Kolchin Seminar in Differential Algebra 
 The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
Alerts: The program for AMS Special Sessions has moved to AMS2017. Authors who would like to post their slides and/or paper may send them to William Sit <wsit@ccny.cuny.edu>.
Last updated on May 14, 2017. For Schedules, lecture notes and additional material, see under (or click):
• Current Schedule • Summer, 2017 • Past Lectures–Spring, 2017 • Past Years
The Kolchin Seminar will be taking a break for the summer, but there will be two international workshops, one in July and another in September (see below). Have a great summer and please do join us next September.
INTERNATIONAL SUMMER WORKSHOPS
The Kolchin Seminars  Kolchin Seminar in Differential Algebra. For 2016 Spring Semester, KSDA meets most Fridays from 10:15 AM to 11:45 AM at the Graduate Center, with occasional talks also from 2:00 PM to 3:30 PM and at Hunter College or other venues on some Saturdays and Sundays. The purpose of these meetings is to introduce the audience to differential algebra and related topics. Most 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 although unannounced normally, has been held regularly since. Occasionally, for various reasons, we may also schedule guest speakers in the afternoon. Informal sessions run from 2:00 pm. to 4:00 p m. in Room 5382 and sometimes start earlier and ends much later. The start time and topics will be announced during the morning sessions (and if not, check with the organizers). All are welcome. 
Unless the contrary is indicated, all meetings will be in Room 5382. This room may be difficult to find; please read the following directions. When you exit the elevator on the 5th floor, there will be doors both to your left and to your right. Go through the doors where you see the computer monitors, then turn left and then immediately right through two glass doors. At the end of the corridor, go past another set of glass doors and continue into the short corridor directly in front of you. Room 5382 is the last room on your right. Security. When you go to the GC you will have to sign in, and it is required that you have some photo ID with you. For directions to the Graduate Center, and for more on security requirements for entering the premise, please click here (updated 2017). For other seminars of the Mathematics Department at the Graduate Center, please click here.  
Hunter College meetings. Occasionally, we also meet on a Saturday and/or Sunday at Hunter College. Hunter College is on 68th Street and Lexington Avenue, where the No. 4,5,6 subways stop. Hunter College has several buildings, including Hunter East (HE), Hunter West (HW), and Hunter North (HN). On weekends, you need to enter from the West Building (a photo ID is required), go up the escalator to the third floor (if necessary, walk across the bridge over Lexington Avenue to the East Building, or across the bridge over 68th Street to the North Building), and take the elevator (ask for direction to the bank of elevators) or escalator to the floor of the meeting room (for example, HN 1036 is on the 10th floor of the North Building). 
February 3, 2017, 10:15–11:30 a.m. Room 5382
Informal Session
RotaBaxter (Type) Algebras
Friday, February 10, 2017, 10:15–11:30 a.m. Room 5382
Jim Freitag, University of Illinois at Chicago
Revisiting the Model Theory of Painlevé EquationsThe Painlevé equations are six families of nonlinear order two ODEs with complex parameters. Around the start of the last century, the equations were isolated for foundational reasons in the analysis of ODEs. Since the 1970s, interest in the equations has steadily increased due in part to their connections with various areas of mathematics (e.g. monodromy of linear differential equations, mathematical physics, and diophantine geometry). In a recent series of works, Nagloo and Pillay established the algebraic independence of solutions of a Painlevé equation, at least when the coefficients are assumed to be transcendental, algebraically independent complex numbers. Later, Nagloo established results of a similar nature for algebraic relations between solutions of equations from different families. In this talk, we will build on the theme of Nagloo and Pillay, answering several questions left open by their work. One of the surprising aspects of the work of Nagloo and Pillay, as well as the present work, is the application of deep structural classification results from model theory to concrete problems on transcendence.
For a review of the lecture, please click video.
Friday, February 17, 2017, 10:15–11:30 a.m. Room 5382
Peter Thompson, Graduate Center, CUNY
NonExistence of Independent Commuting DerivationsLet K be a field. A derivation d_{1} on K[x_{1},…,x_{n}] is said to be integrable if there exist derivations d_{2}, …, d_{n} such that all d_{i} commute pairwise and the set of d_{i} is linearly independent over K[x_{1},…,x_{n}]. Let K be a field of characteristic 0. We present a class of derivations on K[x, y] that is not integrable.
