NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	UNIVERSITY OF ILLINOIS
Initial Amendment Date:	July 5, 1996
Latest Amendment Date:	January 9, 1998
Award Number:	9626856
Award Instrument:	Continuing Grant
Program Manager:	Alvin I. Thaler DMS  Division Of Mathematical Sciences MPS  Directorate for Mathematical and Physical Sciences
Start Date:	July 1, 1996
End Date:	June 30, 1999 (Estimated)
Total Intended Award Amount:	$115,000.00
Total Awarded Amount to Date:	$115,000.00
Funds Obligated to Date:	FY 1996 = $35,000.00 FY 1997 = $45,000.00 FY 1998 = $35,000.00
History of Investigator:	David Marker (Principal Investigator) marker@uic.edu
Recipient Sponsored Research Office:	University of Illinois at Chicago 809 S MARSHFIELD AVE M/C 551 CHICAGO IL  US  60612-4305 (312)996-2862
Sponsor Congressional District:	07
Primary Place of Performance:	University of Illinois at Chicago 809 S MARSHFIELD AVE M/C 551 CHICAGO IL  US  60612-4305
Primary Place of Performance Congressional District:	07
Unique Entity Identifier (UEI):	W8XEAJDKMXH3
Parent UEI:
NSF Program(s):	FOUNDATIONS
Primary Program Source:	app-0196  app-0197  app-0198
Program Reference Code(s):	OTHR, 0000
Program Element Code(s):	126800
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

9626856 Marker In studying the real numbers with exponentiation, van den Dries, Macintyre and Marker gave an algebraic construction of a nonstandard model. This model has proved useful in understanding the asymptotic behavior of functions. Marker's work concentrates on the relationship between formally solving differential equations in this model and the problem of finding actual solutions at infinity. Marker will also work on problems in the model theory of differential fields. This is a fascinating area, requiring a sophisticated mixture of ideas from model theory, differential algebra and algebraic geometry. (Through the work of Buium and Hrushovski, differential algebra has had significant number theoretic applications.) Marker studies the geometry of solutions to algebraic differential equations, concentrating on the use of model theoretic dimensions to develop intersection theory. Model theory is a branch of mathematical logic where one studies mathematical structures by looking at their properties expressed in formal languages. For example, in the real numbers one can say that every number is a cube, but one cannot say that every bounded set has a least upper bound. Nonstandard models of the real numbers share all their formal properties. Studying nonstandard models often leads to fresh insights into the real numbers themselves. Recently, model theorists have been studying the exponential function. These investigations have led to important new insights into the geometry of solution sets of equations involving exponentials. This work has already found applications in analysis, control theory and neural networks. ***

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