Mathematical Sciences: Presidential Young Investigator Award
{'Code': '03040200', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'Ralph M. Krause', 'PO_EMAI': None, 'PO_PHON': None}
Hopkins has recently been working out consequences of the               nilpotence theorem in stable homotopy theory.  The nilpotence                   theorem determines the prime ideals of the sphere spectrum                      thought of as a ring up to homotopy.  The sphere is analogous to                the ring of integers, being an initial object in a suitable                     category of rings.  This analogy becomes more striking when one                 observes that the "completions" of the sphere at its prime ideals               acquire natural compact topologies.  He would like to assemble                  these completions into a global object analogous to the ring of                 adeles in number theory.  Such an object would be appropriate for               getting at large-scale aspects of stable homotopy theory.  One                  goal would be to construct L-series with analytic properties                    measuring the homotopy groups of finite complexes and the                       homomorphisms induced by maps between them.                                             Fundamental to this is a family of cohomology theories                  generalizing K-theory, and a basic problem is to represent them                 geometrically with "generalized vector bundles".  A theorem of                  Atiyah relates the K-theory of the classifying space of a finite                group to its character ring.  D. Ravenel and Hopkins have                       recently generalized the notion of character and proved an                      analogue of Atiyah's result for these exotic K-theories.  It is                 hoped that construction will shed light on what "generalized                    vector bundles" should be.                                                              All of this activity falls under the general heading of                 exploring and perfecting algebraic techniques for treating                      qualitative features of spaces, e.g. connectedness, knottedness,                stability, and so forth.  Such features can be every bit as                     important for applications as are the usual features examined by                methods of classical analysis.
{'FirstName': 'Michael', 'LastName': 'Hopkins', 'PI_MID_INIT': 'J', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Michael J Hopkins', 'EmailAddress': 'MJH@MATH.HARVARD.EDU', 'NSF_ID': '000405428', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Princeton University', 'CityName': 'PRINCETON', 'ZipCode': '085442001', 'PhoneNumber': '6092583090', 'StreetAddress': '1 NASSAU HALL', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'New Jersey', 'StateCode': 'NJ', 'CONGRESSDISTRICT': '12', 'CONGRESS_DISTRICT_ORG': 'NJ12', 'ORG_UEI_NUM': 'NJ1YPQXQG7U5', 'ORG_LGL_BUS_NAME': 'THE TRUSTEES OF PRINCETON UNIVERSITY', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '126700', 'Text': 'TOPOLOGY'}
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