Mathematical Sciences: K-Theory, C*-Algebras and Index      Theory for Elliptic Operators
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The central mathematical concept in this project is that of                index.  In its most basic manifestation, index has to do with the               number of extra conditions that must be imposed on a solution to                a differential equation in order to specify it unambiguously.  A                famous theorem of Atiyah and Singer asserts that this sort of                   index is given in the simplest case by a winding number (counting               the number of times a path in the plane winds around the origin)                and more generally by an analogous higher-dimensional topological               invariant.  The Atiyah-Singer Theorem and its numerous progeny                  all have the form:  lefthand side (index of a differential                      operator) equals righthand side (something you can compute from                 the ingredients that go into the operator).  Progress consists in               achieving a more and more sophisticated understanding of what the               righthand side really is.  Nowadays, this involves geometry,                    topology, and functional analysis.                                                   The most advanced point of view, represented by Professors                 Kaminker and Ji among others, computes index in the K-theory of                 an appropriate algebra of operators. Classically, this would be                 commutative, an algebra of functions on the underlying manifold,                but in the presence of additional structure a more complicated                  algebra is called for.  The investigators will seek to understand               the algebra and its K-theory better, and to extract new                         invariants for differential operators from elements of the K-                   groups.
{'FirstName': 'Jerome', 'LastName': 'Kaminker', 'PI_MID_INIT': 'A', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Jerome A Kaminker', 'EmailAddress': 'kaminker@math.iupui.edu', 'NSF_ID': '000457088', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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