We give a diagrammatic description of the categories of modules coming from the conformal inclusions \( (sl(N),N) \; < \; (so(N^2-1),1) \). A small variant on this construction has uniform generators and relations which are rational functions in \( q = e^{2\pi i/4N} \), which allows us to construct a new continuous family of tensor categories at non-integer level which interpolate between these categories. This is the second example of such an interpolation category for families of conformal inclusions after Zhengwei Liu's interpolation categories \( (sl(N), N + 2) \; < \; (sl(N(N+1)/2),1) \) which he constructed using his classification Yang-Baxter planar algebras. Our approach is different from Liu's, we build a two-color skein theory, with one strand coming from \( X \) the image of defining representation of \( \mathfrak{sl}_N \) and the other strand coming from an invertible object \( g \) in the category of local modules, and trivalent vertex coming from a map \( X \otimes X^* \to g \). This is joint work with Cain Edie-Michell.

For Frobenius maximal parabolic subalgebras of sl(n), we prove that the multiplicities of the eigenvalues of the algebra's principal element form a unimodal and symmetric sequence about 1/2, and that the related spectrum on the full matrix algebra is symmetric and unimodal about zero. The proof uses Panyushev reduction, fence posets, good gradings, and Frobenius duality. This is joint work with John Irving and Mitja Mastnak of Saint Mary's University.

For Frobenius maximal parabolic subalgebras of sl(n), we prove that the multiplicities of the eigenvalues of the algebra's principal element form a unimodal and symmetric sequence about 1/2, and that the related spectrum on the full matrix algebra is symmetric and unimodal about zero. The proof uses Panyushev reduction, fence posets, good gradings, and Frobenius duality. This is joint work with John Irving and Mitja Mastnak of Saint Mary's University.

A partially ordered set is a set equipped with an order. A good example to have in mind is the collection of all subsets of a set, ordered by inclusion. A partially ordered set is called a lattice if for every pair of elements there is a unique smallest element greater than both, called their join, and a unique greatest element less than both, called their meet. (In the case of the poset of subsets of a set, the join is the union and the meet is the intersection.) A lattice is called distributive if the operations of meet and join satisfy a distributive law (just like the distributive law that is familiar from arithmetic). I will present recent work with Viktória Klász and René Marczinzik (arXiv:2501.09447) in which we give a characterization of whether or not a lattice is distributive in terms of linear algebra. A well-studied permutation of the elements of a distributive lattice, known as rowmotion, will also turn out to make a natural appearance from this point of view..

Abstract TBD.

Abstract TBD.

Abstract TBD.

Abstract TBD.

Abstract TBD.

Abstract TBD.

Abstract TBD.

Abstract TBD.

Abstract TBD.

Abstract TBD.