Given a hyperplane arrangement, we associate two projective varieties: the wonderful compactification and the matroid Schubert variety. Adiprasito, Huh and Katz used the Hodge-Riemann relations of the wonderful compactification (and their combinatorial generalizations) to prove that the coefficients of their characteristic polynomials form a log-concave sequence. In a joint work with Huh, we proved Dowling and Wilson's Top-heavy conjecture for realizable matroids by applying the hard Lefschetz theorem to the matroid Schubert varieties. In a more recent work with Braden, Huh, Matherne and Proudfoot, we proved the Top-heavy conjecture to arbitrary matroids. In this talk, I will go over some of the key ideas about the proof of the Top-heavy conjectures. If time permits, I will also mention some on-going works with my students Colin Crowley and Connor Simpson towards generalizations to type A Coxeter matroids.

In the world of $(\infty,1)$-categories, it is well-known that the composition of specified morphisms is well-defined up to a contractible choice. The situation for $(\infty,2)$-categories is more subtle, as there are many potential ways of composing 2-cells. In a joint work in progress with Hackney, Riehl, Rovelli, we prove a uniqueness statement for the composition of so-called pasting diagrams, which I will explain during the talk.

The zeta functions are a pillar of number theory. Zeta functions have been objects of great interest for number theorists due to their beauty, mystery, and power. I will discuss this study in the simplest case: the Hurwitz zeta functions. Recently, some surprising direct connections between the special values of Hurwitz zeta functions and power sums were found. In my talk, I will introduce these discoveries. In particular, we will see that special values of Hurwitz zeta functions have some nice integral representations. This is joint work with Jan Minac and Nguyen Duy Tan.