Mathematical Sciences: Number Theory, Arithmetic Geometry,  and Transcendence
{'Code': '03040300', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'Gary Cornell', 'PO_EMAI': None, 'PO_PHON': None}
This research is in many areas of number theory.  Montgomery               will pursue a number of basic questions in analytic number theory               and related areas of harmonic analysis, diophantine approximation               and the geometry of numbers.  Most of these questions involve                   sets of numbers or vectors which are almost independent.  The                   object is to exploit the almost independence appropriately in                   order to achieve the desired aim.  Masser will study some                       diophantine problems which lend themselves into the following                   groupings: (a) using transcendence techniques to examine                        isomorphism classes of elliptic curves; (b) to obtain upper                     bounds for the number of points of small height on a fixed                      abelian variety; (c) to establish precise conditions for the                    algebraic independence of values of certain Mahler-type power                   series at algebraic points; and (d) to study the multiplicities                 of linear recurrences for a finite field.  Milne's research                     object is to produce a comprehensive set of reciprocity laws                    describing how automorphisms of the complex numbers act on                      automorphic functions automorphic forms, their special values,                  their "Fourier-Jacobi series, and the Eisenstein series attached                to cusp forms on boundary components.  Keating's research object                is to understand Igusa curves better by studying their function                 fields including applications to coding theory and factorization                theory.  He proposes to use Drinfeld's theory of extensions of                  function fields to produce a more explicit characterization of                  the function fields.                                                                 This research is in very broad areas of number theory, the                 study of the properties of the integers.  Montgomery's focus is                 on that part of the subject where classical analysis is used to                 understand these deep questions.  Masser's research concentrates                on using transcendence techniques (again analytic but of a very                 different type than Montgomery's techniques) to study problems                  concerning solutions of equations in integers.  Milne combines                  algebraic and modern analytic techniques to study special                       functions that arise in number theory.  Keating studies special                 domains that are important in number theory.
{'FirstName': 'Hugh', 'LastName': 'Montgomery', 'PI_MID_INIT': 'L', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Hugh L Montgomery', 'EmailAddress': 'hlm@umich.edu', 'NSF_ID': '000294573', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1264', 'Text': 'ALGEBRA,NUMBER THEORY,AND COM'}
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