In this talk we will discuss the connection between invariant evolutions of polygons and completely integrable discrete systems via polygonal geometric invariants. We will give examples and show how some open problems for bi-Hamiltonian structures of discrete systems were made easier and solved using this correspondence. If time allows we will discuss some open problems. The talk is meant for general mathematical audiences, but it might get a bit more technical towards the end.

Gloria Marí Beffa grew up in Málaga, Spain, where she did her undergraduate studies. She moved to the Twin Cities, completing her PhD at the University of Minnesota in 1991 on the Poisson Geometry of the Virasoro Algebra under the direction of Jack Conn. After a postdoctoral position at UC Davis and a visiting position at UW-Madison, she worked as academic staff at UW-Madison until 2006, teaching and doing administration, while working on her research outside her official job during her “free time”. In 2006 she was hired as Associate Professor with tenure, was promoted to Full Professor in 2010, became Chair of the Department in 2014 and Divisional Associate Dean for the Natural, Physical and Mathematical Sciences in the College of Letters and Sciences in 2018, a position she still holds. She is also the Interim Associate Dean for Research in the College, a position she took at the beginning of the COVID crisis in 2020. She has had multiple internal awards (2006 L&S Academic Staff Excellence Mid-career award, 2008 Best L&S Faculty advisor and 2012 Philip R. Certain Award for outstanding research, teaching and service contributions to the College) and was a Simons Fellow during 2012–13.