This is joint work with Joel Nagloo and Alexey Ovchinnikov.
Friday, February 24, 2017, 10:15–11:30 a.m. Room 5382
Informal Session
Informal sessions at the Kolchin Seminar are open to all and attendees may bring short presentations and questions for discussion. In this informal session we expect to explore the algebraic approach to integral equations along the recent development in integral differential algebras of Rosenkranz et al.
Friday, March 3, 2017, 10:15–11:45 a.m. Room 5382
Peter Thompson, Graduate Center, CUNY
NonExistence of Independent Commuting Derivations, Part IILet K be a field. A derivation d_{1} on K[x_{1},…,x_{n}] is said to be integrable if there exist derivations d_{2}, …, d_{n} such that all d_{i} commute pairwise and the set of d_{i} is linearly independent over K[x_{1},…,x_{n}]. Let K be a field of characteristic 0. We present a class of derivations on K[x, y] that is not integrable.
This is joint work with Joel Nagloo and Alexey Ovchinnikov.
Friday, March 3, 2017, 2:00–4:00 p.m. Room 5382
Richard Gustavson (Graduate Center, CUNY)^{*}
Elimination for Systems of Algebraic Differential EquationsWe develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, whether the given system of differential equations has a solution. We first look solely at the "algebraic data" of the system of differential equations through the theory of differential kernels to provide a new upper bound for proving the consistency of the system. We then prove a new upper bound for the effective differential Nullstellensatz, which determines a sufficient number of times to differentiate the original system in order to prove its inconsistency. Finally, we study the RosenfeldGröbner algorithm, which approaches differential elimination by decomposing the given system of differential equations into simpler systems. We analyze the complexity of the RosenfeldGröbner algorithm by computing an upper bound for the orders of the derivatives in all intermediate steps and in the output of the algorithm.
*This talk will be the doctoral dissertation defense of the speaker in the Mathematics Department at the Graduate Center. All are welcome.
Friday, March 10, 2017, CUNY Math Fest Day, NO SEMINAR
Friday, March 17, 2017, 10:15–11:30 a.m. Room 5382
Informal Session
Instead of the previously announced talk (see below), Peter Thompson will instead participate in an informal session, which is open to all and attendees may bring short presentations and questions for discussion.
Peter Thompson, Graduate Center, CUNY
NonExistence of Independent Commuting Derivations, Part IIILet K be a field. A derivation d_{1} on K[x_{1},…,x_{n}] is said to be integrable if there exist derivations d_{2}, …, d_{n} such that all d_{i} commute pairwise and the set of d_{i} is linearly independent over K[x_{1},…,x_{n}]. Let K be a field of characteristic 0. We present a class of derivations on K[x, y] that is not integrable.
This is joint work with Joel Nagloo and Alexey Ovchinnikov.
Friday, March 24, 2017, 10:15–11:30 a.m. Room 5382
Due to a family emergency, the talk by William Keigher has been CANCELED.Until further notice, we will have instead an
Informal Session.Informal sessions at the Kolchin Seminar are open to all and attendees may bring short presentations and questions for discussion.
Friday, March 31 2017, 10:15–11:30 a.m. Room 5382
Informal Session.Informal sessions at the Kolchin Seminar are open to all and attendees may bring short presentations and questions for discussion. In this session, the topic is expected to be an introduction to ultrafilters.
Kolchin Research and Training Workshop II, April 21–23, 2017.
For the three daily programs, please click April 21, April 22, April 23.
Friday, April 7, 2017, 10:00–17:20. All lectures in Room 5382, Graduate Center
 10:00–10:50 Jason Bell, University of Waterloo, Canada
The DixmierMoeglin Equivalence and DgroupsThe DixmierMoeglin equivalence is a result that gives a characterization of annihilators of simple modules in a ring and is the first basic step in understanding the irreducible representations of an algebra. We investigate a differentialalgebraic geometric analogue of this equivalence and show that it holds for Dgroups. We use this to show that the classical DixmierMoeglin equivalence holds for a certain class of Hopf algebras.
This is joint work with Omar Leon Sanchez and Rahim Moosa.
For a review of this lecture, please click video.
