Kolchin Seminar in Differential Algebra

Academic Year 2016–2017 Last updated on January 31, 2020. Other years: 2005–2006   2006–2007   2007–2008   2008–2009   2009–2010   2010–2011   2011–2012   2012–2013   2013–2014   2014–2015   2015–2016   2017–2018   2018–2019   2019–Fall

• 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) which was held from September 30th (Friday) to October 4th (Tuesday), 2016 in New York City. For a review of the lectures at DART-VII, please click DART-VII.
  • three Research and Training Workshops during the Spring Semester, 2017:
    • Workshop I: April 7–9, 2017
    • Workshop II: April 21–23, 2017
    • Workshop III: May 5 and May 7, 2017
• Profs. Alexander Levin (The Catholic University of America) and Omar León Sánchez (University of Manchester, UK) organized four Special Sessions, on Differential and Difference Algebra as part of the AMS Eastern Sectional Meeting on May 6–7, 2017 (Saturday and Sunday) at Hunter College. For a list of presentations and slides of some talks, please click May 6 and May 7

Friday, August 26, 2016, 10:15–11:45 a.m. Room 5382

Richard Churchill, Hunter College and the Graduate Center, CUNY
Model Theory as Ordinary Mathematics

Contemporary formulations of group theory, topology, algebraic geometry, differential algebra, etc. are generally done in terms of set theory/category theory, but formulations of Model Theory seem far more dependent on formal logic. I will sketch how that subject can be presented in a set-theoretic/category-theoretic framework.

Friday, September 2, 2016, 10:15–11:45 a.m. Room 5382

Informal Session. No scheduled seminar talk.

Friday, September 2, 2016, 2:00–3:30 p.m. Room 6417

Hans Schoutens, The City University of New York
A Model Theory of Affine n-Space via Differential Algebra

For abstract, click here.

This is a cross-listing from CUNY Logic Workshop.

Friday, September 9, 2016, 10:15–11:45 a.m. Room 5382

Informal Session. No scheduled seminar talk.

Friday, September 16, 2016, 10:15–11:45 a.m. Room 5382

Reid Dale, University of California at Berkeley
An Introduction to Pillay's Differential Galois Theory (Part 1)

In a series of papers from the 1990s and early 2000s, Pillay used the machinery of model-theoretic binding groups to give a slick geometric account and generalization of Kolchin's theory of strongly normal extensions and constrained cohomology. This series of two talks is intended to be expository, with its main goal being to introduce and frame the relevant model-theoretic notions of internality and binding groups within the context of differential algebra, as well as to go through Pillay's argument that his generalized strongly normal extensions arise from logarithmic differential equations defined over algebraic D-groups.

For a review of Part 1, please click video.

Friday, September 16, 2016, 12:30–1:45 p.m. Room 6417

Reid Dale, University of California at Berkeley
An Introduction to Pillay’s Differential Galois Theory (Part 2)

Abstract: See Part 1.

This series of two lectures is a joint seminar with Model Theory Seminar.

For a review of Part 2, please click video.

Friday, September 23, 2016, 10:15–11:45 a.m. Room 5382

Emma Previato, Boston University
Integrable Systems as Strongly Normal Differential Extensions

The notion of algebraically completely integrable Hamiltonian system was introduced in the 1970s. In particular, the flows of motion can be completed to complex algebraic tori. We show that they can be realized by Picard-Vessiot extension, generalizing—in a sense—Kolchin's example where the group of the extension is an elliptic curve. It is valuable to note that the base field can be transcendental, and to give a suitable interpretation of the Casimir functions that may occur in the original phase space.

This is joint work with A. Buium, who first (1986) observed this fact for the Euler system.

For a review of the lecture, please click video.

Friday, September 30--Tuesday, 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, October 7, 2016, 10:15–11:45 a.m. Room 5382

No Seminar Scheduled.

Friday, October 21, 2016, 10:15–11:45 a.m. Room 5382

Silvain Rideau, University of California at Berkeley
Imaginaries in Valued Differential Fields I: Finding Prolongations

In 2000, Scanlon described a theory of existentially closed differential fields where the derivation is contractive: v(d(x)) ≥ v(x), for all x. He also proved a quantifier elimination result for this theory. Around the same time, Haskell, Hrushovski and Macpherson classified all the quotients of definable sets by definable equivalence relations in an algebraically closed valued field by proving elimination of imaginaries (relative to certain quotients of the linear group). In analogy with the pure field situation where elimination of imaginaries for differentially closed fields can be derived from elimination of imaginaries in the underlying algebraically closed field, it was conjectured that Scanlon's theory of existentially closed contractive valued differential fields also eliminated imaginaries relatively to those same quotients of the linear group.

In this talk, I will describe the first part of the proof that this result indeed holds. Our main goal will be to explain a construction that can be interpreted as finding "generic" prolongations for valued differential constructible sets.

