Kolchin Seminar in Differential Algebra 

The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
Last updated on February 23, 2015. For
Schedules,
lecture notes and additional material, see under (or click):
• Current Schedule • Spring, 2015
•
Past Lectures–Spring,
2015 •
Past Years
Friday, February 27, 2015, 10:15–11:45 a.m. Room 5382
Annette Maier, Technische Universität Dortmund, Germany
Computing Difference Galois Groups over 𝔽_{q}(s,t)We consider linear difference equations σ(y) = Ay over (𝔽_{q}(s,t), σ), where σ(s) = s^{q} and σ acts trivially on 𝔽_{q}(t). The difference Galois group G of such an equation is a linear algebraic group defined over 𝔽_{q}(t). In the talk, I will present criteria that provide upper and lower bounds on G depending on A. The lower bound criterion asserts that G contains conjugates of certain reductions ¯A of A. These criteria can be applied to partly solve the inverse difference Galois problem over (𝔽_{q}(s,t), σ), namely every semisimple, simplyconnected linear algebraic group H defined over 𝔽_{q} is a difference Galois group over (𝔽_{qi}(s,t), σ) for some i ∈ ℕ. This can be seen as a difference analogue of Nori's theorem in finite Galois theory which states that H(𝔽_{q}) is a Galois group over 𝔽_{q}(s).
[If your browser (such as Chrome) is not able to display blackboard bold (using unicode), the missing symbol is {\mathbb F}, symbol for finite fields.]
Friday, March 6, 2015, 10:15–11:45 a.m. Room 5382
Michael Wibmer, RWTH Aachen University, Germany
Difference Algebraic GroupsDifference algebraic groups are the discrete analog of differential algebraic groups. These groups occur naturally as the Galois groups of linear differential or difference equations depending on a discrete parameter. The talk will start with a brief introduction to difference algebra and difference algebraic geometry. Then I will present some basic results on difference algebraic groups, i.e., groups defined by algebraic difference equations. In particular, I will introduce some numerical invariants, such as the limit degree, and discuss two possible definitions of the identity component of a difference algebraic group. Finally, I will explain the role of these concepts in a decomposition theorem for étale difference algebraic groups.
Friday, March 13, 2015, 10:15–11:45 a.m. Room 5382
Gal Binyamini, University of Toronto
Bezouttype Theorems for Differential FieldsWe consider the following problem: given a set of algebraic conditions on an ntuple of functions and their first ℓ derivatives, admitting finitely many solutions in a differentially closed field, give an upper bound for the number of solutions. I will present estimates in terms of the degrees of the algebraic conditions, or more generally the volumes of their Newton polytopes (analogous to the Bezout and BKK theorems). The estimates are singlyexponential with respect to n and ℓ and have the natural asymptotic with respect to the degrees or Newton polytopes. This result sharpens previous doublyexponential estimates due to Hrushovski and Pillay.
I will give an overview of the geometric ideas behind the proof. If time permits I will also discuss some diophantine applications.
Friday, March 20, 2015, 10:15–11:45 a.m. Room 5382
WaiYan Pong, California State University Dominguez Hills
Title: TBA
Friday, March 27, 2015, 10:15–11:45 a.m. Room 5382
James Freitag, University of California at Berkeley
Title: TBA
Fridays, April 3 and 10, 2015, No Seminar (Spring Recess)
Friday, April 17, 2015, 10:15–11:45 a.m. Room 5382
William Keigher, Rutgers University at Newark
Interlacing of Hurwitz Series
Friday, April 24, 2015, 10:15–11:45 a.m. Room 5382
William Simmons, University of Pennsylvania
Title: TBA
Friday, May 1, 2015, 10:15–11:45 a.m. Room 5382
Omar Sanchez, McMaster University
Title: TBA
Friday, May 8, 2015, 10:15–11:45 a.m. Room 5382
Abraham D. Smith, Fordham University
Title: TBA
Friday, May 15, 2015, 10:15–11:45 a.m. Room 5382
Victor Kac, Massachusetts Institute of Technology
Title: TBA
Friday, May 15, 2015, 2:00 p.m.–3:30 p.m. Room 5382
Laurent Poinsot, Computer Science Laboratory of ParisNorth University (LIPN)
Title: TBA
Friday, May 22, 2015, 10:15–11:45 a.m. Room 5382
Lou van den Dries, University of Illinois
Title: TBA
Kolchin Seminar in Differential Algebra. For 2015 Spring 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 January 14, 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, February 6, 2015, 10:15–11:45 a.m. Room 5382
Julia Hartmann, University of Pennsylvania, and RWTH Aachen University
Differential Galois Groups over Laurent Series FieldsWe apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of characteristic zero. More precisely, we show that any linear algebraic group (i.e., affine group scheme of finite type) over such a Laurent series field does occur as the differential Galois group of a linear differential equation with coefficients in any such function field (of one or several variables). This is joint work with David Harbater and Annette Maier and generalizes previous results for split groups.
Friday, February 13, 2015, 10:15–11:45 a.m. Room 5382
William Sit, City College of New York
Revisiting Term Rewriting in AlgebraTermrewriting systems are an essential part of symbolic computations in algebra (including differential algebra and RotaBaxter algebra). We introduce a class of termrewriting systems on free modules and proved some general results on confluence, termination and convergence. Definitions and examples will be given and this topic is suitable for graduate students. No prior knowledge of differential algebra is needed for the talk, although, in an effort to answer a question Rota posed in the 1970s, the results are applied to a class of algebras known as RotaBaxter Type algebras, which, with Differential Type algebras, provides examples of linear operators on associative algebras.
This is a preliminary report and joint work with Xing Gao, Li Guo, and Shanghua Zheng. For lecture notes, please click here.
Friday, February 20, 2015, 10:15–11:45 a.m. Room 5382
William Keigher, Rutgers University at Newark
Category Theory Meets the First Fundamental Theorem of CalculusIn recent years, algebraic studies of the differential calculus in the form of differential algebra and the same for integral calculus in the form of RotaBaxter algebra have been merged together to reflect the close relationship between the two calculi through the First Fundamental Theorem of Calculus. In this paper we study this relationship from a categorical point of view in the context of distributive laws. The monad giving RotaBaxter algebras and the comonad giving differential algebras are constructed. Then a mixed distributive law of the monad over the comonad is established. As a consequence, we obtain monads and comonads giving the composite structures of differential and RotaBaxter algebras. This is joint work with Li Guo and Shilong Zhang.
This talk may extend to the afternoon session beginning at 2:00 pm.
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