 11:00–11:50 Michael Singer, North Carolina State University
Walks, Difference Equations and Elliptic CurvesIn 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)?
In a seminal paper, BousquetMélou and Mishna attach a group to any walk in the quarter plane and make the conjecture that a walk has a 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 nonsingular, 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, Bernardi, BousquetMélou, and Raschel proved that 9 of the 51 such walks have a generating series which is hyperalgebraic.
In this talk, I will discuss how difference Galois theory can be used to show that the remaining 42 walks have a generating series which is not hyperalgebraic, leading to a new proof for, and generalizing, the results of Kurkova/Raschel and giving insight into the recent work of Bernardi, BousquetMélou, and Raschel. This is joint work with T. Dreyfus, C. Hardouin and Julien Roques.
For a review of this lecture, please click video.
 14:00–14:50 Michael Wibmer, University of Pennsylvania
Groups Defined by Algebraic Difference EquationsDifference equations are a discrete analog of differential equations. The algebraic theory of difference equations, also known as difference algebra, enhances our understanding of the solutions of difference equations in much the same way as commutative algebra and algebraic geometry enhances our understanding of the solutions of algebraic equations.
The protagonists in this talk are subgroups of the general linear group defined by a system of algebraic difference equations in the matrix entries. These groups have a rich structure theory, to some extent analogous to the theory of linear algebraic groups.
Groups defined by algebraic difference equations occur naturally as the Galois groups of linear differential equations depending on a parameter. I will explain how structure results for these groups can be applied in the study of the relations among the solutions of a linear differential equation and their transforms under various operations like scaling or shifting.
 15:15–16:05 Liang Zhao, Graduate Center (CUNY)
Fast Algorithms on Randomized Structured MatricesRandomization of matrix computations has become a hot research area in the big data era. Sampling with randomly generated matrices has enabled fast algorithms to perform well for some most fundamental problems of numerical algebra with probability close to 1. Empirically, however, randomized structured matrices are used because of their significantly lower computational cost. The talk will illustrate our recent development on fast algorithms with randomized structured matrices that facilitates efficient and stable lowrank matrix approximation, Gaussian elimination without pivoting, and neural network computations for image classification.
 16:30–17:20 Richard Gustavson, Graduate Center (CUNY)
Effective Methods in Differential Elimination TheoryWe develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. Differential elimination has many applications outside of mathematics, such as parameter estimation for system modeling in cellular biology. In this talk I will discuss a special case of differential elimination, namely the question of consistency, i.e. determining when a system of differential equations has a solution. I use the “algebraic data” of the system to produce new upper bounds for several effective methods for testing the consistency of a system of differential equations, including the effective differential Nullstellensatz and the RosenfeldGröbner algorithms.
 18:00–20:00 Venue: TBA
Dinner
Kolchin Research and Training Workshop I, April 7–9, 2017.
For the three daily programs, please click April 7, April 8, April 9.
Saturday, April 8, 2017, 10:00–17:45, All meetings in Room 6215 at Baruch College
 10:00–10:45 OneonOne Discussions
 11:00–11:45 OneonOne Discussions
 13:30–14:15 OneonOne Discussions
 14:30–15:20 Peter Thompson, Graduate Center (CUNY)
Upper Bounds on Commuting Polynomial DerivationsA system of ordinary differential equations corresponds to a derivation on a ring. If a second derivation commutes with the first and the two are independent, then this data can be used to solve the system of equations. We present a class of derivations on a polynomial ring each of whose members has the property that there is an upper bound on the degree of any derivation commuting with it.
 16:00–16:50 Eli Amzallag, Graduate Center (CUNY)
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 the 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. This is joint work with Andrei Minchenko and Gleb Pogudin.
 17:00–17:45 OneonOne Discussions
Sunday, April 9, 2017, 9:15–12:30, All meetings in Room 6215 at Baruch College
 09:15–10:00 OneonOne Discussions
 10:15–12:30 Panel Discussions
Friday, April 14, 2017, Spring Recess, No Seminar
Friday, April 21, 2017, 10:00–17:30, Graduate Center
 10:15–11:05, Room 5382
Carlos Arreche, North Carolina State University
Galois Theories for Functional EquationsFunctions defined by systems of differential and difference equations are a principal focus of study in many areas of mathematics and physics. Understanding the algebraic properties of such functions is essential in many of their physical and mathematical applications. A fruitful approach to discovering these properties is through Galois theory, which produces a geometric object, called the Galois group, that encodes the sought properties of the solutions. I will explain how this approach is used to compute the functional relations satisfied by some concrete special functions and generating series arising in combinatorics, and describe some of my contributions in this area.
