Kolchin Seminar in Differential Algebra 
 The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
Last updated on November 24, 2015. For Schedules, lecture notes and additional material, see under (or click):
• Current Schedule • Fall, 2015 • Past Lectures–Fall, 2015 • Past Years
Friday, November 27, 2015, Thankgiving Weekend, NO SEMINAR
Friday, December 4, 2015, 10:15–11:45 a.m. Room 5382
William Keigher, Rutgers University at Newark
Interlacing of Hurwitz SeriesWe begin by reviewing some definitions and properties of the ring of Hurwitz series. Given a commutative ring A with identity, the ring of Hurwitz series over A can be thought of as consisting of all sequences (a_{n}) with entries a_{n} in A. This set HA of all sequences in A can be given a natural addition, multiplication, and derivative operator to make HA into a differential ring. In the case where ℚ ⊆ A, HA = A[[t]], the ring of formal power series over A. After describing some important properties of HA, we will define the notion of interlacing and show some important properties of interlacing, including a result on the solutions of n^{th} order linear differential equations over A.
This talk should be accessible to all graduate students.
Friday, December 11, 2015, 10:15–11:45 a.m. Room 5382
Matthew HarrisonTrainor, University of California at Berkeley
DifferentialAlgebraic Jet Spaces and InternalityWe will show that the generic type of a differentialalgebraic jet space satisfies a strengthening of almost internality to the constants, called preserving internality to the constants. This is a notion which was first introduced by Moosa and Pillay as a modeltheoretic abstraction of a phenomenon in complexanalytic geometry. In contrast with the complexanalytic case, only a generic analogue holds in the differentialalgebraic case: there is a finitedimensional differentialalgebraic variety X with a subvariety Z that is internal to the constants, such that the restriction of the differentialalgebraic tangent bundle of X to Z is not almost internal to the constants.
Kolchin Seminar in Differential Algebra. For 2015 Fall Semester, KSDA meets most Fridays from 10:15 AM to 11:45 AM at the Graduate Center, with occasion talks also from 2:00 PM to 3:30 PM and at Hunter College, on some Saturdays. 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 will be continued. Occasionally, for various reasons, we may also schedule guest speakers in the afternoon. Informal sessions, which run from 2:00–4:00 pm, normally go unannounced or are announced at the end of the morning sessions (please check with 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 September 1, 2015).  
Hunter College meetings. Occasionally, we also meet on a Saturday at Hunter College, Room E920. Hunter College is on 68th Street and Lexington Avenue, where the No. 4,5,6 subways stop. You need to enter from the West Building (a photo ID is required), go up the escalators to the third floor, walk across the bridge over Lexington Avenue to the East Building, and take the elevator before the Library to the 9th floor. Room 920 is located in a northeast corner. 
Friday, September 11, 2015, 10:15–11:45 a.m. Room 5382
Xing Gao (Lanzhou University, China and Rutgers University at Newark)
IntegroDifferential Algebra of Combinatorial SpeciesThe concept of structure is fundamental and recurring in all branches of mathematics, as well as in computer science. Informally, a combinatorial species is a class of finite structures on arbitrary finite sets which is closed under arbitrary "relabellings" along bijections. Various combinatorial operations can be defined on species of structures, such as addition, multiplication, substitution, differentiation and integration, giving rise to combinatorial algebras. Roughly speaking, an integrodifferential algebra (R,d, P) is an algebraic abstraction of the familiar setting of derivatives and integrals in analysis, where one employs a notion of differentiation d together with a notion of integration P on some (real or complex) algebra of functions. In this talk, we give an integrodifferential algebra structure on combinatorial species of structures.
The lecture is available: including slides, and video at CUNY, or on Youtube.
Friday, September 18, 2015, 10:15–11:45 a.m. Room 5382
Michael Wibmer, University of Pennsylvania
Strongly Étale Difference Algebras and Babbitt's DecompositionWe introduce a certain class of difference algebras whose role in the study of difference equations is analogous to the role of étale algebras in the study of algebraic equations. We use these difference algebras to deduce an improved version of Babbitt's decomposition theorem. We also present applications to difference algebraic groups and the compatibility problem. This is joint work with Ivan Tomasic.
For a review of the talk, please click video.
Friday, September 25, 2015, NO SEMINAR (Tuesday schedule for CUNY)
Friday, October 2, 2015, 10:15–11:45 a.m. Room 5382
XiaoShan Gao, Academy of Mathematics and Systems Science, Beijing, China
Differential and Difference Chow Form, Sparse Resultant, and Toric VarietyIn this talk, I will give a survey on the recent work on differential and difference Chow forms, sparse resultants, and toric varieties. Chow forms are used as canonical representations as well as coordination for algebraic cycles. Sparse resultants are powerful tools for elimination of sparse polynomial systems. Chow forms and sparse results are connected through toric varieties. More precisely, for a given set A of monomial supports, the Chow form of the toric variety defined by A is the Asparse resultant. We will show how these results are extended to differential algebra and difference algebra.
For a review of the lecture, please click video and slides.
