# Mathematics Colloquia and Seminars

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### Multivariate special functions and duality from the point of view of enumerative geometry

**Distinguished Lecture Series**

Speaker: | Andrei Okounkov, Columbia University |

Related Webpage: | http://www.math.columbia.edu/~okounkov/ |

Location: | 1147 MSB |

Start time: | Thu, May 16 2019, 3:10PM |

Special functions in this lecture series will be generalizations of characters of irreducible representations of Lie groups, spherical functions of symmetric spaces, and more general Macdonald-type hypergeometric functions. They are not all that special from the point of view of analysis, because they satisfy certain linear differential or q-difference equations. However, these difference equations have a very delicate structure involving roots, coroots, and the like. In particular, they enjoy a certain powerful Langlands-like duality, which in the Lie groups context would interchange the arguments of the functions, that is, an element in the maximal torus of the groups G with the label of the function, which has to do with the highest weight and the dual torus. In this lecture series, my goal is to explain this phenomenon from the point of view of enumerative geometry and related modern high-energy physics. This seems to be a natural generality in which to consider these questions, in particular, it is much broader than the domain of the traditional Lie theory.

This is a third lecture in the series.