Vladimir Berkovich
The Matthew B. Rosenhaus Professor of Mathematics
My general area of research is algebraic
geometry whose objects of study are algebraic varieties, i.e., geometrical objects
that can be defined by algebraic equations. Due to both the algebraic and geometric
aspects of algebraic varieties, one can study their geometrical properties using
methods of algebra and, on the other hand, if such a variety is defined, say,
by algebraic equations with rational coefficients, one can get information on
the rational solutions of those equations using geometrical methods and intuition.
The classical way for the latter is to consider the set of all solutions in
the complex (or real) affine space, which is a nice geometrical object. Another
way is to use so called p-adic numbers which were discovered a hundred years
ago and since then became an indispensable tool in number theory. P-adic numbers
are obtained from rational numbers by the same procedure of completion as in
the construction of real numbers, but using a different distance between rational
numbers which is associated with every prime p. It was discovered much later
that p-adic numbers give rise to nice geometrical objects in the same way as
complex numbers give rise to the classical complex analytic spaces. My work
is concentrated in the study of those geometrical objects. P-adic analytic spaces
are no less important than their classical counterparts and found applications
in number theory, algebraic geometry and mathematical physics.
Recent Publications
- Smooth p-adic analytic spaces are locally contractible. II, in Geometric Aspects of Dwork Theory, Walter de Gruyter & Co., Berlin, 2004, 293-370.
- Integration of one-forms on p-adic analytic spaces, Annals of Mathematics Studies 162, Princeton University Press, Princeton and Oxford, 2007, 156 pp.
- A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures, to appear in Algebra, Arithmetic and Geometry, Volume I: In Honor of Y.I. Manin, Birkhäuser, Boston, 2008, 49-68.