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Area of Research |
Number Theory, Algebraic Geometry, Cryptography, Partial Differential Equations & Mathematics Education |
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Degrees |
B.A., University of Pennsylvania 1981, PhD. University of Cambridge 1985 |
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Joined ASU in |
2008 |
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Here is a link to my CV My mathematical research is in arithmetical algebraic geometry, with applications to cryptography. More specifically, |
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I am interested in the theory of algebraic cycles on algebraic varieties defined over various types of fields. Recall that an algebraic cycle of codimension n on an algebraic variety X is a formal linear combination of closed integral subvarieties of codimension n. Two algebraic cycles are rationally equivalent if their difference is a sum of divisors of functions on codimension (n-1)-subvarieties on X. The group of algebraic cycles of codimension n modulo rational equivalence is called the Chow group of codimension n cycles. When n=1, this group is often called the divisor class group or Picard group if X is smooth, and its structure is understood reasonably well. If n>1, these groups are not understood well at all and much of my work has been dedicated to elucidating their structure. The theory of algebraic cycles has an intimate relationship with class
field theory, and I devoted some of my early work to studying this. In the mid 1990’s, I wrote a long survey
paper on this subject, Abelian class field theory of
arithmetic schemes, that appeared in In recent years, I have become interested in p-adic Hodge theory, which puts much more structure on the p-adic étale cohomology of a smooth projective variety defined over a p-adic field. This is similar to Hodge theory for a complex variety, but includes the action of the Galois group. About ten years ago, I became interested in cryptography in joint work with Ming-Deh Huang. We study the discrete log problem for the multiplicative group of a finite field and for elliptic curves over finite fields. In more recent work, we study the open problem of producing efficient multilinear pairings, which would have great application in cryptography. Here are some recent papers: A multilinear generalization of the Tate pairing Etale cohomology of varieties with totally degenerate reduction |
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