For a review of this lecture, please click video.

Friday, October 14, 2016, Tuesday Schedule, No Seminar.

Friday, October 21, 2016, 12:30–1:45 p.m. Room 6417

Silvain Rideau, University of California at Berkeley
Imaginaries in Valued Differential Fields II: Computing Canonical Bases

In 2000, Scanlon described a theory of existentially closed differential fields where the derivation is contractive: v(d(x)) ≥ v(x), for all x. He also proved a quantifier elimination result for this theory. Around the same time, Haskell, Hrushovski and Macpherson classified all the quotients of definable sets by definable equivalence relations in an algebraically closed valued field by proving elimination of imaginaries (relative to certain quotients of the linear group). In analogy with the pure field situation where elimination of imaginaries for differentially closed fields can be derived from elimination of imaginaries in the underlying algebraically closed field, it was conjectured that Scanlon's theory of existentially closed contractive valued differential fields also eliminated imaginaries relatively to those same quotients of the linear group.

In this talk, I will describe the second part of the proof that this result indeed holds. Our goal will be to explain a result, joint with Pierre Simon, on definable types in enrichments of NIP theories, which is crucial to prove elimination of imaginaries. We show that under certain hypothesis if a type in some NIP theory T is definable in an enrichment of T, then it is already be definable in T.

These two talks are jointly organized with Model Theory Seminar.

Friday, October 28, 2016, 10:15–11:45 a.m. Room 5382

Informal Session. No scheduled seminar talk.

Friday, November 4, 2016, 10:15–11:45 a.m. Room 5382

Informal Session
No Seminar.

Due to a conflict of schedule, the previously announced talk by Joel Nagloo has been canceled and rescheduled to November 11.

Friday, November 11, 2016, 10:15–11:45 a.m. Room 5382

Joel Nagloo, Bronx Community College (CUNY)
On the Algebraic Independence Conjecture for the Generic Painlevé Equations

In this talk we explain how one can 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.

For a review of the lecture, please click video.

Friday, November 18, 2016, 10:15–11:45 a.m. Room 5382

Gleb Pogudin, Johannes Kepler University
Jet Ideals and Products of Ideals in Differential Rings

We will start with the following question emerged recently in the algorithmic studies of algebraic differential equation: assume that you have several ideals (not necessarily differential) in a differential algebra, how are the product of their derivatives and derivatives of their product related? This question turned out to be closely related to the classical membership problem for the differential ideal [x₁···x_n] in the algebra of differential polynomials in x₁, …, x_n.
Both these questions can be approached using the jet schemes of the ideal (x₁···x_n) in the polynomial algebra in x₁, …, x_n. In the talk we will describe the structure of these jet schemes and use the obtained results to solve initial problems in differential algebra.

For a review of the lecture, please click video-1 and video-2.

Friday, December 2 and 9, 2016, 10:15–11:45 a.m. Room 5382

William Sit, The City College (CUNY)
Rota-Baxter Type Operators, Rewriting Systems, and Gröbner-Shirshov Bases, Part I and II

G.-C. Rota asked for a complete list of all possible algebraic identities that can be satisfied by a linear operator on an associative algebra over a field. Known as Rota's Problem, or Rota's Classification Problem on linear operators, it remains unsolved for decades and only a few were known to Rota. In this talk, which will be in two parts, we present 14 Rota-Baxter type identities and their common properties. These identities, first obtained by symbolic computation, can be uniformly characterized as a class in several ways. For example, an operator identity is of Rota-Baxter type if and only if for all free operator algebra on a well-ordered set, a certain (uniformly defined) rewriting system is convergent, if and only the identity is compatible with a certain (uniformly constructed) monomial order, and if and only if the identity determines (uniformly) a Gröbner-Shirshov basis with respect to the monomial order. We obtain (uniformly) a construction of the free operator algebra satisfying any given Rota-Baxter type identity on any given well-ordered set.

We explain the background, definitions, basics, and main results, sketch our approach and the methods used to prove the results, and discuss open problems.

This is joint work with Shanghua Zheng, Xing Gao, and Li Guo.

For a copy of the slides, please click Part I and Part II

December 16, 2016 to January 31, 2017 Winter Break, no seminars.

February 3, 2017, 10:15–11:30 a.m. Room 5382

Informal Session
Rota-Baxter (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 Equations

The 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-1 and video-2.

