Kolchin Seminar in Differential Algebra
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Academic Year 2014–2015
May 22, 2015 was our last meeting of the year. Please note that the 6th International Workshop in Differential Algebra and Related Topics (DART VI, 2015) will be held August 10–14, 2015 in Beijing. Have a great summer! KSDA will resume in September, 2015.

Last updated on May 26, 2015. For Schedules, lecture notes and additional material, see under (or click):
 •  Current Schedule  •  Spring, 2015   •   Past Lectures–Spring, 2015  •  Past Years

Friday, May 22, 2015, 10:15–11:45 a.m. Room 5382

Lou van den Dries, University of Illinois at Urbana-Champaign
Differential-Henselian Fields

I will discuss valued differential fields. What parts of valuation theory go through for these objects, under what conditions? Is there a good differential analogue of Hensel's Lemma? Is there a reasonable notion of differential-henselization? What about differential-henselianity for systems of algebraic differential equations in several unknowns?

I will mention results as well as open questions. The results are part of joint work with Matthias Aschenbrenner and Joris van der Hoeven, and have turned out to be useful in the model theory of the valued differential field of transseries.

For a review of the talk, please click video.

Friday, May 22, 2015, 12:30–1:45 p.m. Room 6417
This is a cross-listing from Model Theory Seminar.

Matthias Aschenbrenner, University of California, Los Angeles
Model-Completeness of Transseries

The concept of a “transseries” is a natural extension of that of a Laurent series, allowing for exponential and logarithmic terms. Transseries were introduced in the 1980s by the analyst Écalle and also, independently, by the logicians Dahn and Göring. The germs of many naturally occurring real-valued functions of one variable have asymptotic expansions which are transseries. Since the late 1990s, van den Dries, van der Hoeven, and myself, have pursued a program to understand the algebraic and model-theoretic aspects of this intricate but fascinating mathematical object. Last year we were able to make a significant step forward, and established a model completeness theorem for the valued differential field of transseries in its natural language. My goal for this talk is to introduce transseries without prior knowledge of the subject, and to explain our recent work.

For a review of the talk, please click video.

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 north-east corner.

Past Lectures, Spring 2015

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 Fields

We 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 Algebra

Term-rewriting systems are an essential part of symbolic computations in algebra (including differential algebra and Rota-Baxter algebra). We introduce a class of term-rewriting 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 Rota-Baxter 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 Calculus

In recent years, algebraic studies of the differential calculus in the form of differential algebra and the same for integral calculus in the form of Rota-Baxter 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 Rota-Baxter 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 Rota-Baxter algebras. This is joint work with Li Guo and Shilong Zhang.

This talk may extend to the afternoon session beginning at 2:00 pm.

Friday, February 27, 2015, 10:15–11:45 a.m. Room 5382

Annette Maier, Technische Universität Dortmund, Germany
Computing Difference Galois Groups over 𝔽q(s,t)

We consider linear difference equations σ(y) = Ay over (𝔽q(s,t), σ), where σ(s) = sq and σ acts trivially on 𝔽q(t). The difference Galois group G of such an equation is a linear algebraic group defined over 𝔽q(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 (𝔽q(s,t), σ), namely every semisimple, simply-connected linear algebraic group H defined over 𝔽q is a difference Galois group over (𝔽qi(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(𝔽q) is a Galois group over 𝔽q(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.]

For a video of this presentation, please click here.

Friday, March 6, 2015, 10:15–11:45 a.m. Room 5382

Michael Wibmer, RWTH Aachen University, Germany
Difference Algebraic Groups

Difference 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.

For a copy of the slides, please click Slides.
For a review of the lecture, please click Video.

Friday, March 13, 2015, 10:15–11:45 a.m. Room 5382

Gal Binyamini, University of Toronto
Bezout-type Theorems for Differential Fields

We consider the following problem: given a set of algebraic conditions on an n-tuple of functions and their first 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 singly-exponential with respect to n and and have the natural asymptotic with respect to the degrees or Newton polytopes. This result sharpens previous doubly-exponential 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.

For a review of the lecture, please click Video.

Friday, March 20, 2015, 10:15–11:45 a.m. Room 5382

Wai-Yan Pong, California State University Dominguez Hills
Applications of Differential Algebra to Algebraic Independence of Arithmetic Functions

We generalize and unify the proofs of several results of algebraic independence of arithmetic functions using a theorem of Ax on differential Schanuel conjecture. Along the way of we investigation, we found counter-examples to some results in the literature.

For a copy of the slides, please click slides.

Friday, March 27, 2015, 10:15–11:45 a.m. Room 5382

James Freitag, University of California at Berkeley
On the Existence of Differential Chow Varieties

Chow varieties are a parameter space for cycles of a given variety of a given codimension and degree. We construct their analog for differential algebraic varieties with differential algebraic subvarieties, answering a question of Gao, Li and Yuan. The proof uses the construction of classical algebro-geometric Chow varieties, the model theory of differential fields, the theory of characteristic sets of differential varieties, the theory of prolongation spaces, and the theory of differential Chow forms. This is joint work with Wei Li and Tom Scanlon.

For a review of the talk, please click video.

