An abelian variety $A$ is indecomposable but not simple if $A$ is isogenous to a product of smaller-dimensional abelian varieties but is not itself a product. We explore how to construct complete families of abelian varieties with this property and discuss implications for the question of understanding the possible fundamental groups of fibered projective varieties.

A stratified smooth variety $M$ admits a Bruhat atlas if it can be covered by open charts isomorphic to opposite Bruhat cells in some Kac-Moody flag manifold via stratified isomorphisms. In this talk, I will present two cases where the Kac-Moody is finite or affine type, which allows explicit computation in coordinates using Bott-Samelson maps and other familiar techniques in Schubert calculus.

Most closed 4-manifolds do not admit symplectic forms, but most admit "near-symplectic forms", certain closed 2-forms which are symplectic outside of a collection of circles. This provides a gateway from the symplectic world to the non-symplectic world. Just like the Seiberg-Witten (SW) invariants, there are invariants in terms of J-holomorphic curves that are compatible with the near-symplectic form. Although the SW invariants don't apply to (potentially exotic) 4-spheres, nor do these spheres admit near-symplectic forms, there is still a way to bring in near-symplectic techniques.

Recall that the cocenter (or trace, or $HH_0$) of an algebra $A$ is the algebra modulo commutators. It is well known that the direct sum of cocenters of symmetric group algebras $Q[S_n]$ forms an algebra isomorphic to the ring of symmetric functions in infinitely many variables. There is a $q$-deformation of this fact: the direct sum of cocenters of type $A$ Hecke algebras forms an algebra isomorphic to the ring of symmetric functions with an additional formal parameter $q$. In this talk I will discuss a categorification of this fact, in which the Hecke algebra gets replaced by the category of Soergel bimodules (or ``Hecke category"). I will present an explicit dg model for the categorical cocenter. Miraculously, the cocenters of Hecke categories can be calculated (in type $A$, anyway) as the derived categories of explicit wreath product algebras. Finally, I plan to sketch how this gives rise to a well-behaved notion of Khovanov-Rozansky link homology for links in a solid torus (which was the primary motivation for this work). This is joint with Eugene Gorsky and Paul Wedrich.

Tropical geometry provides a convenient algebraic framework for matroids, and in this talk I'll present recent work with a student, Jacob Manaker, where we use this language to explore representations of groups on tropical linear spaces, rather than on vector spaces over a field. This brings up some intriguing combinatorial aspects of classical representation theory, though we are only at the first steps of this story and hope to interest others in joining this project.

A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers - connections on the projective line with extra structure. I shall describe a deformation of this correspondence for $\operatorname{SL}(N)$. I will introduce a difference equation version of opers called $q$-opers and prove a $q$-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted $q$-opers with regular singularities on the projective line. The so-called quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model may be viewed as a special case of the $q$-Langlands correspondence. Some applications of $q$-opers to the equivariant quantum K-theory of the cotangent bundles to partial flag varieties will be discussed as well as generalizations of our constructions to an arbitrary simply connected complex simple Lie group $G$.