Kolchin Seminar in Differential Algebra

Current Schedule

Friday, May 10, 2013 at 10:15 a.m. Room 5382

Raymond Hoobler, Graduate Center (CUNY)
Deligne Revisited III: Gauss-Manin Connections and Regular Singular Points

Let (E, ∇ ) be a bundle with an integrable connection on a smooth, not necessarily complete variety X over ℂ. The notion of regular singular points for ∇  is a kind of finiteness condition that readily provides local solutions for the system of differential equation defined by ∇. In previous lectures, I have defined this condition. In this talk, I will show, for a given a smooth map between smooth varieties  f : X → Y, that Rⁱf_*((E, ∇ )) has a natural integrable connection
ℵ : Rⁱf_*((E, ∇ )) → Rⁱf_*((E, ∇ )) ⊗ Ω¹_X/Y, 
which is known as the Gauss-Manin connection. I will sketch a proof that if (E, ∇ ) has only regular singular points, then so does (Rⁱf_*((E, ∇ )), ℵ ).

Friday, May 10, 2013 at 2:00-3:30 p.m. Room 5382

Carlos Arreche, Graduate Center, CUNY
PPV Groups and Differential Transcendence

We will apply parameterized Picard-Vessiot theory to give simple necessary and sufficient criteria for the ∂/∂t-transcendence of the solutions to a parameterized second-order linear differential equation of the form ∂²Y/∂ x²= p ∂Y/∂x, where p∈C(t,x) and C is a field of characteristic zero. These criteria imply, in particular, the differential transcendence of the incomplete Gamma function Γ(t,x) over ℂ(t,x), generalizing a result of Johnson, Reinhart, and Rubel (1995).

Have a good summer vacation, and see you in September or at one of the summer conferences.

Kolchin Seminar in Differential Algebra. KSDA meets most Fridays from 10:15 AM to 11:45 AM at the Graduate Center. The purpose of these meetings is to introduce the audience to differential algebra. The 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. It normally goes from 2:00-5:00 pm (please check with organizers). All are welcome. Unless the contrary is indicated, all morning and afternoon 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, please click here, and for more on security requirements for entering the premise, please click here.

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 2013

Friday, February 1, 2013 at 10:15 a.m. Room 5382

Friday, February 8, 2013 at 10:15 a.m. Room 5382

Richard Churchill, Hunter College and Graduate Center, CUNY
A Set-Theoretic Approach to Model Theory

Although one always employs logic in proofs, the foundations of many branches of mathematics appear to be predominantly set-theoretic: one defines a topological space to be a pair (X, τ) consisting of a set X and a collection τ of subsets satisfying certain well-known properties; one defines a group to be a pair (G, μ) consisting of a set G and a subset μ ⊂ G × G × G  as the binary operation satisfying certain well-known properties (of course, for a group one needs a bit more to handle the identity); etc. There are advantages to this commonality, particularly if one is well-versed in category theory: one can move from one area to the other and still have a fairly good idea of what the major problems are and the sort of techniques one might expect to see. In contrast, in Model Theory, the foundation appears to be heavily based on logic, and as a result the language and terminology can seem foreign to those who work in more widely publicized areas of mathematics. Rather than "sets of groups", one hears about "sets of formulas"; rather than products (Cartesian, fibered, direct, or semi-direct), one hears of "ultraproducts"; rather than "reducing to a simpler case", one is told about "eliminating quantifiers".

In this talk I will indicate how some of the basic ideas of Model Theory can be formulated set-theoretically, that is, in the topological and algebraic spirit indicated above.

For lecture notes and slides, please click Revised as of April 4, 2013.

Friday, February 15, 2013 at 10:15 a.m. Room 5382

Roman Kossak*, Graduate Center, CUNY
On the Existence of Sets

I will review the axioms of ZFC. I will focus on the axiom schema of replacement. I will say a bit about its history and discuss some of its consequences.

