–15:30, Sessions III and IV
 08:00–08:45, Thomas Dreyfus*, University Lyon, France; Charlotte Hardouin, University Toulouse, France; Julien Roques, University Grenoble, France; and Michael Singer, University Raleigh, USA
1317: On the Nature of the Generating Series of Random Walks in the Quarter PlaneIn the recent years, the nature of the generating series of the walks in the quarter plane has attracted the attention of many authors. The main questions are: are they algebraic, holonomic (solutions of linear differential equations) or at least hyperalgebraic (solutions of algebraic differential equations)? This problem was first considered in a seminal paper, where BousquetMélou and Mishna attach a group to any walk in the quarter plane and make the conjecture that a walk has an holonomic generating series if and only if the associated group is finite. They proved that, if the group of the walk is finite, then the generating series is holonomic, except, maybe, in one case, which was solved positively by Bostan, van Hoeij and Kauers. In the infinite group case, Kurkova and Raschel proved that if the walk is in addition non singular, then the corresponding generating series is not holonomic. This work is very delicate, and relies on the explicit uniformization of a certain elliptic curve. Recently, it has been proved that 9 of the 51 such walks have a generating series which is hyperalgebraic. In this talk, we will prove, using difference Galois theory, that the remaining 42 walks, have a generating series which is not hyperalgebraic.
For a review of the presentation, please click slides.
 09:00–09:20, Taylor Dupuy, University of Vermont
11365: Deforming DerivativesWe will talk about algebraic parameter spaces of rings with extra operations. In particular we will talk about deforming derivative operations into difference operations and what this means algebraically.
 09:30–09:50, Gleb Pogudin, Institute for Algebra, Johannes Kepler University
39211: On the Effective Difference NullstellensatzWhile modelling a discretetime system, it is natural to assign a sequence of numbers in which the i^{th} number is equal to the value of the parameter at the i^{th} moment in time to every parameter of the system. There are usually several parameters with some relations among them. For every i^{th} moment in time, these relations can be written as equations in the values of the parameters at this moment and some neighboring moments. It is assumed that these equations are the same for all moments in time up to shifting the indices. A natural question to ask is whether such an infinite system of equations corresponding to the model has a solution. In this talk, we will describe cases in which this problem can be solved algorithmically using effective upper bounds. This is joint work with Alexander Levin and Alexey Ovchinnikov.
 10:00–10:20, Joel C R Nagloo, CUNY Bronx Community College
33370: On the Algebraic Independence Conjecture for the Generic Painlevé Equations.In this talk, we explain how the Riccati equations can be used to show that the solutions (and derivatives) of the generic sixth Painlevé equation are algebraically independent over ℂ(t). This extends recent progress made on the third Painlevé equations and hence fully proves the algebraic independence conjecture for the generic Painlevé equations.
 10:30–10:50, Alexander Levin, The Catholic University of America
12167: Dimension Quasipolynomials of Inversive Difference Field Extensions with Weighted TranslationsLet K be an inversive difference field with basic translations σ_{1}, … σ_{m} that are assigned positive integer weights w_{1}, … w_{m}, respectively. Let Γ denote the set of all power products τ = σ^{k1}_{1}···σ^{km}_{m} (k_{i} ∈ ℤ), let the order of such a power product be defined as ord_{w}τ = ∑^{m}_{i=1} w_{i} k_{i}, and for every r ∈ ℕ, let Γ(r) = {τ ∈ Γ  ord_{w}τ ≤ r}. We prove that if L is a finitely generated inverse difference field extension of K with a set of difference generators η = {η_{1},…, η_{n}}, then the function ϕ_{η}(r) = tr deg_{K} K(⋃^{n}_{i=1} Γ(r)η_{i}) is a quasipolynomial in r that can be expressed as an alternating sum of certain Ehrhart quasipolynomials. We also determine some difference birational invariants of this quasipolynomial and give a generalization of the obtained results to the case of multivariate dimension quasipolynomials associated with partitions of the set of basic translations.
For a review of the presentation, please click slides.
 14:00–14:20, James Freitag, University of Illinois at Chicago
14447:Model theory and Transformations of Painlevé EquationsWe will discuss how to use model theory to prove some classification results on transformations of Painlevé equations.
 14:30–15:15, Julia Hartmann, University of Pennsylvania
12358: Differential Torsors and Differential Embedding ProblemsWe introduce the notion of a differential torsor, which allows us to state and prove a converse to Kolchin’s structure theorem for PicardVessiot rings. This is used to obtain a patching result for PicardVessiot rings. As an application, we deduce the solvability of differential embedding problems over one variable complex function fields. (Joint work with A. Bachmayr, D. Harbater, and M. Wibmer.) (
Kolchin Seminar in Differential Algebra. KSDA meets most Fridays from 10:15 AM to 11:45 AM at the Graduate Center. The purpose of these meetings is to introduce the audience to differential algebra. The lectures will be suitable for graduate students and faculty and will often include open problems. Presentations will be made by visiting scholars, local faculty, and graduate students. Kolchin Afternoon Seminar in Differential Algebra. This informal discussion series began during the Spring Semester of 2009 and will be continued. It normally goes from 2:005:00 pm (please check with organizers). All are welcome. Other Academic Years
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