For a review of this lecture, please click video. 12:30–14:00, Room 6417
Anand Pillay, Notre Dame University
On the Existence of Embedded Differential Galois ExtensionsWe identify a family of differential fields (U,D) with the feature that if K is a differential subfield of U and the field of constants of K is an elementary substructure of the field U, then any linear DE over K has a PicardVessiot extension L in U (and analogously for logdifferential equations and strongly normal extensions).
This is jointly organized with the Model Theory Seminar.
For a review of this lecture, please click video. 14:15–15:30, Room 5382
David Harbater, University of Pennsylvania
Patching in Differential Galois TheoryPatching methods have been used to make progress in ordinary Galois theory, especially to solve versions of the inverse Galois problem. More recently, this approach has been introduced to differential Galois theory in order to solve analogous problems. Doing so has required the introduction of new ideas in order to deal with the fact that PicardVessiot extensions are generally of infinite degree. This talk will review the use of patching in ordinary Galois theory and then describe its recent use in differential Galois theory in joint work with A. Bachmayr, J. Hartmann, and M. Wibmer.
For a review of this lecture, please click video. 15:45–16:30 OneonOne Sessions
 16:45–17:30 OneonOne Sessions
 18:00–19:30 Dinner, Venue changed to: Ali Baba Turkish Cuisine at 862 Second Avenue (between 46th and 47th Street), New York, NY 10017; Tel:2128888622)
Saturday, April 22, 2017, 10:00–18:00, All meetings in Room 6215 at Baruch College
 10:00–10:50, James Freitag, University of Illinois at Chicago
Model theory, Differential Equations, and TranscendenceThis is a talk about how to use model theory and differential algebra to prove transcendence results for solutions of certain classical differential equations. The talk will focus on solutions of the Painlevé equations, but we will mention several other classical functions as well.
 11:00–11:50, Mengxiao Sun, Graduate Center (CUNY)
Complexity of Triangular Representations of Algebraic SetsWe study the representation of the radical of a polynomial ideal or its corresponding affine variety by triangular sets. The motivation of this study is to turn recent theoretical bounds for effective differential elimination and Nullstellensatz into bounds for practical algorithms. Agnes Szanto proposed an algorithm to compute such a triangular representation. We present the first numerical bounds for the degrees of the polynomials and the number of components in the output of the algorithm. This is joint work with Eli Amzallag, Gleb Pogudin, and Ngoc Thieu Vo.
For a copy of the slides, please click slides.
 13:30–14:20, Greg Cousins, University of Notre Dame
Large FieldsThe notion of a large field was introduced by Pop in 1996 in the paper Embedding problems over large fields. It turns out that a large field is a field over which many Galois theoretic problems have satisfying solutions. For example, if K is a large field, then every finite group appears as the Galois group of some Galois extensions F/K(x) (that is, the inverse Galois problem is true over K(x) if K is large). More generally, every finite split embedding problem is solvable over K(x) if K is large. It turns out that the class of large fields encompasses many familiar examples: algebraically closed fields, separably closed fields, real closed fields, PAC fields, pseudoreal closed fields, PpC fields and many more are all large fields. In this talk, we hope to introduce some history of large fields, some of their arithmetic and geometric properties as well as how they fit into the framework of the model theory of fields.
 14:35–15:40 OneonOne Sessions
 16:00–18:00 Panel Discussion
Sunday, April 23, 2017, 9:15–13:15, All meetings in Room 6215 at Baruch College
 9:15–10:00 OneonOne Sessions
 10:00–10:45 OneonOne Sessions
 11:00–12:30 Panel Discussion
 12:45–13:15 OneonOne Sessions
Friday, April 28, 2017, 10:15–11:30 a.m. Room 5382
Informal Session.Informal sessions at the Kolchin Seminar are open to all and attendees may bring short presentations and questions for discussion.
