Kolchin Seminar in Differential Algebra

Academic Year 2014–2015 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   2015–2016   2016–2017   2017–2018   2018–2019   2019–Fall

Sad News:  On June 17, 2014, Professor Richard M. Cohn passed away at the age of 94. The New York Times published a brief obituary on June 22, 2014. Memorial Service at 11am, July 24, 2014 at Riverside Memorial Chapel, 180 W. 76th Street, NYC. William Sit, who delivered an eulogy on behalf of KSDA, Phyllis Cassidy, William Keighter, John Miller, and John Nahay attended the funeral service.

Good News: At a lunch meeting on September 5, 2014, the organizers unanimously welcomed Alice Medvedev of CCNY to join the Organizing Committee.

Alerts:

• Applications of Computer Algebra (ACA) 2014 was held at Fordham University, New York, from July 9 through 12, 2014. The conference was organized by Robert H. Lewis (General and Local Chair), Tony Shaska and Illias Kotsireas (Program Co-chairs) with an Advisory Committee composed of Eugenio Roanes-Lozano, Stanly Steinberg,and Michael Wester. Interested parties were invited to organize special sessions. Alexey Ovchinnikov organized a Special Session on Computational Differential and Difference Algebra.

Friday, September 5, 2014, 1:00 –2:30 p.m. Room 5382

Richard Churchill, Graduate Center and Hunter College, CUNY
Kolchin's Proof that Differential Galois Groups are Algebraic

Kolchin's 1948 proof that differential Galois groups are algebraic appeals to a formulation of algebraic geometry which is no longer in fashion. Modern proofs require considerable knowledge of the Grothendieck approach to that subject, which takes considerable time to digest. In this talk I hope to convince those attending that Kolchin's proof can be understood in contemporary terms with only minor appeals to algebraic geometry, i.e. the definition of an algebraic set and the Hilbert Basis Theorem.

For lecture notes, please click here.

Friday, September 12, 2014, 12:30 –2:00 p.m. Room 5382

James Freitag, University of California at Berkeley
Effective Bounds For Finite Differential-Algebraic Varieties (Part I)

Given a differential algebraic variety over a partial differential field, can one give a bound for the degree of its Zariski closure that depends only on the order and degree of the differential polynomials (but not the parameters) which determine the variety? We will discuss the general theory of prolongations of differential algebraic varieties as developed by Moosa and Scanlon, and use this theory to reduce the problem to a combinatorial problem (which will be discussed in detail in the second part of the talk). Along the way we will give numerous examples of the usefulness of the result, some of an arithmetic flavor. We will also describe some other applications of the theory of prolongations.
This is joint work with Omar Sanchez. For a video recording of the talk, please click here.

Friday, September 12, 2014, 2:15 –3:45 p.m. Room 5382

Omar Leon Sanchez, McMaster University
Effective Bounds For Finite Differential-Algebraic Varieties (Part II)

We will talk about the difficulties that commutativity entails when trying to find points of the form (a, d₁(a),...,d_m(a)) in algebraic subvarieties of prolongations. We discuss how to deal with these issues by passing to higher order prolongations (where the order is "uniform"). We use this to establish effective bounds for finite differential-algebraic varieties.
This is joint work with James Freitag. For a video recording of the talk, please click here.

Friday, September 19, 2014, 12:30 –2:00 p.m. Room 5382

Ronnie Nagloo, Graduate Center of CUNY
Model Theory and the Painlevé Equations

The Painlevé equations are nonlinear 2^nd order ODEs and come in six families P₁, …, P₆, where P₁ consists of the single equation y''=6y²+t, and P₂, …, P₆ come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20^th century but have arisen in a variety of important physical applications, including for example random matrix theory and general relativity.
To view the video recording of this talk, please click here and here.

Tuesday (Friday Schedule), September 23, 2014, 12:30 –2:00 p.m. Room 5382

Raymond Hoobler, CCNY and Graduate Center of CUNY
A Differential Poincaré Lemma

Let X be a smooth scheme over a field k of characteristic 0. I will show that the complex
0 →  O^Δ_X  → O_X  → Ω¹ _X/k → …
constructed from the de Rham complex of X is exact in the d-finite topology if Δ consists of all "derivations" of X. In particular, this means that the d-finite cohomology of a variety matches the singular cohomology of X with complex coefficients and agrees with the étale cohomology of X with torsion coefficients. Most of the talk will be an outline of the necessary steps that lead up to this result. Potential applications will be described.
To view a video recording of this talk, please click here.

Friday, September 26 and October 3, 2014, School Holidays, no seminar.

Thursday,  October 2, 2014, 1:00 –2:00 p.m. Room 6/113 North Academic Center, CITY COLLEGE

David Marker, University of Illinois at Chicago
Model Theory and Exponentiation

Methods from mathematical logic have proved surprisingly useful in understanding the geometry and topology of sets definable in the real field with exponentiation. When looking at the complex exponential field, the definability of the integers is a seemingly insurmountable impediment, but a novel approach due to Zilber leads to a large number of interesting new questions.
This is a cross-listing from Model Theory Seminar and the CCNY Mathematics Colloquium. You are welcome to join the speaker for lunch at noon in the Faculty Dining Hall on the 3rd Floor of the North Academic Center, CCNY.

