The Springer fiber of a linear operator $X$ is the subvariety of the flag variety that is "fixed" by $X$. Hessenberg varieties are a generalization of Springer fibers: they consist of the flags that are "moved" by $X$ only to a certain extent, as measured by a second parameter $H$. The geometry of Springer fibers and Hessenberg varieties encodes deep information about representations of the symmetric group. However, the varieties themselves are not well understood. In this talk, we introduce Springer fibers and Hessenberg varieties, describe some of their combinatorial and representation-theoretic context, and sketch some results about cell decompositions (including closure relations) in certain cases.

Given four random red lines in 3-space, how many blue lines touch all four red? The answer is two, and this is the first nontrivial question in "Schubert calculus". Hilbert's 15th problem was to give this theory a solid foundation, which we now see as the cohomology ring of the Grassmannian of 1-planes in 3-space (or k-planes in affine n-space). There are many variations, all of which are easy to study algebraically, but only a few of which are understood combinatorially. In the late '90s Terry Tao and I proved one could count "puzzles" in place of counting actual subspaces, and I solved similar problems with puzzles, some only conjecturally. In the last couple of years, through joint work with Paul Zinn-Justin, the geometry behind puzzles has become clearer: they are actually calculations on Nakajima quiver varieties (though for this talk I will mainly need spaces of diagonalizable complex matrices with fixed spectrum).

The study of the birational automorphisms of the plane has a history of more than a hundred years. These automorphisms are invertible maps defined by polynomials, and several significant results have been established over the field of complex numbers, or more generally over perfect fields. Over a finite field, we call such a map bijective if it induces a bijection on the points defined over the ground field. Given an abstract permutation, can we always realize it via a bijective map? In this talk, I will give an almost full answer to this question. This is joint work with Shamil Asgarli, Masahiro Nakahara, and Susanna Zimmermann.

Computing volumes of moduli spaces has significance in many fields. For instance, the celebrated Witten's conjecture regarding intersection numbers on moduli spaces of curves has a fascinating connection to the Weil-Petersson volumes, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. In this talk I will introduce an analogue of Witten's intersection numbers on moduli spaces of holomorphic differentials to compute the Masur-Veech volumes induced by the flat metric of the differentials. This is joint work with Moeller, Sauvaget, and Zagier (arXiv:1901.01785).