Fridays, April 3 and 10, 2015, No Seminar (Spring Recess)

Friday, April 24, 2015, 10:15–11:45 a.m. Room 5382

William Simmons, University of Pennsylvania
A Differential Algebra Sampler

We discuss several problems involving differential algebraic varieties and ideals in differential polynomial rings. The first one is the completeness of projective differential varieties. We consider examples showing the failure to generalize (even in the finite-rank case) of the classical "fundamental theorem of elimination theory". We also treat identification of complete differential varieties and a connection to the differential catenary problem. We finish by examining what proof-theoretic techniques have to say about the constructive content of results such as the Ritt-Raudenbush basis theorem and differential Nullstellensatz. Our remarks include work with James Freitag and Omar León-Sánchez as well as ongoing work with Henry Towsner.

For a review of the talk, please click video.
For a copy of the revised slides, please click slides.

Friday, May 1, 2015, 10:15–11:45 a.m. Room 5382

Omar Sanchez, McMaster University
A Differential Hensel's Lemma for Local Algebras

We will discuss a differential version of the classical Hensel's lemma on lifting solutions from the residue field (working on a local artinian differential algebra over a differentially closed field). We will also talk about some generalizations; for example, one can remove the locality hypothesis by assuming finite dimensionality. If time permits, I will give an easy application on extensions of generalized Hasse-Schmidt operators. This is joint work with Rahim Moosa.

For a review of the talk, please click video.

Friday, May 8, 2015, 10:15–11:45 a.m. Room 5382

Abraham D. Smith, Fordham University
The Variety of Involutive Differential Systems via Guillemin Form

A PDE (or an exterior differential ideal) is called "involutive" when it admits analytic families of analytic solutions. The curious property of involutivity has proven very useful in analysis, geometry, and homological algebra. By writing the property explicitly in terms of algebras of almost-commuting matrices, we obtain the *ideal* of the variety of involutive PDEs. This should allow us to study the moduli of involutive PDEs and to uncover invariant structures beyond the basic notions of elliptic, hyperbolic, and parabolic. This topic is ripe for productive collaboration between differential geometers and computational algebraists. (References: 1410.6947 and 1410.7593.)

For a review of the talk, please click video.

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

Victor Kac, Massachusetts Institute of Technology
Non-commutative Geometry and Non-commutative Integrable Systems

In order to develop a theory of non-commutative integrable systems, one needs to develop such notions as non-commutative vector fields and evolutionary vector fields, non-commutative de Rham and variational complexes, non-commutative Poisson algebras and Poisson vertex algebras, etc. I will discuss these notions in my talk.

For a review of the talk, please click video.

Friday, May 15, 2015, 2:00 p.m.–3:30 p.m. Room 5382

Laurent Poinsot, Computer Science Laboratory of Paris-North University (LIPN) and French Air Force Academy (CReA)
Jacobi Algebras, in-between Poisson, Differential, and Lie Algebras

In the non-differential setting there is a functorial relation between Lie algebras and associative algebras: any algebra becomes a Lie algebra under the commutator bracket, and, conversely, to any Lie algebra is attached a universal associative envelope. In the realm of differential algebras, there are two such adjoint situations. The most obvious is obtained by lifting the above correspondence to differential algebras. The second connection, on the contrary, is proper to the differential setting. Any commutative differential algebra admits the Wronskian bracket [x, y] := xy' −x'y as a Lie bracket, and to any Lie algebra is provided a universal differential and commutative associative envelope.

A natural question is to know under which conditions a given Lie algebra embeds into its differential envelope. While an answer is known—by the Poincaré-Birkhoff-Witt theorem—for the non-differential setting, there is yet no such solution in the differential case. In the first part of this talk, after having briefly recalled the above construction, I will present some classes of Lie algebras for which the canonical map to their differential algebra is one-to-one.

Note that differential commutative algebras not merely are Lie algebras, but, with help of their Wronskian bracket, also Lie-Rinehart algebras [1], the algebraic counterpart of a Lie algebroid. However, this Lie-Rinehart structure on a differential commutative algebra is just a consequence of a more abstract structure, namely that of a Jacobi algebra. A Jacobi algebra [2] is a commutative algebra A together with a Lie bracket [-,-] (called the Jacobi bracket) which satisfies the following version of the Leibniz rule:

[ab, c] = a[b, c] + b[a, c] − ab[1A, c],   for all a, b, c in A.

A Jacobi bracket provides a derivation and an alternating biderivation. Hence forgetting one or the other of those differential operators provides a differential or a Poisson algebra, and these relations are functorial.

In the second part of the talk I will present some of the functorial relations between Jacobi, differential, and Lie algebras, such as, e.g., the Jacobi envelope of a Lie algebra. I will also explain that the Lie algebra of global smooth sections of a line bundle E over a smooth manifold M (i.e., a vector bundle over M each fibre of which is one-dimensional) embeds, when E is trivial, into its Jacobi envelope.

[1] G. S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108, pp. 195–222 (1963).
[2] J. Grabowski, Abstract Jacobi and Poisson structures. Quantization and star-products, Journal of Geometry and Physics 9, pp. 45–73 (1992).

For a copy of the slides, please click slides.
For a review of the talk, please click video. Please disregard the comment by Sit beginning at 1:25:10 about the Jacobi algebra axiom: For a fixed c, the operator [−,c] is not an operator of differential type with parameter [1A, c]. The axiom for an operator d of differential type with parameter λ  is d(xy)=xd(y)+ y(x)d + λd(x)d(y), not d(xy)=xd(y)+ y(x)d + λxy.

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