# "Pentagramma Mirificum". Hirzebruch lecture by Sergey Fomin on Friday, November 8, University Club Bonn

**Pentagramma Mirificum** (the miraculous pentagram) is a beautiful geometric construction studied by Napier and Gauss. Its algebraic description yields the simplest instance of cluster transformations, a remarkable family of recurrences which arise in diverse mathematical contexts, from representation theory and enumerative combinatorics to theoretical physics and classical geometry (Euclidean, spherical, or hyperbolic). The lecture will explore some of the most basic and concrete examples of cluster transformations, and briefly discuss their properties such as periodicity, integrability, Laurentness, and positivity.

**Sergey Fomin** studied at St. Petersburg State University where he received his doctorate in 1982 with Leonid Ossipov and Anatoli Verschik. He was a lecturer at St. Petersburg Electrotechincal University (LETI) from 1982 to 1991. From 1992 to 2000 he was on the faculty at the Massachusetts Institute of Technology (Assistant Professor 1993, Associate Professor 1996) before moving to a position at the University of Michigan in 1999, where he has been Robert M. Thrall Collegiate Professor since 2007. He was also a scientist at the Institute of Computer Science and Automation of the Russian Academy of Sciences in St. Petersburg from 1991 to 2005. He was a visiting scientist at the Mittag-Leffler-Institut (1992, 2005), MSRI, the Isaac Newton Institute in Cambridge, the University of Strasbourg (IRMA), the Hausdorff Institute in Bonn, and the Erwin Schrödinger Institute for Mathematical Physics in Vienna. In 2012 he became a fellow of the American Mathematical Society. He was invited speaker at the International Congress of Mathematicians 2010 in Hyderabad. In 2018 he received the Leroy P. Steele Prize of the American Mathematical Society.

His research is on combinatorics with applications to geometry, algebra, and representation theory, where he introduced cluster algebras with Andrei Zelevinsky. He has also worked in ennumerative geometry (Schubert calculus) and mathematical physics (e.g. Yang-Baxter equation, Bethe ansatz).

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