Type of Document Dissertation Author Tripathi, Girja Shanker URN etd-11152010-145059 Title Orthogonal Grassmannians and Hermitian K-Theory in A^1-Homotopy Theory of Schemes Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee

Advisor Name Title Schlichting, Marco Committee Chair Cohen, Daniel C. Committee Member Delzell, Charles N. Committee Member Hoffman, Jerome W. Committee Member Sundar, Padmanabhan Committee Member Srivastava, Ashok Dean's Representative Keywords

- Orthogonal Grassmannians
- A^1-Homotopy Theory
- Algebraic Geometry
- Hermitian K-Theory
Date of Defense 2009-04-29 Availability unrestricted AbstractIn this work we prove that the hermitian K-theory is geometrically representable inthe A^1 -homotopy category of smooth schemes over a field. We also study in detail a

realization functor from the A^1 -homotopy category of smooth schemes over the field

R of real numbers to the category of topological spaces. This functor is determined

by taking the real points of a smooth R-scheme. There is another realization functor

induced by taking the complex points with a similar description although we have

not discussed this other functor in this dissertation. Using these realization functors we

have concluded in brief the relation of hermitian K-theory of a smooth scheme over

the real numbers with the topological K-theory of the associated topological space

of the real and the complex points of that scheme: The realization of hermitian

K-theory induced taking the complex points is the topological K-theory of real

vector bundles of the topological space of complex points, whereas the realization

induced by taking the real points is a product of two copies of the topological

K-theory of real vector bundles of the topological space of real points.

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