Blog Archives

Topic Archive: differential algebra

Ronnie Nagloo
CUNY Graduate Center
Dr. Joel (Ronnie) Nagloo studies model theory and differential algebra. He holds a Ph.D. from Leeds University, completed under the supervision of Anand Pillay and Frank Nijhoff. After an initial postdoctoral position at the CUNY Graduate Center, he is now an Assistant Professor in mathematics at Bronx Community College.
Model theory seminarKolchin seminar in Differential AlgebraFriday, September 12, 201412:30 pmGC 5382

James Freitag

Effective Bounds For Finite Differential-Algebraic Varieties (Part I)

University of California, Berkeley

Given a differential algebraic variety over a partial differential field, can one give bounds for the degree of the Zariski closure which depend 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.

NERDS: New England Recursion & Definability SeminarSunday, March 30, 20143:15 pmAcademic Center, Room 113, Olin College

Russell Miller

Spectra of differentially closed fields

City University of New York

The spectrum Spec(M) of a countable structure M is the set of all Turing degrees of structures isomorphic to M. This topic has been the focus of much research. Here we describe the spectra of countable differentially closed fields of characteristic 0: they are precisely the preimages { d : d’ ∈ Spec(G) } of spectra of arbitrary countable graphs G, under the jump operation. To establish this, we describe the proofs of two theorems: one showing how to build the appropriate differential field K from a given graph G, and the other showing that every low model of the theory DCF0 is isomorphic to a computable one. The latter theorem (which relativizes, to give the main result above) resembles the famous result of Downey and Jockusch on Boolean algebras, but the proof is different, yielding a Δ2 isomorphism between the low model and its computable copy; moreover, our first theorem shows that the extension of the result to the low4 case for Boolean algebras does not hold for DCF0.

This is joint work by Dave Marker and the speaker.

Moshe Kamensky
Hebrew University of Jerusalem
Moshe Kamensky is a postdoc in model theory at the Mathematics department of The Hebrew University in Jerusalem, Israel, working with Ehud Hrushovski.
Kolchin seminar in Differential AlgebraFriday, February 21, 201410:15 amGC 5382

Moshe Kamensky

Picard-Vessiot structures

Hebrew University of Jerusalem
Model theory seminarFriday, March 7, 201412:30 pmGC 6417

Russell Miller

Turing degree spectra of differentially closed fields

City University of New York

The spectrum Spec(M) of a countable structure M is the set of all Turing degrees of structures isomorphic to M. This topic has been the focus of much research. Here we describe the spectra of countable differentially closed fields of characteristic 0: they are precisely the preimages { d : d’ ∈ Spec(G) } of spectra of arbitrary countable graphs G, under the jump operation. To establish this, we describe the proofs of two theorems: one showing how to build the appropriate differential field K from a given graph G, and the other showing that every low model of the theory DCF0 is isomorphic to a computable one. The latter theorem (which relativizes, to give the main result above) resembles the famous result of Downey and Jockusch on Boolean algebras, but the proof is different, yielding a Δ2 isomorphism between the low model and its computable copy; moreover, our first theorem shows that the extension of the result to the low4 case for Boolean algebras does not hold for DCF0.

This is joint work by Dave Marker and the speaker. The slides for this talk are available here.

Kolchin seminar in Differential AlgebraFriday, November 15, 201310:15 amGC 5382

David Marker

Logarithmic-Exponential Series

University of Illinois at Chicago

I will survey some old work of van den Dries, Macintyre and myself. We construct an algebraic nonstandard model of the theory of the real exponential field. There is a natural derivation on the LE-series which is compatible with the exponential and the archimedean valuation.

Model theory seminarFriday, October 25, 201312:30 pm

Alf Dolich

LE-Series

The City University of New York

In this expository talk I will discuss the construction of the field of LE-series after van den Dries, Macintyre, and Marker. The field of LE-series is an ordered differential field extending the field of real Laurent series which also has a well-behaved exponential function. The field of LE-series is closed under a host of operations, in particular it is closed under formal integration as well as compositional inverse (once composition has been properly interpreted). As such this field may be viewed, at least conjecturally, as providing a universal domain for ordered differential algebra as witnessed in Hardy fields.

Kolchin seminar in Differential AlgebraFriday, October 18, 201310:15 amGC 5382

Tom Scanlon

D-Fields as a Common Formalism for Difference and Differential Algebra

University of California - Berkeley

In a series of papers with Rahim Moosa, I have developed a theory of D-rings unifying and generalizing difference and differential algebra. Here we are given a ring functor D whose underlying additive group scheme is isomorphic to some power of the additive group. A D-ring is a ring R given together with a homomorphism f : R → D(R). A first motivating example is when D(R) = R[ε]/(ε2), so that the data of D-ring is that of an endomorphism σ:R → R and a σ-derivation ∂:R → R (that is, ∂(rs) = ∂(r)σ(s)+σ(r)∂(s)). Another example is when D(R) = R, where a D-ring structure is given by an endomorphism of R.

We develop a theory of prolongation spaces, jet spaces, and of D-algebraic geometry. With our most recent paper, we draw out the model theoretic consequences of this work showing that in characteristic zero, the theory of D-fields has a model companion, which we call the theory of D-closed fields, and that many of the refined model theoretic theorems (eg the Zilber trichotomy) hold at this level of generality. As a complement, we show that no such model companion exists in characteristic p under a mild hypothesis on D.

David Marker
University of Illinois at Chicago
Professor Marker holds the position of LAS Distinguished Professor in the Department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago. He conducts research in model theory and it applications, particularly in applications to real algebraic geometry and real analytic geometry, exponentiation and differential algebra. His excellent textbook Model Theory: an Introduction is widely studied.
Russell Miller
City University of New York
Russell Miller is professor of mathematics at Queens College of CUNY and also at the CUNY Graduate Center.  He conducts research in mathematical logic, especially computability theory and its interaction with other areas of mathematics, as in computable model theory. He received his doctorate from the University of Chicago in 2000, as a student of Robert Soare, and subsequently held a postdoctoral position at Cornell University until 2003, when he came to CUNY.