Special day in differential and difference algebra, 22nd April 2014, 9am to 6pm, Simons Auditorium.
The videos of the talks are available here.
A. Buium, L. Di Vizio, C. Hardouin, R. Moosa, T. Scanlon and M. Singer.
9:30 - 10:20 am: Alexandru Buium (U. New Mexico) Curvature on the integers The talk explains how one can introduce an arithmetic analogue of connection and curvature
on principal bundles (over flat space). In the arithmetic framework functions on manifolds are replaced by integer numbers, partial derivatives are replaced by Fermat quotients with respect to different primes, and linear connections with values in classical Lie algebras are replaced by non-linear objects naturally attached to the corresponding classical groups.
Our main results are vanishing/non-vanishing theorems for the curvatures of these connections.
This is joint work in progress, with Malik Barrett.
10:30 - 11:20 am: James Freitag (UC Berkeley) Superstability and central extensions of algebraic groups
Altinel and Cherlin proved that any perfect central extension of an algebraic group over an algebraically closed field which happens to be of finite Morley rank as a group is actually a finite central extension and is itself an algebraic group. We will prove a generalization of their result in the infinite rank setting with an additional hypothesis on the center of the group, while giving an example which shows the necessity of this hypothesis. The inspiration for the work comes from differential algebra; namely, this work answers a question of Cassidy and Singer in the (more general) setting of superstable groups.
11:30 am - 12:20 pm: Rahim Moosa (U. Waterloo) Differential-algebraic tangent spaces and internality to the constants
Motivated on the one hand by phenomena in bimeromorphic geometry (so compact complex manifolds) and on the other by model-theoretic consideration arising from the study of the canonical base property, I asked some years ago whether the Kolchin tangent bundle of a finite dimensional differential-algebraic variety has the property that the restriction to any C-algebraic subvariety is C-algebraic. Here C is the field of constants in a saturated differentially closed field of characteristic zero, and C-algebraic means being generically in finite-to-finite correspondence with the C-points of an algebraic variety over C. Actually, one is interested not only in the Kolchin tangent bundle but also its higher order incarnations appearing in the work of Pillay and Ziegler. In recent work with Zoé Chatzidakis and Matthew Harrison-Trainor, we show that while the answer to the question as stated is no, an appropriate generic formulation has a positive answer. I will give motivations for the problem and discuss some aspects of its solution.
2:00 - 2:50 pm: Thomas Scanlon (UC Berkeley) Algebraic differential equations from covering maps
We show as a consequence of two model theoretic theorems, elimination of imaginaries in differentially closed fields and the Peterzil-Starchenko GAGA theorem, that in wide generality for analytic covering maps π :U → X(C) of algebraic varieties that there are differential constructible functions χ defined on X which "nearly invert'' π in the same way in which the logarithmic derivative nearly inverts the exponential function. As consequences we deduce the finite dimensionality of the Kolchin closure of the Hecke orbits of arbitrary points in moduli spaces of abelian varieties generalizing a theorem of Buium.
3:00 - 3:20pm : Tea break
3:20 -4:05pm: Michael Singer (U. North Carolina) Direct and Inverse Problems for Parameterized Linear Differential Equations
I will give an introduction to a Galois theory for differential equations of the form Y_x = A(x,t)Y where A(x,t) is an mxm matrix with entries that are functions of the principal variable x and the parameter t . The Galois groups in this theory are linear differential algebraic groups, that is, groups of mxm matrices whose entries are functions of t satisfying some fixed set of differential equations. This theory has been successfully used to decide when solutions of such equations satisfy additional differential equations with respect to the parameters and in particular questions of integrability. I will describe recent work of Minchenko, Ovchinnikov and myself concerning the direct problem (calculating the Galois group of a given equation) and the inverse problem (which groups can occur as Galois groups).
4:10 - 4:55pm: Lucia di Vizio (Versailles Saint Quentin) Linear Differential Equations with a discrete parameter: Theory
5 - 5:45pm: Charlotte Hardouin (Toulouse) Linear Differential Equations with a discrete parameter: Applications
(Common abstract): We will discuss an analogous situation to the one described by M. Singer, apart from the fact that we consider a discrete parameter. In this case the theory produce a Galois group, which is a difference scheme and can be non-\sigma-reduced. We will explain how we get around this difficulty and the implications on the theory.