Higher-Dimensional Analogs of Stable Curves

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DMS-0401795Valery AlexeevThe project builds on PI's work towards generalizations of stable n-pointed curves to higher dimensions: stable pairs of dimension two and stable pairs with group action: toric, abelian, semiabelian and reductive. He will investigate several new applications of stable pairs and their moduli, including toric Torelli map; characterization of non-regular periodic polyhedral tilings; toric degenerations of reductive varieties with applications to mirror symmetry.The main subject of algebraic geometry - the primary field of the investigator's work - is to describe solutions of polynomial equations that are of fundamental importance to mathematics and its applications to science and engineering. Stable curves, whose higher-dimensional generalizations the investigator will continue to study, have seen exciting applications in number theory and in string theory in physics.