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 $X$.

Rank-Metric Lattices (RML in short) were introduced in 2023 as the q-analogue of Higher-Weight Dowling Lattices. They are special families of geometric lattices whose elements are the Fq^m-linear subspaces of F^n,q^m having a basis of vectors with rank weight bounded from above, ordered by inclusion.

In this talk, we investigate the properties of RMLs from a finite geometry perspective. We compute the Whitney numbers of the first kind for some of these lattices, providing a recursive formula. In the second part of the talk, we present asymptotic results on the density of some classes of cardinality-optimal rank-metric codes.

A root system is a finite subset of a Euclidean space that satisfies certain simple geometric properties. Each element of a root system is called a root, each root gives rise to a reflection map on the ambient Euclidean space, and together the reflections generate a group known as a Weyl group. In this talk we will explore maximal sets of pairwise orthogonal roots in root systems of types E7, E8 or Dn for n even. We call such maximal sets n-roots, and we show that the reflections in the Weyl group act on the n-roots in a particularly nice way, endowing them with the structure of a so-called quasiparabolic set. The quasiparabolic structure can be understood via a connection between n-roots and perfect matchings, and we will discuss how this structure helps reveal new insights about the Fano plane, quantum isomorphisms of graphs, and certain irreducible representations of the Weyl group. (Joint work with Richard Green.)

I compute the knot Floer complex for the regular fiber of Σ(2,3,7), and I show that its Seifert genus and genus in a self homology cobordism agree. A step used for this result was providing upgrades to the surgery formula for knot lattice homotopy. Ozsv\'ath, Stipsicz, and Szab\'o showed that knot lattice homology satisfies a surgery formula similar to the one relating knot Floer homology and Heegaard Floer homology. I provide an iterable version of this formula that given the doubly filtered knot lattice space with involutive maps for a generalized algebraic knot produces the corresponding data for the dual knot post-surgery.

The notion of conformal blocks on stable curves defined by modules over a vertex operator algebra (VOA) is a generalization of the WZNW-conformal blocks defined by modules over affine Lie algebras. The recent advances in the theory of VOA-conformal blocks shed new light on the representation theory of VOAs. The factorization theorem and vector bundle property of the sheaves of VOA-conformal blocks lead to new and short proof of some deep theorems in the theory VOAs, like the associativity of fusion rules, generalized Verlinde formula, and the modular invariance of intertwining operators. In this talk, we will briefly discuss these applications. This talk is based on a project with Xu Gao.

Moduli spaces of marked curves form an important class of examples of moduli spaces whose geometry can be studied in terms of some combinatorial data. There are related moduli spaces of admissible covers of marked curves whose branch points are the marked points of the base curve, where the genus, degree, and monodromy data of the covers are fixed. This allows us to study families of such covers in terms of combinatorial data, yielding a powerful computation tool for doing intersection theory on these moduli spaces. In this talk, I will introduce these moduli spaces and then discuss a project (joint with Seamus Somerstep and Renzo Cavalieri) in which we compute the first Chern class of the Hodge bundle on spaces of cyclic admissible covers.

A central question in number theory is finding the set of integer or rational number solutions to a given system of polynomial equations. This question, called the Diophantine problem, has more than 1700 years of history and is still an active area of research. In this talk, we will start with finding rational numbers x and y satisfying the equation -139y²=x³+10x²-20x+8. One key ingredient is heights, which measure the arithmetic complexity of such rational solutions. While we explore this question, I will introduce some key players in number theory and their interactions: elliptic curves, modular curves, modular forms, L-functions, and the more general height theory. In particular, we will see a connection between the heights of rational points and the derivatives of L-functions, known as the Gross-Zagier formula. One direction of my work is a generalization of this formula, where a key step involves constructing Green’s functions on graphs. Another direction is concrete computations of rational points on some modular curves when the set of rational points is known to be finite. I will introduce my past, current, and possible future work in these two directions.