Friday, February 17, 2017, 10:15–11:30 a.m. Room 5382

Peter Thompson, Graduate Center, CUNY
Non-Existence of Independent Commuting Derivations

Let K be a field. A derivation d₁ on K[x₁,…,x_n] is said to be integrable if there exist derivations d₂, …, d_n such that all d_i commute pairwise and the set of d_i is linearly independent over K[x₁,…,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
Non-Existence of Independent Commuting Derivations, Part II

Let K be a field. A derivation d₁ on K[x₁,…,x_n] is said to be integrable if there exist derivations d₂, …, d_n such that all d_i commute pairwise and the set of d_i is linearly independent over K[x₁,…,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 Equations

We 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 Rosenfeld-Gröbner algorithm, which approaches differential elimination by decomposing the given system of differential equations into simpler systems. We analyze the complexity of the Rosenfeld-Gröbner 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
Non-Existence of Independent Commuting Derivations, Part III

Let K be a field. A derivation d₁ on K[x₁,…,x_n] is said to be integrable if there exist derivations d₂, …, d_n such that all d_i commute pairwise and the set of d_i is linearly independent over K[x₁,…,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 ultra-filters.

Kolchin Research and Training Workshop I, April 7–9, 2017.

For the three daily programs, please click April 7, April 8, April 9.

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 Dixmier-Moeglin Equivalence and D-groups

  The Dixmier-Moeglin 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 differential-algebraic geometric analogue of this equivalence and show that it holds for D-groups. We use this to show that the classical Dixmier-Moeglin 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 Curves

  In 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, Bousquet-Mélou 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 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, Bernardi, Bousquet-Mélou, 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, Bousquet-Mélou, 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 Equations

  Difference 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 Matrices

  Randomization 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 low-rank 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 Theory

  We 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 Rosenfeld-Gröbner algorithms.

• 18:00–20:00 Venue: TBA
  Dinner
  

Saturday, April 8, 2017, 10:00–17:45, All meetings in Room 6-215 at Baruch College

• 10:00–10:45  One-on-One Discussions
  
• 11:00–11:45  One-on-One Discussions
  
• 13:30–14:15  One-on-One Discussions
  
• 14:30–15:20  Peter Thompson, Graduate Center (CUNY)
  Upper Bounds on Commuting Polynomial Derivations

  A 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 Algorithm

  We 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 model-theoretic language in his explanation of the algorithm's various steps. In a 2015 paper, Feng described the steps using differential-algebraic notions in place of model-theoretic 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  One-on-One Discussions
  

Sunday, April 9, 2017, 9:15–12:30, All meetings in Room 6-215 at Baruch College

• 09:15–10:00  One-on-One Discussions
  
• 10:15–12:30  Panel Discussions
  

Friday, April 14, 2017, Spring Recess, No Seminar

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 21, 2017, 10:00–17:30, Graduate Center

• 10:15–11:05, Room 5382
  Carlos Arreche, North Carolina State University
  Galois Theories for Functional Equations

  Functions 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 Extensions

  We 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 Picard-Vessiot extension L in U (and analogously for log-differential 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 Theory

  Patching 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 Picard-Vessiot 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  One-on-One Sessions
  
• 16:45–17:30  One-on-One 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:212-888-8622)
  

Saturday, April 22, 2017, 10:00–18:00, All meetings in Room 6-215 at Baruch College

• 10:00–10:50, James Freitag, University of Illinois at Chicago
  Model theory, Differential Equations, and Transcendence

  This 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 Sets

  We 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 Fields

  The 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, pseudo-real 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  One-on-One Sessions
  
• 16:00–18:00  Panel Discussion
  

Sunday, April 23, 2017, 9:15–13:15, All meetings in Room 6-215 at Baruch College

• 9:15–10:00  One-on-One Sessions
  
• 10:00–10:45  One-on-One Sessions
  
• 11:00–12:30  Panel Discussion
  
• 12:45–13:15  One-on-One 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 Urbana-Champaign
  The Wild Behavior of Truncation in Hahn Fields

  We define what is a mathematical structure from a model-theoretic point of view. We then introduce the Model-Theoretic 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 Model-Theoretic Universe.

  For a review of the slides, please click slides.

• 11:00–11:50,  Allen Gehret, University of Illinois at Urbana-Champaign
  A Tale of Two Liouville Closures

  H-fields are ordered differential fields which serve as an abstract generalization of both Hardy fields (ordered differential fields of germs of real-valued functions at +∞) and transseries (ordered valued differential fields such as 𝕋 and 𝕋_log). A Liouville closure of an H-field K is a minimal real-closed H-field extension of K that is closed under integration and exponential integration. In 2002, Lou van den Dries and Matthias Aschenbrenner proved that every H-field 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 H-fields called λ-freeness. 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 Numbers

  The additive group of integers is a well-studied 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 model-theoretic 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 Equations

  Systems of differential and difference equations arise naturally as a way of modeling real-world 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  One-on-One 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 One-on-One 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 One

For fixed integers r and m₀, … m_r, the difference equation

∏^r_i=0 (σⁿⁱ(x))^m_i = 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 zⁱ 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.

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