Friday, October 10, 2014, 12:30 –1:45 p.m. Room 5382

Ronnie Nagloo, Graduate Center of CUNY
Geometrically Trivial Strongly Minimal Sets in DCF₀

In this talk we look at the problem of describing the `finer' structure of geometrically trivial strongly minimal sets in DCF₀. In particular, I will talk about the ω-categoricity conjecture, recently disproved in its general form by James Freitag and Tom Scanlon, and the unimodularity conjecture, a weakening of the above conjecture and which came to life after the work on the second Painlevé  equations.

Friday, October 10, 2014, 2:00 –3:30 p.m. Room 6417

Alice Medvedev, City College, CUNY
Model Theory of Difference Fields, Part I

I'll begin by setting up the first-order language and axioms of difference fields, and give some interesting examples, including Frobenius automorphisms of fields in positive characteristic and difference equations from analysis that give the subject its name. Difference-closed fields, a natural analog of algebraically closed fields, have a nice model theory, starting with almost-quantifier elimination. Further model-theoretic notions—algebraic closure, elementary equivalence, forking independence—all have elementary purely algebraic characterizations that I will explain. The model theory of difference fields has been used in arithmetic geometry in several exciting ways (Hrushovski's results on the Manin-Mumford Conjecture; his twisted Lang-Weil estimates; several people's work on algebraic dynamics) that I will probably not explain in detail.

This is a cross-listing from CUNY Logic Workshop. This talk will be continued in the Model Theory Seminar the following week.

Friday, October 17, 2014, 10:45 –12:15 p.m. Room 5382

Alice Medvedev, City College, CUNY
Model Theory of Difference Fields, Part II

This is a continuation of my talk at the CUNY Logic Workshop last week. The necessary background will be summarized briefly at the beginning of this one.

ACFA, the theory of difference closed fields, is a rich source of explicit examples showing forking independence and nonorthogonality, the distinction between stable and simple theories, and the distinction between one-based and locally-modular groups. To present these examples, I will introduce difference varieties, which are the basic building blocks of definable sets in ACFA, and sigma-varieties, a more tractable special case of these.

This is a cross-listing from Model Theory Seminar.

Friday, October 17, 2014, 2:00 –3:30 p.m.  Room 6417

Ronnie Nagloo, Graduate Center of CUNY
On Transformations in the Painlevé Family

The Painlevé equations are nonlinear 2^nd order ODEs and come in six families P₁, …, P₆, where P₁ consists of the single equation y''=6y²+t, and P₂, …, P₆ come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20^th Century but have arisen in a variety of important physical applications. In this talk I will explain how one can use Model Theory to answer the question on existence of algebraic relations among solutions of different Painlevé equations from the families P₁, …, P₆.

This is a joint seminar with CUNY Logic Workshop.

Friday, October 24, 2014, 12:30 –2:00 p.m. Room 5382

Li Guo, Rutgers University at Newark
Rota-Baxter Operators on Polynomial Algebras, Integration, and Averaging 0perators

The Rota-Baxter operator is an algebraic abstraction of integration. Following this classical connection, we study the relationship between Rota-Baxter operators and integration in the case of the polynomial ring k[x], where both concepts make sense. We consider two classes of Rota-Baxter operators: monomial ones and injective ones. For the first class, we apply averaging operators to determine monomial Rota-Baxter operators. For the second class, we make use of the double product on Rota-Baxter algebras.
If time permits, we will talk more about averaging operators, which crop up naturally from this study.
This is a joint work with Markus Rosenkranz and Shanghua Zheng.

For a copy of the slides, please click here.

Friday, October 31, 2014, 12:30 –1:45 p.m. Room 5382

Anand Pillay, University of Notre Dame
Interpretations and Differential Galois Extensions

We prove a number of results around finding strongly normal extensions of a differential field K, sometimes with prescribed properties, when the constants of K are not necessarily algebraically closed. The general yoga of interpretations and definable groupoids is used (in place of the Tannakian formalism in the linear case).
This is joint work with M. Kamensky.
For a video recording of this talk, please click here.

Friday, October 31, 2014, 2:00 –3:30 p.m. Room 6417

Anand Pillay, University of Notre Dame
Mordell-Lang and Manin-Mumford in Positive Characteristic, Revisited

We give a reduction of function field Mordell-Lang to function field Manin-Mumford, in positive characteristic. The upshot is another account of or proof of function field Mordell-Lang in positive characteristic, avoiding the recourse to difficult results on Zariski geometries.
This work is joint with Benoist and Bouscaren.

This is a cross listing from CUNY Logic Workshop.

