Mei-Chu ChangProfessor of Mathematics, EmeritusMathematics Dept mcch@ucr.edu(951) 827-5094
Arithmetic combinatorics and applications to number theory
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AWARD NUMBER
008401-002
FUND NUMBER
33276
STATUS
Closed
AWARD TYPE
3-Grant
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AWARD EXECUTION DATE
7/14/2016
BEGIN DATE
7/15/2016
END DATE
6/30/2018
AWARD AMOUNT
$125,000
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Sponsor Information
SPONSOR AWARD NUMBER
SPONSOR
SPONSOR TYPE
FUNCTION
Organized Research
PROGRAM NAME
Proposal Information
PROPOSAL NUMBER
16030206
PROPOSAL TYPE
New
ACTIVITY TYPE
Basic Research
PI Information
PI
Chang, Mei-Chu
PI TITLE
Other
PI DEPTARTMENT
Mathematics
PI COLLEGE/SCHOOL
College of Nat & Agr Sciences
CO PIs
Project Information
ABSTRACT
This research project concerns arithmetic combinatorics, an interdisciplinary field of research with many emerging applications. Progress on questions in number theory and theoretical computer science requires new insights and methods; part of the success in this direction over recent years relies on novel techniques at the interface of algebra and combinatorics. Central to these developments is the so-called arithmetic combinatorics of finite fields, which has undergone significant recent advances and opened new challenges. This research project focuses on several questions motivated by current research in the area, including study of the orders of points on varieties and counting solutions to algebraic equations with constraint variables.
Combinatorial problems in finite fields continue to offer many challenges. Of particular interest in this research project are questions involving orders of points on varieties over finite fields (e.g. recent developments related to the Markoff surface). Part of the motivation for the work is to investigate the problem of strong approximation for Markoff triples and to give estimates on the number of solutions of equations when the variables are restricted one way or another, for instance to multiplicative groups. When classical techniques do not apply, general sum product theory in finite fields and residue rings may be useful. Sum-product results in various settings are of interest in their own right as they lead to new results in analytic number theory, in particular, estimates on Gauss sums and short character sums.(Abstract from NSF)
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