NCSU Algebra & Combinatorics Seminar

Spring 2025






Title: The equivariant $\gamma$-positivity of matroid Chow rings


Abstract: Chow rings and augmented Chow rings of matroids played important roles in the settlement of the Heron-Rota-Welsh conjecture and the Dowling-Wilson top-heavy conjecture. Their Hilbert series have been extensively studied since then. It was shown by Ferroni, Mathern, Steven, and Vecchi, and indepedently by Wang, that the Hilbert series of Chow rings of matroids are $\gamma$-positive using inductive arguement followed from the semismall decompositions of the Chow ring of matroids. However, they do not have an interpretation for the coefficients in the $\gamma$-expansion. Recently, Angarone, Nathanson, and Reiner further conjectured that Chow rings of matroids are equivariant $\gamma$-positive under the action of groups of matroid automorphisms. In this talk, I will give a proof of this conjecture without using semismall decomposition, showing that both Chow rings and augmented Chow rings of matroids are equivariant $\gamma$-positive. Moreover, we obtain explicit descriptions for the coefficients of the equivariant $\gamma$-expansions. Then we consider the special case of uniformmatroids which extends Shareshian and Wachs Schur-$\gamma$-positivity of Frobenius characteristics of the  cohomologies of the permutahedral and the stellahedral varieties.



Title: Quasipolynomiality in multigraded homological algebra

Abstract: Families of modules or vector spaces indexed by an integer n, for instance arising from powers of ideals, often have numerical invariants that grow polynomially.  These invariants might be lengths, Betti or Bass numbers, or regularity, or combinations of these with deeper homological constructions. Sometimes the growth is only quasipolynomial: there are polynomials P_1,...,P_r such that the numerical invariant takes the value P(n) = P_i(n) whenever n is congruent to i (mod r). What drives this kind of growth and periodicity?  Joint work with Hailong Dao, Jonathan Montaño, Christopher O'Neill, and Kevin Woods provides a general answer in the multigraded setting -- that is, for families of monomial ideals and other finely graded modules over affine semigroup rings.  The theory is clarified by thinking so generally that the case of multiple parameters comes for free; when the families are indexed not merely by a single integer but by many, a numerical function is quasipolynomial when its values agree with one of several polynomials, one for each coset of a given integer lattice. The proofs and constructions rest on foundations from applied topology, specifically tame modules in persistent homology, combined with Presburger arithmetic.  No familiarity with either of these theories is assumed.




Title: Universal quantum computation using Ising anyons from a non-semisimple topological quantum field theory


Abstract: We propose a framework for topological quantum computation using newly discovered nonsemisimple analogs of topological quantum field theories in 2+1 dimensions. These enhanced theories can be used to construct more sensitive knot invariants. In our work, we show that they also lead to more powerful computational models: while the conventional theory of Ising anyons is not universal for quantum computation via braiding of quasiparticles, the non-semisimple theory introduces new anyon types that extend the Ising framework. By adding just one new anyon type, universal quantum computation can be achieved through braiding alone. 


Title: Adjoint modalities in combinatorial Hopf algebras

Abstract: In studying Hopf algebras $H=\bigplus_{n\geq0} \mathbb{Z}P_n$ built on combinatorial gadgets $P_\bullet$ with a natural poset structure, it is natural (and often fruitful) to use the poset to define new bases for $H$. Here I introduce a variation on the common technique, when in the presence of poset maps $f_n:P_n \to Q_n$. After introducing the basic framework, I'll give lots of examples and a couple consequences. As the title suggests, the former requires the maps $f_n$ to be part of an *adjoint modality.* Based on joint work with Marcelo Aguiar. (In progress.)



Title: Positive formula for Jack polynomials and genus expansion of Jack characters


Abstract: In this talk, I describe an explicit formula for the power-sum expansion of Jack polynomials. This formula resembles the so-called genus expansion originating from random matrix theory; each coefficient is a generating function of certain bipartite graphs embedded into two-dimensional surfaces. Our formula is naturally parametrized by the shifted parameter $b := \alpha-1$, instead of the classical Jack parameter $\alpha$, and we provide a combinatorial/topological meaning of this reparametrization. If time permits, I will explain how our result implies Lassalle's conjecture from 2008 on the integrality and positivity of Jack characters in Stanley's coordinates.


Title: Log-concavity of the Alexander polynomial

Abstract: Almost a century after the introduction of the Alexander polynomial, it still presents us with tantalizing questions, such as Fox’s conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $\Delta_L(t)$ of an alternating link $L$ are trapezoidal. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsv\’ath and Szab\’o (2003) for the case of genus 2 alternating knots, among others. We settle Fox’s conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of $\Delta_L(t)$, where $L$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are trapezoidal. This talk is based on joint work with Elena Hafner and Alexander Vidinas.