Kolchin Research and Training Workshop III, May 5 and 7, 2017.
Also, Special Sessions on Differential and Difference Algebra at the AMS Sectional Meeting at Hunter College on May 6 and May 7.
For the three combined daily programs, please click May 5, May 6, May 7.
Friday, May 5, 2017, 10:00–17:40, Room 5382 at the Graduate Center (Kolchin Workshop III)
 10:00–10:50, Santiago Camacho, University of Illinois at UrbanaChampaign
The Wild Behavior of Truncation in Hahn FieldsWe define what is a mathematical structure from a modeltheoretic point of view. We then introduce the ModelTheoretic Universe and some of the classification properties in it. We shift our attention to the specific structure of Hahn fields equipped with a monomial group, a valuation ring and an additive complement to the valuation ring. We show how the notion of Truncation in Hahn fields is robust, in the sense that it is preserved under many different kinds of field extensions, and even a few differential field extensions. We finally proceed to show how Hahn fields with the aforementioned structure violate many of the "tameness'' conditions and finally find its place in the ModelTheoretic Universe.
For a review of the slides, please click slides.
 11:00–11:50, Allen Gehret, University of Illinois at UrbanaChampaign
A Tale of Two Liouville ClosuresHfields are ordered differential fields which serve as an abstract generalization of both Hardy fields (ordered differential fields of germs of realvalued functions at +∞) and transseries (ordered valued differential fields such as 𝕋 and 𝕋_{log}). A Liouville closure of an Hfield K is a minimal realclosed Hfield extension of K that is closed under integration and exponential integration. In 2002, Lou van den Dries and Matthias Aschenbrenner proved that every Hfield K has exactly one, or exactly two, Liouville closures, up to isomorphism over K. Recently (in \arxiv.org/abs/1608.00997), I was able to determine the precise dividing line of this dichotomy. It involves a technical property of Hfields called λfreeness. In this talk, I will review the 2002 result of van den Dries and Aschenbrenner and discuss my recent contribution.
 13:00–13:50, Gabriel Conant, University of Notre Dame
Stability and Sparsity in Sets of Natural NumbersThe additive group of integers is a wellstudied example of a stable group, whose definable sets can be easily and explicitly described. However, until recently, very little has been known about stable expansions of this group. In this talk, we examine the relationship between modeltheoretic stability of expansions of the form (ℤ,+,0,A), where A is a subset of the natural numbers, and the number theoretic behavior of A with respect to asymptotic structure and density of sumsets.
 14:00–14:50, Gleb Pogudin, Johannes Kepler University
Effective Bounds for Differential and Difference EquationsSystems of differential and difference equations arise naturally as a way of modeling realworld processes. Differential equations are usually used in the situation of continuous time, while difference equations correspond to models with discrete time.
We will consider the following two questions about such a system:
1. Checking Consistency: How to determine if the system has a solution? Answering this question would provide us with a way of checking if our model is feasible.
2. Elimination: How to find equations that follow from our system but involve only variables from a given subset? Algorithms answering this question are useful in modeling situations in which we can measure values only of some of the variables occurring in the system.One way to approach these questions is to reduce the problem to checking the consistency or performing an elimination for a system of polynomial equations. Such a reduction relies on effective upper bounds for the number of variables this polynomial system will involve.
In this talk, I will discuss several known bounds obtained in my recent work and explain how they can be used in order to solve the above problems. 15:00–16:30 OneonOne Sessions
 16:40–17:40 Panel Discussions
 18:00–19:30 Dinner at Ravagh
Saturday, May 6 and 7, 2017
AMS Spring Eastern Sectional Meeting #1129
Differential and Difference Algebra: Recent Developments, Applications, and Interactions
Special Sessions I–IV
Sunday, May 7, 2017, 15:30–TBD, Room HN504, Hunter College North Building (Kolchin Research and Training Workshop III)
 15:30–TBD
Panel Discussion and OneonOne Sessions
Friday, May 12, 2017, 10:15–11:30 a.m. Room 5382
Alice Medvedev, City College (CUNY)
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.
2005–2006
2006–2007
2007–2008
2008–2009
2009–2010
2010–2011
2011–2012
2012–2013
2013–2014
2014–2015
2015–2016
2016–2017
Hosted by 
