VIRGINIA TECH GEOMETRY & TOPOLOGY SEMINAR (FALL 2025)

The geometry of Hilbert schemes is intimately tied to algebra and combinatorics. In his proof of the Macdonald Positivity Conjecture, Haiman showed through a derived equivalence that the equivariant K-theory of Hilbert schemes of points in the plane is described by the ring of symmetric functions with two parameters. In particular, the fixed point classes correspond to modified Macdonald polynomials. Later, relating to the proof of the Shuffle Conjecture by Carlsson-Mellit, Carlsson-Gorsky-Mellit described the equivariant K-theory of parabolic flag Hilbert schemes using algebraic methods. I will discuss recent work in collaboration with Dan Orr where we describe the fixed point classes of the equivariant K-theory of parabolic flag Hilbert schemes in terms of partially-symmetric modified Macdonald polynomials.

Fusion categories are rich mathematical objects that generalize finite groups and their representation categories. Where do fusion categories appear in topology? What does it mean for a fusion category to act on an algebra? Do these actions generalize group actions on algebras? This question has been studied by many in the context of operator algebras but poses an interesting question for an associative algebra A. In this talk, we will define an action of a fusion category on an algebra and show how it relates to group actions on algebras. Then, as an application of this theory, we will apply our results to study actions of fusion categories on path algebras.

The Khovanov-Rozansky skein lasagna module was introduced by Morrison-Walker-Wedrich as an invariant of 4-manifold with a framed oriented link in the boundary. I will discuss an extension of the skein lasagna theory to 4-manifolds with codimension 2 corners, its behavior under gluing, and applications to trisections of 4-manifolds. This is joint work with Sarah Blackwell and Slava Krushkal.

The representation theory of vertex operator algebras (VOAs) provides a natural way to construct line bundles on moduli spaces of curves with marked points. When these line bundles are positive, the associated morphisms to projective varieties can shed light on the geometry of the moduli space itself. In this talk, I will discuss several criteria for ensuring positivity of these line bundles arising from VOA representations.

CSM-Thom polynomials are ℏ-deformations of the classical Thom polynomial, which was introduced to unify problems in classical enumerative geometry. The CSM-Thom polynomials are powerful universal polynomials which capture the geometry of the singularity locus of any map (sharp information), given only (soft) topological information about the map. We prove new formulas for the CSM classes of various contact multisingularities using interpolation, a method recently developed by Koncki-Rimányi inspired by the stable envelope construction of geometric representation theory. As an application, we show how these new formulas can be used to obtain geometric information about singularity loci of maps of projective spaces, utilizing a theorem of Aluffi-Ohmoto.

To learn about the representation theory of quantum SL(2), one can study the Temperley-Lieb category, whose basis morphisms are planar matchings. A generalization to the SL(3) case gives rise to the SL(3) web category, whose basis morphisms are given by non-elliptic webs - planar trivalent graphs with girth greater or equal to 6. It is shown by Fontaine-Kamnitzer-Kuperberg that the dual graphs of SL(3) basis webs are embedded isometrically into the SL(3) affine building, which is defined as a simplicial complex with vertices given by elements in the affine Grassmannian. In this talk, I will review background materials and talk about a generalization of the embeddings to the SL(4) case, based on upcoming joint work with Gaetz-Striker-Swanson.

Stratified vector bundles are like vector bundles over a stratified space, but which may drop rank over the lower strata. They arise naturally in many contexts. As a first step to see if we can "do algebraic topology" with them, we investigate the extent to which they exhibit homotopy invariance. We will see that this leads naturally to Nash blow-up constructions, and that we can answer questions about their homotopy invariance in terms of these constructions. If there is time we will discuss how this in turn guides the way to a potentially fruitful definition of stratified K-groups and Chern-Mather classes of stratified vector bundles. 

Topological branched covers of the 2-sphere with only finitely many postcritical points (now commonly called Thurston maps) came under scrutiny in complex dynamics about 45 years ago because of the interest of Adrien Douady and John Hubbard in matings of quadratic polynomials. William Thurston gave a topological characterization of when such a branched cover is equivalent to a rational map. Thurston's characterization has been central to the field, but even decades later it can be difficult to work with because of the subtlety of the obstructions. I'll give an overview of the area and discuss some of the problems we're working on now that are related to this. The talk should be widely accessible.

The (equivariant) quantum K-theory ring of a flag variety is a Frobenius algebra equipped with a perfect pairing called the quantum K-metric. It is known that in the classical K-theory ring for a given flag variety, the ideal sheaf basis is dual to the Schubert basis with regard to the sheaf Euler characteristic pairing. We define a quantization of the ideal sheaf basis for the equivariant quantum K-theory of cominuscule flag varieties. Cominuscule flag varieties allow for combinatorial descriptions of their Schubert calculus, beyond that of a general flag variety. In this talk we will give an overview of these cohomology theories and discuss why we care about the dual basis for these various intersection pairings. Time permitting, we will discuss the combinatorics of cominuscule varieties that allow one to write the Schubert expansion of any quantized ideal sheaf very quickly by hand.

A pair of elliptic curves $E_1, E_2$ over a field $k$ are $n$-isogenous if there exists a non-constant rational map $\phi: E_1 \to E_2$ defined over the algebraic closure $\overline{k}$ of degree $n$ that preserves the group structure. *Usually* there is zero or one $n$-isogenies between two elliptic curves, but this is not always the case. The modular polynomial $\Phi_{n}(X,Y) \in \mathbb{Z}[X,Y]$ of level $n$ is a bivariate symmetric polynomial $\Phi_n(X, Y) \in \mathbb{Z}[X, Y]$ whose $k$-roots $(j_1, j_2)$ correspond to pairs of $n$-isogenous elliptic curves over $\overline{k}$ with $j$-invariants $j_1, j_2$. The modular curve $X_0(n)$ is a smooth projective curve which is a coarse moduli space for pairs of $n$-isogenous elliptic curves and is birationally equivalent to the affine curve defined by $\Phi_n(X, Y) = 0$. The singular points of $\Phi_n(X,Y)=0$ correspond to elliptic curves with more than one $n$-isogeny between them, and by resolving the singularity we can recover the full set of $n$-isogenies. In this talk we give background on $\Phi_n$ and $ X_0(n)$ and demonstrate how to recover isogenies from singular points of $\Phi_n(X,Y)$.

Fano mirror symmetry is a duality between Fano manifolds and Landau-Ginzburg models. Predictions include: Certain D-modules associated with given mirror pairs are isomorphic, and certain integral structures on their spaces of flat sections correspond to each other. I will discuss this picture using the projective line. Then I describe the current progress for a larger class of Fano manifolds called flag varieties.