Topics in Arithmetic Geometry and K-theory

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This proposal concerns questions in higher dimensional Arakelov theory andalgebraic K-theory. The PI intends to extend results of previous work withSoule on motivic weight complexes of algebraic varities to the arithmetic case.In collaboration with J. Hu, he intends to use the deformation to the normalcone techniquess for Arakelov theory (developed by J. Hu in his thesis) to thecase of stacks. He also intends to study questions in arithmetic geometryrelated to the Mordell conjecture using methods from differential algebra andArakelov theory. This proposal deals with arithmetic geometry and algebraic K-theory.Arithmetic geometry is the study of the properties of equations withcoefficients that are whole numbers, and using methods both from numbertheory (the study of properties of whole numbers) and algebraic geometry, whichstudies geometric figures that can be defined by the simplest of equations,namely polynomials. Algebraic K-theory studies properties of linear equationsusing methods from geometry. The questions and phenomena which arise fromcombining number theory and geometry serve as driving forces in much ofcontemporary mathematics research. Moreover, arithemetic geometry hascontributed to many applications including cryptography and theoreticalcomputer science.