| Kolchin Seminar in Differential Algebra | |
| The Graduate Center 365 Fifth Avenue, New York, NY 10016-4309 General Telephone: 1-212-817-7000 |
During the Spring, 2016 semester, Professors Alice Medvedev (City College) and Alexey Ovchinnikov (Queens College and the Graduate Center) organized a thematic workshop on differential algebra and related topics on April 8–10 and a second one on May 13–15.
Friday, May 13, 2016, Workshop at The Graduate Center
David Marker, University of Illinois at Chicago
Differential Fields—A Model Theorist's View
In his book Saturated Model Theory, Gerald Sacks described differentially closed fields as "the least misleading” example of an Ω-stable theory. His remark was particularly prescient as many interesting model theoretic phenomena arise naturally in differential algebra. Model theory has been strangely effective in both solving and generating questions in differential algebraic geometry. I will survey some aspects of this interaction.
For a review of this talk, please click video.
Rahim Moosa, University of Waterloo, Canada
The Dixmier-Moeglin Problem for D-Varieties
Following Buium, by a D-variety we mean an algebraic variety V over an algebraically closed field k equipped with a regular section s: V→ TV, where TV is the tangent bundle of V. (The category of D-varieties (V, s) is equivalent to the category of finite dimensional differential-algebraic varieties over the constants.) There are natural notions of D-rational map and D-subvariety. Motivated by problems in noncommutative algebra, we are led to ask under what conditions (V, s) has a maximum proper D-subvariety over k. (Model-theoretically, this asks when the generic type is isolated.) A necessary condition is that (V, s) does not admit a nonconstant D-constant, that is, a D-rational map from (V, s) to the affine line equipped with the zero section. When is this condition sufficient? I will discuss this rather open-ended problem, including some known cases.
This is a cross-listing from Model Theory Seminar.
For a review of this talk, please click video.
Thomas Scanlon, University of California at Berkeley
Trichotomy Principle for Partial Differential Fields
The Zilber trichotomy principle gives a precise sense in which the structure on a sufficiently well behaved one-dimensional set must have one of only three possible kinds: disintegrated (meaning that there may be some isolated correspondences, but nothing else), linear (basically coming from an abelian group with no extra structure), or algebro-geometric (essentially coming from an algebraically closed field). This principle is true in differentially closed fields when "one dimensional'' is understood as "strongly minimal'' (proven by Hrushovski and Sokolovic using the theory of Zariski geometries and then by Pillay and Ziegler using jet spaces).
In this lecture, I will explain in detail what the trichotomy principle means in differential algebra, how the reduction to the linear case works, and then how one might approach the open problems.
This is a cross-listing from Logic Workshop.
For a review of this talk, please click video-1 and video-2.
Efim Zelmanov, University of California, San Diego
Groups with Identities
We will discuss groups satisfying pro-p and pro-unipotent identities : examples, theory and possible applications.
This is a cross listing from the New York Group Theory Seminar. The lecture will be followed by a wine and cheese reception with group theorists in Room 4214 and dinner at 6:00 pm at Ravagh Persian Grill, at 11 E. 30th St., New York, NY 10016; Tel:212-696-030.
Saturday, May 14, 2016, Workshop at Hunter College, West Building (HW)
Ronnie Nagloo, Bronx Community College (CUNY)
Around Strong Minimality and the Fuchsian Triangle Groups
From the work of Freitag and Scanlon, we now know that the differential equations satisfied by the Hauptmoduls for (genus 0) arithmetic subgroups of SL2(ℤ) are strongly minimal, geometrically trivial and non Ω-categorical. In this talk we look at the problem of showing similar results for Fuchsian triangle groups.
For a review of the talk, please click video.
Carlos Arreche, North Carolina State University
Projectively Integrable Linear Difference Equations and Their Galois Groups
A difference-differential field (of characteristic 0) is a field k equipped with an automorphism σ and a derivation δ that commute with each other. A linear difference equation is integrable if its solutions also satisfy a linear differential system of the same size. The difference equation is projectively integrable if it becomes integrable "modulo scalars". Based on recent results of R. Schaefke and M. Singer, we show that when k = C(x) and σ is either a shift, q, or Mahler operator, the difference-differential Galois group G attached to a projectively integrable difference equation has a very special form. These results have applications for the direct problem of computing G for a given linear difference equation, as well as for the inverse problem of deciding which linear differential algebraic groups occur as difference-differential Galois groups for such equations. This is joint work with Michael Singer.
For a review of the talk, please click video.
James Freitag, University of California at Berkeley
Integrability and Co-order One Subvarieties
Given an affine differential variety V over some finitely generated differential field K, if there is a (nonconstant) map over K from the variety to the constants, then there is an infinite family of co-order one subvarieties of V over K given by the fibers of the map. Surprisingly, there is a strong converse to this result: if V has infinitely many co-order one subvarieties over K, then there is a map over K from V to the constant field. Special cases of this result have been previously proven by Hrushovski and (in slightly different language) Jouanolou and Ghys. Besides being a problem of foundational interest in differential algebra, the result has various applications, which we will touch on. Bell, Lanois, Leon-Sanchez, and Moosa proved a special case of the result and used it to solve open problems in Poisson algebra. We will explain how to use the above general form of the theorem to answer an open problem of Hrushovski and Scanlon. This is joint work with Rahim Moosa.
For a review of the talk, please click video.
Organized (or informal) collaborations and discussions, including mentoring, to be followed by Pizza afterwards.
Sunday, May 15, 2016, Workshop at Hunter College, West Building (HW)
Alexandru Buium, University of New Mexico
Arithmetic Differential Geometry
The aim of this talk is to explain how one can develop an arithmetic analogue of classical differential geometry; this analogue will be referred to as arithmetic differential geometry. In this new geometry, the ring of integers will play the role of a ring of functions on an infinite dimensional manifold; the role of coordinate functions on this manifold will be played by the prime numbers; the role of partial derivatives of functions with respect to the coordinates will be played by the Fermat quotients of integers with respect to the primes; the role of metrics (respectively 2-forms) will be played by symmetric (respectively antisymmetric) matrices with integral coefficients; and the role of connection (respectively curvature) attached to metrics or 2-forms will be played by certain adelic (respectively global) objects attached to matrices as above. One of the main conclusions of our theory will be that (the "manifold" corresponding to) the ring of integers is "intrinsically curved;" the study of this curvature is then one of the main tasks of the theory.
For a review of the talk, please click video.
Gleb Pogudin, Moscow State University
On the Existence of Surjective Projection of a d-Dimensional Differential Algebraic Variety onto 𝔸d
Noether normalization lemma is a standard tool in commutative algebra and algebraic geometry. Stated geometrically, it says that for every affine algebraic variety of dimension d, there exists a surjective map onto the d-dimensional affine space.
We prove a differential analogue of this fact. More precisely, we prove that for every differential algebraic variety of dimension d, there exists a surjective map onto the affine d-dimensional space.
For a review of the talk, please click video.
Malik Barrett, University of New Mexico
The Search for Arithmetic Holonomy
In arithmetic differential geometry, we view the spectrum of a ring of integers as an arithmetic analogue of an infinite dimensional manifold with directions given by p-derivations indexed by odd primes p. We then consider the affine group scheme GLn as a principle bundle equipped with a remarkable connection. One question, among many, that can be asked is: what is the analogous notion of holonomy? In this talk I’ll briefly review the above foundation, as well as the classical relationship between monodromy, Galois representations, and holonomy. I’ll conclude by discussing the current state of an arithmetic description of holonomy.
For a review of the talk, please click video.
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