TBA

Hessenberg varieties are subvarieties of flag varieties invented in the early 1990s by de Mari and Shayman.  De Mari and Shayman were motivated by questions in applied linear algebra, but, very quickly, people realized that Hessenberg varieties are very interesting objects of study from the point of view of algebraic groups.  

I got interested in Hessenberg varieties because of their connection to questions in combinatorics, in particular, the Stanley-Stembridge conjecture.  I'll explain this conjecture, now a theorem due to Hikita, and I will explain some of my work with Tim Chow, which resolved a conjecture of Shareshian and Wachs connecting Hessenberg varieties directly to Stanley-Stembridge. (I'll also try to say a few words about Hikita's work and the very exciting state the field is in now.)  Then I'll explain joint work with Escobar, Hong, Lee, Lee, Mellit and Sommers on the moduli of Hessenberg varieties. 

Who doesn't like one of these three: geometry, topology, and combinatorics? Sperner's lemma, a combinatorial statement that is equivalent to the Brouwer fixed point theorem in topology, is amazing and powerful. I'll explain why, give heartwarming old and new proofs, and present some generalizations to polytopes that has surprised me with diverse applications: to the study of triangulations, to fair division problems, and the Game of Hex.

Holomorphic Dynamics (in a narrow sense) is the theory of the iteration of rational maps on the Riemann sphere. It was founded in the classical work by Fatou and Julia around 1918. After about 60 years of stagnation, it was revived in the 1980s, bringing together deep ideas from Conformal and Hyperbolic Geometry, Teichmüller Theory, the Theory of Kleinian Groups, Hyperbolic Dynamics and Ergodic Theory, and Renormalization Theory from physics, illustrated with beautiful computer-generated pictures of fractal sets (such as various Julia sets and the Mandelbrot set). We will highlight some landmarks of this story.

Holonomy is a prime example of mathematical intuition and creativity - it generalises our school knowledge about the sum of angles in a triangle and led to  `Berger’s holonomy theorem’ from 1954 which turned out to be a  most successful research programme for differential geometry for over 50 years. We are going to tell the story of this development, how holonomy relates to curvature and advanced symmetry concepts, including a small detour to theoretical physics and what Calabi-Yau manifolds have to do with it. We conclude by a small outlook to recent results.