Vaughan Jones showed how to associate links in the $3$-sphere to elements of Thompson’s group $F$ and proved that $F$ gives rise to all link types. This talk will discuss two recent extensions of Jones’ work– the first is a method of building annular links from Thompson’s group $T$, which contains $F$ as a subgroup, and the second is a method of building $(n,n)$-tangles from $F$ . Annular links from $T$ arise from Jones’s unitary representations of the Thompson groups, and tangles from $F$ give rise to an action of $F$ on Khovanov’s chain complexes. This talk includes joint work with Slava Krushkal and Yangxiao Luo.

 I will describe some recent work in collaboration with Peng Cheng and Ruben Minasian on the classic topic of string theory compactification on a Calabi-Yau 3-fold $X$ and mirror symmetry. I will discuss the physical role of torsion subgroups of integral cohomology groups of $X$, as well as the implication that such non-trivial groups have for mirror symmetry.  An intriguing part of the story is the relationship between orbifold discrete torsion and the topology of a flat abelian gerbe on $X$.

The cosmetic crossing conjecture posits that switching a non-trivial crossing in a knot diagram always changes the knot type. This question is closely related to cosmetic surgery problems for three-manifolds, and has seen significant progress in recent years. We will discuss the conjecture, and present new obstructions to cosmetic crossing changes for a family of links that includes all alternating knots.

The general rule for the interactions between tropical geometry and moduli spaces of course is the following: everything you may wish is going to work like a charm in genus zero, and break down horribly in higher genus. This is the case for the tautological intersection theory of psi classes, a class of fundamental objects in the geometry of moduli spaces of curves: the generating function of their intersection numbers has made waves, pun intended, when it was noticed that it is a tau function for the KdV hierarchy. Back to tropical geometry: in genus zero tropical psi classes have been first defined by Mikhalkin in the early 2000's, then through the work of Kerber-Markwig and Katz it was shown that intersection numbers of tropical psi classes agree with their algebraic counterparts. In work with A.Gross and H.Markwig (2021), we were able to make sense of tropical psi classes in higher genus, by making the tropical moduli space of curves into a stack for families of tropical curves with an affine structure. This is a combinatorial theory that recovers the algebraic intersection numbers, but can also produce results that do not have a counter part in algebraic geometry. To this end, in recent work with A.Gross we answer the question of when we can show that tropical psi classes are tropicalizations. In order to even make sense of the statement, we had to introduce a notion of tropicalization for families of curves based on the Picard theory of the base.

The ordinary Grothendieck group of varieties is a group with simple generators and relations built out of varieties analogous to the construction of the integers from the natural numbers. Called the ``poor man's motives,'' this group recovers Hodge numbers, even for open, singular varieties, and point counts over finite fields.

Log schemes are morally pairs of a scheme with a divisor (X, D). Used for compactifications and degenerations of varieties, they admit a log Hodge theory due to Kato-Usui. This theory is not yet understood for open or singular varieties, so we define a log Grothendieck group to get log Hodge numbers. Unfortunately, this does not work! We modify log Hodge numbers to get weaker well-defined invariants from our theory, leaving lots of open questions. This is joint with Andreas Gross, David Holmes, Pim Spelier, Jesse Vogel.

Calabi-Yau hypersurfaces in toric spaces of general type (described by certain non-convex, flip-folded and otherwise multilayered "multitopes") may be smoothed by rational anticanonical sections. Nevertheless, gauged linear sigma model phases and an increasing number of their classical and quantum data are just as computable as for their siblings encoded by reflexive polytopes, and they all have transposition mirrors. Moreover, explicit deformations of such general-type embeddings are diffeomorphic to regular hypersurfaces in Fano toric varieties, jointly realized as generalized complete intersections in products of projective spaces. Showcasing Calabi-Yau hypersurfaces in Hirzebruch scrolls shows this class of constructions to be infinitely vast, yet amenable to several well-founded algebro-geometric methods of analysis. This talk will include joint work with Per Berglund, as reported in part: arXiv:1606.07420, arXiv:1611.10300, arXiv:2205.12827 and arXiv:2403.07139.

Toric varieties are algebraic varieties with a group action which can be defined using convex polyhedra. They have rich structure due to their combinatorial nature and can be useful in generating examples and testing conjectures. Gorenstein varieties are varieties which can be thought of as being "close to smooth" in many instances. We will study toric varieties coming from partially ordered sets (posets) and determine in many cases when they are Gorenstein. No prior knowledge of toric geometry will be assumed, and in our setting the Gorenstein property will be reduced to combinatorial, computational, and linear algebraic questions.

I will describe some recent work in collaboration with Peng Cheng and Ruben Minasian on the classic topic of string theory compactification on a Calabi-Yau 3-fold $X$ and mirror symmetry. I will discuss the physical role of torsion subgroups of integral cohomology groups of $X$, as well as the implication that such non-trivial groups have for mirror symmetry.  An intriguing part of the story is the relationship between orbifold discrete torsion and the topology of a flat abelian gerbe on $X$.