Simmons_William.pdf (743.96 kB)

Completeness of Finite-Rank Differential Varieties

Download (743.96 kB)
thesis
posted on 24.10.2013, 00:00 by William D. Simmons
Differential algebraic geometry offers tantalizing similarities to the algebraic version as well as puzzling anomalies. This thesis builds on results of Kolchin, Blum, Morrison, van den Dries, and Pong to study the problem of completeness for projective differential varieties. The classical fundamental theorem of elimination theory asserts that if V is a projective algebraic variety defined over an algebraically closed field K and W is any algebraic variety defined over K, then the projection VxW -> W takes Zariski-closed sets to Zariski-closed sets. Differential varieties defined by differential polynomial equations over a differentially closed field are more complicated. We give the first example of an incomplete finite-rank differential variety, as well as new instances of complete differential varieties. We also explain how model theory yields multiple versions of Pong's valuative criterion for completeness and reduces the differential completeness problem to one involving algebraic varieties over the complex numbers.

History

Advisor

Marker, David

Department

Mathematics, Statistics, and Computer Science

Degree Grantor

University of Illinois at Chicago

Degree Level

Doctoral

Committee Member

Baldwin, John Goldbring, Isaac Verschelde, Jan Wood, Carol

Submitted date

2013-08

Language

en

Issue date

24/10/2013