Friday, October 2, 2015, 12:30–1:45 p.m. Room 6417
This is a crosslisting from Model Theory Seminar
Richard Gustavson, The Graduate Center (CUNY)
Effective bounds for the Existence of Differential Field ExtensionsWe present a new upper bound for the existence of a differential field extension of a differential field (K; D) that is compatible with a given field extension of K. In 2014, Pierce provided an upper bound in terms of lengths of certain antichain sequences of ℕ ^{m} equipped with the product order. Pierce’s theory has interesting applications to the model theory of fields with m commuting derivations, and his results have been used when studying effective methods in differential algebra, such as the effective differential Nullstellensatz problem. We use a new approach involving Macaulay’s theorem on the Hilbert function to produce an improved upper bound. In particular, we see markedly improved results in the case of two and three derivations.
This is joint work with Omar Leon Sanchez.
Friday, October 9, 2015, 10:15–11:45 a.m. Room 5382
Eugenia Cheng, University of Sheffield and School of the Art Institute of Chicago
Operads: from loop spaces to ncategoriesThis talk was cancelled by the speaker due to a severe mosquito attack. We thank Prof. Li Guo of Rutgers University for presenting a brief introduction to operads.
It is known that ngroupoids are too strict to model homotopy ntypes, and instead we must use "weak ngroupoids", whose axioms are only satisfied up to equivalence. The problem is to make this definition in a coherent way; these equivalences must satisfy some other axioms of their own, possibly only up to equivalence. In this talk we will discuss how to approach this problem using operads. Operads were originally introduced as a powerful tool for handling the "weak monoid" structure of loop spaces, and we will show how to harness this same power to handle the structure of weak ncategories according to the work of Trimble and Batanin, with further developments by Leinster and Cheng. The talk will be introductory; in particular no knowledge of ncategories will be assumed.
Friday, October 16, 2015, 10:15–11:45 a.m. Room 5382
Alice Medvedev (session chair), The City College of New York
L. diVizio and C. Hardouin: Intrinsic Approach to Galois Theory—qDifference EquationsAs an experimental project for the Fall semester, we will designate several Fridays (mornings and occasionally afternoons) discussing (mainly Part IV of) the above paper.
From the Introduction: "The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of qdifference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of qdifference equations and we use it to establish the relations among the different theories in the literature."
Graduate students and faculties with an interest in Hopf algebras, Tannakian categories, algebraic groups, difference equations, or differential algebra are welcome to join us and participate in the discussions!
The first meeting will be chaired by Alice Medvedev (CCNY). We shall go through the scope of the paper and choose the topics of common interest as well as their prerequisites. We shall then agree on a plan, which is to include the dates and session chairs of future meetings.
Friday, October 23, 2015, 10:15–11:45 a.m. Room 5382
Mengxiao Sun, Graduate Center
L. diVizio and C. Hardouin: Intrinsic Approach to Galois Theory—qDifference EquationsThis is the first study/discussion of selected topics from the paper. This Friday's topic will be the basics of qdifference equations based on Chapter I.
From the Introduction: "The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of qdifference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of qdifference equations and we use it to establish the relations among the different theories in the literature."
Friday, October 30, 2015, 10:15–11:45 a.m. Room 5382
Mengxiao Sun, Graduate Center
L. diVizio and C. Hardouin: Intrinsic Approach to Galois Theory—qDifference EquationsThis is the second study/discussion of selected topics from the paper. This Friday, the speaker will continue on the basics of qdifference equations based on Chapter I.
From the Introduction: "The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of qdifference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of qdifference equations and we use it to establish the relations among the different theories in the literature."
Friday, November 6, 2015, 10:15–11:45 a.m. Room 5382
Richard Gustavson and Eli Amzallag, Graduate Center (CUNY)
Effective Difference NullstellensatzThe difference Nullstellensatz says that in a difference closed pseudofield R (which is a difference ring with some extra properties) a system of polynomial difference equations F = 0 has a common solution in R if and only if the difference ideal generated by F does not contain 1. The effective difference Nullstellensatz tasks us with finding an upper bound on the number of times we must apply the automorphism to the elements of the system in order to obtain this differencealgebraic relation. It is currently unknown if such a bound exists. In this talk, we will present an introduction to the problem along with our approach to solving it, modeled after recent breakthroughs in the differential case.
Friday, November 13, 2015, 10:15–11:45 a.m. Room 5382
Joel Nagloo and Peter Thompson, Graduate Center (CUNY)
On Degree Bounds for Integrability of Differential EquationsIn this talk we look at the question of existence of degree bounds for integrability of differential equations. We discuss the failure of the "ultraproduct" methods for bounds in this context and look at well known nonintegrable ODEs such as the Painlevé equations.
Friday, November 20, 2015, 9:45–10:30 a.m. Room 5382
Eli Amzallag and Richard Gustavson, Graduate Center (CUNY)
Model Theory of Difference FieldsIn this talk, we will give a brief introduction to the model theory of difference closed fields, called ACFA. This theory comes from the seminal paper by Z. Chatzidakis and E. Hrushovski (Trans. Amer. Math. Soc. 351 (1999), 29973071). We will describe the axioms of ACFA and discuss some of the elementary properties of this theory.
Friday, November 20, 2015, 10:30–12:00 noon. Room 5382
Richard Churchill (session chair), Graduate Center and Hunter College (CUNY)
L. diVizio and C. Hardouin: Intrinsic Approach to Galois Theory—qDifference EquationsThis Friday will be the third study/discussion session on the paper.
From the Introduction: "The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of qdifference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of qdifference equations and we use it to establish the relations among the different theories in the literature."
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