We present a uniform description of the center of the Temperley–Lieb algebra \(TL_n(\delta)\) for all nonzero complex parameters \(\delta\). In particular, we show that the center has dimension \[dim(Z(TL_n(\delta)) = 1 +\lfloor\frac{n}{2}\rfloor\] for every\(n\) and \(\delta\), and we describe a basis arising from a natural filtration closely tied to the cellular structure of \(TL_n(\delta)\). Our approach makes use of the diagrammatic and algebraic presentation of Temperley–Lieb algebras, their representation theory, and deformation theoretic methods.

In studying Hopf algebras \(H=\bigoplus_{n\geq0} \mathbb{Z}P_n\) 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 in the presence of poset maps \(f_n:P_n \to Q_n\). After introducing the basic framework, I'll give several 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.).

The notion of superselection sectors has appeared often in algebraic quantum field theory and has been adapted to various settings, including conformal nets and quantum spin systems. Here, we describe a general mathematical framework that describes quantum spin systems and conformal nets. Specifically, we define a net of von Neumann algebras associated to a poset and define superselection sectors corresponding to that net. We show that if the poset satisfies geometric axioms, then the superselection sectors form a braided monoidal category. This is joint work with Anupama Bhardwaj, Tristen Brisky, Chian Yeong Chuah, Kyle Kawagoe, Joseph Keslin, and David Penneys..

In 2002, Aval, F. Bergeron, and N. Bergeron discovered that the quotient of the usual polynomial ring \(Q[x_1,...,x_n]\) by the (ideal generated by) quasisymmetric polynomials had dimension equal to the nth Catalan number.  This rhymes with the dimension formula for the quotient by symmetric polynomials, \(n!\), which is the shadow of a deeper connection between invariant theory and algebraic geometry.  In this talk I will summarize my recent and ongoing work with N. Bergeron, P. Nadeau, H. Spink, and V. Tewari, which finds a similar geometric phenomenon for the quasisymmetric case, including  "quasisymmetric analogues" of the flag variety and Schubert polynomials.  Beyond this, I will describe a wider family of algebra-geometric objects infused with Catalan combinatorics, and the potential generalizations of quasisymmetric polynomials present in this work.

Here is where the abstract will appear.

Abstract

Here is where the abstract will appear.

Here is where the abstract will appear.