Friday, November 7, 2014, 12:30 –2:00 p.m. Room 5382

Michael Singer, North Carolina State University
The General Solution of A First Order Differential Polynomial

This is the title of a 1976 paper by Richard Cohn in which he gives a purely algebraic proof of a theorem (proved analytically by Ritt) that gives a bound on the number of derivatives needed to find a basis for the radical ideal of the general solution of such a polynomial. I will show that the method introduced by Cohn can be used to give a modern proof of a theorem of Hamburger stating that a singular solution of such a polynomial is either an envelope of a set of solutions or embedded in an analytic family of solutions depending on whether or not it corresponds to an essential singular component of this polynomial. I will also discuss the relation between this phenomenon and the Low Power Theorem. This will be an elementary talk with all these terms and concepts defined and explained.
This is joint work with Evelyne Hubert.
For a video recording of this talk, please click here.

Friday, November 14, 2014, 12:30 –2:00 p.m. Room 5382

Rahim Moosa, University of Waterloo
Differential Varieties with Only Algebraic Images

Consider the following condition on a finite-dimensional differential-algebraic variety X: whenever X→Y is a dominant morphism, and dim(Y) < dim(X), then Y is (a finite cover of) an algebraic variety in the constants. This property is a specialisation to differentially closed fields of a model-theoretic condition that itself arose as an abstraction from complex analytic geometry. Non-algebraic examples can be found among differential algebraic subgroups of simple abelian varieties. I will give a characterisation of this property that involves differential analogues of "algebraic reduction" and "descent". This is joint work with Anand Pillay.

For a partial video recording of this talk, please click here.

Friday, November 14, 2014, 2:15 –3:45 p.m. Room 5382

Uma Iyer, Bronx Community College, CUNY
Weight Modules of an Algebra of Quantum Differential Operators

Generalized Weyl Algebras (GWAs) were independently introduced by V. Bavula and A. Rosenberg. These algebras have been widely studied. In particular, weight modules over the GWAs have been also studied. We study weight modules over a particular algebra of quantum differential operators which contains a GWA of rank 1. This is joint work with V. Futorny.

For a video recording of this talk, please click here.

Saturday, November 15, 2014, 11:00 a.m.–12:30 p.m. Hunter College Rm HW217

Andrey Minchenko, Weizmann Institute
On a Problem of Computing Parameterized Picard-Vessiot Group

We will deal with the problem of computing parameterized Galois groups of differential equations. At present, one can determine whether the group is unipotent or reductive, and compute the group algorithmically in both cases. We will consider the obstacles we face in our attempt to solve the general case (when the group is neither unipotent nor reductive) and explain how some of them may be dealt with.

Friday, November 21, 2014, 12:30 –2:00 p.m. Room 5382

Joseph Gunther, Baruch College and the Graduate Center, CUNY
Difference Algebraic Geometry

We'll examine the foundations of scheme-style difference algebraic geometry, as developed by Hrushovski to prove a generalization of the Lang-Weil estimates for the number of points of a variety over finite fields. This means working over not just an arbitrary ring, but over an arbitrary ring with a self-map. We'll consider constructions and facts from standard algebraic geometry, and see how they work (or don't) in the difference algebra setting.

Friday, November 28, 2014, Thanksgiving Week, no seminar.

Friday, December 5, 2014, 12:30 –2:00 p.m. Room 5382

Carlos Arreche, North Carolina State University
On the Computation of the Difference-Differential Galois Group for a Second-Order Linear Difference Equation

Consider the difference field C(x) with automorphism φ: x ↦ x +1, and assume that the field of constants C is algebraically closed and of characteristic zero. The difference Galois theory of van der Put and Singer associates a linear algebraic group over C to a linear difference equation with respect to φ. This Galois group measures the algebraic relations amongst the solutions to the difference equation. There is an algorithm to compute the difference Galois group corresponding to a second-order linear difference equation over C(x), due to Hendriks. More recently, Hardouin and Singer have developed a Galois theory for difference-differential equations that associates a linear differential algebraic group to a linear difference equation, and this Galois group measures the differential-algebraic relations amongst the solutions. In this talk, I will describe ongoing work towards extending Hendriks' algorithm to compute the difference-differential Galois group associated to a second-order linear difference equation over C(x).

Friday, December 12, 2014, 12:30 –2:00 p.m. Room 5382

Wei Li, KLMM, Chinese Academy of Sciences, and University of California at Berkeley
Sparse Difference Resultant

In this talk, we first define sparse difference resultant for a Laurent transformally essential system of difference polynomials and give a simple criterion for the existence of sparse difference resultant which requires only linear algebraic techniques. Then we discuss the basic properties of the sparse difference resultant, in particular, give its order bound in terms of the Jacobi number and degree bound. We show the projective difference space is not transformally complete. If time permits, as a special case, we introduce the difference resultant and give the precise order and degree, a determinant representation, and a Poisson-type product formula for the difference resultant.
This is joint work with Xian-Shan Gao and Chun-Ming Yuan.

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.

For a copy of the slides, please click here.

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, simply-connected 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.]

For a video of this presentation, please click video-1 and video-2.

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-1 and video-2.

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 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 singly-exponential with respect to n and ℓ 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-1 and video-2.

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 review of the talk, please click video and 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[1_A, 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.

References:
[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 [1_A, 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.

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.

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