**Abstract**

The last book of Euclid contains the description of Platonic solids and their symmetries. The formal definition of a collection of symmetries, called a group in mathematics, first appeared in the work of Galois, in 19th century, in his study of symmetries of roots of polynomials. The speaker will recall a major result of 20th century - classification of finite simple groups and discuss examples: permutations, linear maps, symmetries of sphere packings. Next, he will discuss continuous symmetries of Euclidian and non-Euclidean geometry. He will review the amazing relation between the largest sporadic finite simple group called Monster, and the modular group that appears as discrete symmetry of non-Euclidean geometry. He will conclude with applications of symmetries/groups to modern mathematics and physics.**About the Speaker**

Prof. Igor Frenkel graduated from St. Petersburg State University with Honors Diploma in Mathematics. He emigrated to US several years later and earned a PhD in Mathematics from Yale University in 1980. After assuming postdoctoral positions at Yale University, Mathematical Sciences Research Institute, and the Institute for Advanced Study, he obtained his first tenure position at Rutgers University. In 1985, he returned back to Yale University and is currently the Sol Goldman Family Professor of Mathematics in the Department of Mathematics.

Prof. Frenkel studies finite and infinite dimensional symmetries and their realizations in algebraic, geometric, and analytic structures. This field of mathematics is called representation theory of groups and algebras. He has published and authored a series of research papers and books, including *Vertex Operator Algebras and the Monster *and *Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations*, which have opened substantial new areas of research in representation theory.

Prof. Frenkel is the recipient of fellowships from the Sloan Foundation and the Solomon R. Guggenheim Foundation. He was elected a Fellow of the American Academy of Arts & Sciences in 2015 and a Member of the US National Academy of Sciences in 2018.**For Attendees' Attention**

Seating is on a first come, first served basis.

Sol Goldman Family Professor of Mathematics, Yale University; IAS Senior Visiting Member