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).

Hausel-Proudfoot varieties are a family of spaces attached to a graph, which behave in many ways like the moduli of local systems or higgs bundles on a Riemann surface. I will give an overview of recent work on their cohomology and their p-adic volumes, joint with Z. Dancso and V. Shende on the one hand and M. Groechenig on the other.

Strominger-Yau-Zaslow conjecture predicts the existence of special Lagrangian fibrations on Calabi-Yau manifolds and provides an recipe for the construction of mirrors via dual torus fibration. Due to the analytic difficulty of the original conjecture, Kontsevich-Soibelman and Gross-Siebert developed an algebraic algorithm to construct the mirror. The symlectic counterpart is the family Floer homology introduced by Fukaya. However, there are not many explicit examples of the family Floer mirrors are computed, due to the lack of control of the holomorphic discs in a given geometry. In this talk, we will provide some log Calabi-Yau surfaces and the idea how to compute the family Floer mirror explicitly. In particular, by comparing with the construction of mirrors in the work of Gross-Hacking-Keel, this motivates some conjectures of the existence of Ricci-flat metric on some log Calabi-Yau surfaces. This is a joint work in progress with Man-Wai Cheung.

This talk will focus on homological mirror symmetry for tori, and some of its unexpected features for abelian varieties with complex multiplication. After reviewing mirror symmetry for elliptic curves and the challenges posed by complex multiplication, I will describe an approach, developed in the PhD thesis of my student Yingdi Qin, to the problem of incorporating coisotropic branes into the Fukaya category of a torus; as well as the motivation for this construction from the perspective of SYZ and homological mirror symmetry. Qin's work also gives an insight into the equivalence between the Fukaya categories of dual symplectic tori (eg elliptic curves with inverse areas), which I will explain if time permits.

Action-angle coordinates are a type of coordinate chart on symplectic manifolds originating from the theory of commutative completely integrable systems. Symplectic toric manifolds are the prototypical example of symplectic manifolds with global action-angle coordinates. Multiplicity-free spaces are the natural non-abelian generalization of toric manifolds. For example, coadjoint orbits of compact Lie groups are multiplicity-free spaces. Unlike toric manifolds, multiplicity-free spaces do not come with natural global action-angle coordinates.

In this talk I will present recent work that constructs action-angle coordinates on "big subsets" of a large family of multiplicity-free spaces. As a corollary, we close a conjecture on the Gromov width of coadjoint orbits in the case of arbitrary regular coadjoint orbits of compact simple Lie groups.

This talk is based on collaboration with Anton Alekseev, Benjamin Hoffman, and Yanpeng Li.

The Strange Duality conjecture suggests that there should be a natural duality between cohomologies of certain tautological bundles on a pair of moduli spaces of stable sheaves coming from two orthogonal Chern characters. In this talk, I will formulate the conjecture more precisely, briefly review the recent result of Marian-Oprea establishing the Strange Duality over elliptic K3 surfaces, and formulate our pointwise generalization. Time permitting, I will outline the formulation of the result in moduli (over the moduli stack of (quasi)polarized K3s) and explain how to extend the result from the elliptic locus.