*Due to our scheduled speaker falling sick, this talk will be given by David Marker of University of Illinois at Chicago instead.

Friday, February 22, 2013 at 10:15 a.m. Room 5382

William Sit, City College of New York, CUNY
Basics of Dimension in Differential Algebra

This will be a review of dimension concepts in differential algebra, with a goal to relate them to those in Model Theory. I will discuss differential transcendence degree, differential type, typical differential dimension, and the differential dimension polynomial (Kolchin polynomial) of a finitely generated differential field extension and state certain properties. I will sketch a proof of the well-ordering theorem on numerical polynomials that included the Kolchin polynomials. Time permitting, examples will be given to suggest connections between some concepts as introduced by Kontrateva et al., Berline and Lasker, Aschenbrenner and Pong, and others.

For lecture slides, please click here.

Friday, March 1, 2013 at 10:15 a.m. Room 5382

Igor Krichever, Columbia University
Analytic Theory of Difference Equations with Rational Coefficients

For a copy of the lecture slides, please click here.
For a paper by the author related to this topic, please visit author's website or click here.

Friday, March 8, 2013 at 10:15 a.m. Room 5382

Omar Leon Sanchez, University of Waterloo
Differential D-groups and Galois Theory

We present differential D-groups as an extension (to infinite-dimension) of algebraic D-groups. Then we develop the Galois theory of logarithmic equations on differential D-groups. This theory generalizes both the parameterized Picard-Vessiot theory and the differential Galois theory of algebraic D-groups.

This talk will continue at 2:00 pm in the afternoon seminar.

Friday, March 15, 2013 at 2:00 p.m. Room 5382
Please note that this talk starts at 2:00 pm.

Richard Gustavson, Graduate Center (CUNY)
Some Open Problems in Differential Galois Theory

In this talk we look at some open problems in differential Galois theory and their partial solutions. The two main problems we examine are the direct problem and the inverse problem. The direct problem asks whether the differential Galois group for a given system of differential equations exists, and if it does, to construct it. The inverse problem asks if a given group is the Galois group of some system of differential equations. If time permits, we will examine some open problems unrelated to the direct and inverse problems, including questions concerning monodromy groups of differential equations and factorization in the ring of linear differential operators.

Friday, March 22, 2013 at 10:15 a.m. Room 5382

Raymond Hoobler, Graduate Center (CUNY)
Deligne Revisited I: The Riemann-Hilbert Correspondence

Let (E, ∇ ) be a bundle with an integrable connection on a smooth variety U over ℂ. Deligne showed that the analytic de Rham cohomology, Hⁱ_DR(U_an (E, ∇ )), agreed with the algebraic de Rham cohomology, Hⁱ_DR(U_alg, (E, ∇ )) if the connection was regular using Hironaka's resolution of singularities. André and Baldassari have more recently given an entirely algebraic proof of this result by using Artin neighborhoods to reduce it to a one dimensional case. In Part I, I will define Artin neighborhoods and outline the strategy for proving the result which rests on establishing several properties of (E, ∇ ). Part II will be devoted to outlining the proof of one of these properties to be selected at the end of Part I.

Friday, March 29, 2013 Spring Break. No seminar.

Friday, April 5, 2013 at 10:15 a.m. Room 5382

Raymond Hoobler, Graduate Center (CUNY)
Deligne Revisited II: Gauss-Manin Connections and Regular Singular Points

Let (E, ∇ ) be a bundle with an integrable connection on a smooth, not necessarily complete variety X over ℂ. The notion of regular singular points for ∇  is a kind of finiteness condition that readily provides local solutions for the system of differential equation defined by ∇. We will define this condition as well as show, for a given a smooth map between smooth varieties  f : X → Y, that Rⁱf_*((E, ∇ )) has a natural integrable connection
ℵ : Rⁱf_*((E, ∇ )) → Rⁱf_*((E, ∇ )) ⊗ Ω¹_X/Y, 
which is known as the Gauss-Manin connection. We will sketch a proof that if (E, ∇ ) has only regular singular points, then so does (Rⁱf_*((E, ∇ )), ℵ ).

