Kolchin Seminar in Differential Algebra
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The Graduate Center
365 Fifth Avenue, New York, NY 10016-4309
General Telephone: 1-212-817-7000

Academic Year 2015–2016
We welcome Professor Xing Gao from Lanzhou University, China, who is currently visiting Professor Li Guo at Rutgers University at Newark. He will be staying for a year, working on, among other topics, Rota-Baxter type algebras and integral differential algebras.

Last updated on October 8, 2015. For Schedules, lecture notes and additional material, see under (or click):
 •  Current Schedule  •  Spring, 2015   •   Past Lectures–Spring, 2015  •  Past Years

Alerts:  The slides for William Keigher's talk on February 20. 2015 has been posted.
The slides and video for the talk by Wai Yan Pong on March 20. 2015 have been posted.

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 n-categories

It is known that n-groupoids are too strict to model homotopy n-types, and instead we must use "weak n-groupoids", 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 n-categories 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 n-categories will be assumed.

Friday, October 16 and 23, 2015, 10:15–11:45 a.m. Room 5382

Alice Medvedev (session chair, October 16), The City College of New York
William Sit (session chair, October 23), The City College of New York

L. diVizio and C. Hardouin: Intrinsic Approach to Galois Theory---q-Difference Equations

As 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 q-difference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of q-difference 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 30, 2015, 10:15–11:45 a.m. Room 5382

Andrey Minchenko, Weizmann Institute
Title: TBA

Friday, October 30, 2015, 12:30–1:45 p.m. Room 6417
This is a cross-listing from Model Theory Seminar

Joel Nagloo, The Graduate Center (CUNY)
Title: TBA

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 north-east corner.

Past Lectures, Fall 2015

Friday, September 11, 2015, 10:15–11:45 a.m. Room 5382

Xing Gao (Lanzhou University, China and Rutgers University at Newark)
Integro-Differential Algebra of Combinatorial Species

The 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 integro-differential 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 integro-differential 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 Decomposition

We 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

Xiao-Shan Gao, Academy of Mathematics and Systems Science, Beijing, China
Differential and Difference Chow Form, Sparse Resultant, and Toric Variety

In 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 A-sparse 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 cross-listing from Model Theory Seminar

Richard Gustavson, The Graduate Center (CUNY)
Effective bounds for the Existence of Differential Field Extensions

We 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.

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