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Professor

Office Location(s): 1858 East Hall Phone: 734.763.0294 barvinok@umich.edu Personal Homepage

I am interested in computational complexity and algorithms in algebra, geometry and combinatorics. One way to understand a structure is to write ``a formula'', but often there are no formulas or they are so complicated that obscure rather than elucidate the structure. Then, instead of a formula, one can think of an algorithm as the key to understand a mathematical object. Constructing algorithms has been my occupation for a number of years. Among them are the algorithms for counting integer points in polyhedra, for solving systems of real polynomial (quadratic) equations and for counting various combinatorially defined objects (such as perfect matchings in graphs). In particular, I am interested in various phenomena of higher-dimensional geometry (such as the ``measure concentration'') and how they can be used to obtain efficient algorithms. High dimensions have long been regarded as an obstacle for computationally efficient solution of a problem. I am interested in situations where high dimensions help.