Friday, April 12, 2013 at 10:15 a.m. Room 5382

David Marker, University of Illinois at Chicago
Canonical Definitions in Differentially Closed Fields

Canonical definitions play an important role in modern model theory. In most structures one must add "imaginary elements" to give canonical definitions. Poizat showed that for differentially closed fields this is unnecessary. This is related to the the existence of definable quotients. I will survey the basic definitions, applications, and Poizat's proof.

Friday, April 19, 2013 at 10:15 a.m. Room 5382

Alexander Levin, Catholic University of America
Dimension Quasi-polynomials in Differential and Difference Algebra

In this talk we consider Hilbert-type functions associated with differential and difference field extensions and systems of algebraic differential and difference equations in the case when the basic derivations or translations are assigned some rational weights. We will show that such functions are quasi-polynomials, which can be obtained as linear combinations of Ehrhart quasi-polynomials. In particular, we will obtain generalizations of the theorems on differential and difference dimension polynomials.

Friday, April 26, 2013 at 10:15 a.m. Room 5382

Philipp Rothmaler, Bronx Community College (CUNY)
What is a Type?

I will first remind the audience of the concept of formula in its various disguises: syntactic object, definable set, sentence (i.e., formula with no free variable) in an expansion by constants, subfunctor of the forgetful functor and, element of the so-called Lindenbaum-Tarski algebra. This latter structure is a Boolean algebra whose Stone space (under Stone duality) is the space of (complete) types. Its importance, in turn, lies in the fact that it is a compact space, which is the content of the celebrated Compactness Theorem and which I will explain if time permits.

My foremost goal though is to define and describe types in much plainer terms, namely as intersections of definable sets and, most importantly, orbits under the action of certain automorphism groups. (The disguises corresponding to the above would be: ultrafilter of formulas, type-definable set (=intersection of definable sets), complete theory in an expansion by constants, subfunctor of the forgetful functor (namely, intersections of the aforementioned subfunctors), and element of the Stone space of the L-T algebra.)

I will indicate, in some simple examples, what forking extensions of types are (i.e., how the various Stone spaces associated with models of a complete theory interact). This may lead us to Lascar rank and a topological definition of stability due to Herzog and myself.

Disclaimer: there will be more pictures on the board than rigorous technical statements and—I apologize—differential algebra.

Please note that Philipp Rothmaler will speak also on Mittag-Leffler objects in definable categories of modules at the CUNY Logic Workshop at 2:00 p.m. (Room 6417, Graduate Center). For details, please visit Logic Workshop.

Friday, May 3, 2013 at 10:15 a.m. Room 5382

James Freitag, University of California at Berkeley
An Application of Differential Completeness

We will discuss generalizations of Kolchin's work on complete differential algebraic varieties. In particular, we will examine notions of generalized Wronskians and linear dependence over general differential algebraic varieties.

Friday, May 3, 2013 at 2:00–3:30 p.m. Room 5382

Li Guo, Rutgers University at Newark
On Integro-Differential Algebras

Integro-differential algebras have been introduced recently in the study of boundary problems of differential equations. We generalize these to integro-differential algebras with a weight, in analogy to differential and Rota-Baxter algebras. We construct free commutative integro-differential algebras with weight generated by a differential algebra. This gives in particular an explicit construction of the free integro-differential algebra on one generator. Properties of the free objects are studied.

If time permits, we will discuss another construction of free integro-differential algebras. This alternative uses the method of Groebner-Shirshov bases and is given in a recent joint work with X. Gao and S. Zheng.

The present work is a joint work with G. Regensburger and M. Rosenkranz.

For a copy of the slides presented, please click here.