In 1997, Friedlander and Suslin introduced the category of strict polynomial functors, as a tool to study the cohomology of finite group schemes. One of the main calculations in their paper was the calculation, in this functor category, of the Yoneda (self-extension) algebra of the Frobenius twist functor. At the very end of their paper they gave - though only for characteristic two - a relatively simple and explicit injective resolution of the Frobenius twist functor, defined in terms of tensor products of symmetric powers, with differentials defined in terms of the natural Hopf structure on those symmetric powers. Later, Troesch showed that this construction could be generalized to characteristic p > 2, although the underlying object was no longer an ordinary cochain complex, but rather a p-complex, i.e., a graded space with "differential" satisfying d^p = 0 rather than the usual d^2 = 0. In this talk, I'll give an overview of some of these constructions, and then discuss work with Jon Kujawa in which we looked at how the Troesch construction generalized to the world of super (i.e., Z/2-graded) vector spaces.

Much of the early work on Fusion Categories was inspired by physicists' desire for rigorous foundations of topological quantum field theory. One effect of this was that base fields other than the complex numbers were rarely considered, if at all. The relevant features of $\mathbb C$ that make the theory work are the fact that it is characteristic zero, and algebraically closed. This talk will focus on the interesting things that can be found when the algebraically closed requirement is removed. The content will start with lots of examples, and slowly accelerate into higher categorical implications.

The Burau representation is a q-analogue of the permutation representation of the symmetric group which admits an action by both the braid and twin group. Both groups are obtained by omitting a relation in the Artin presentation of the symmetric group. The braid group omits the involution relation, and the twin group omits the cubic relation. The centralizers of tensor powers of the Burau representation for these groups yield new instances of Schur-Weyl duality. The centralizers are combinatorially described in terms of partial permutation and partial Brauer algebras. This is joint work with Stephen Doty.

Recent work of Zakharevich and Campbell has focused on developing the K-theoretic machinery to study scissors congruence problems and applying these tools to the Grothendieck ring of varieties. In this talk we will discuss a new application of their framework to study the so called cut-and-paste invariants of manifolds. Namely, we will describe a K-theory spectrum, which recovers the classical groups SK_n ("schneiden und kleben" is German for "cut and paste") as its zeroth homotopy group. We will also explain how the Euler characteristic, which is an example of a cut-and-paste invariant, fits into this new setup. This is joint work with R. Hoekzema, M. Merling, L. Murray, and C. Rovi.

The meridional rank conjecture of Cappell and Shaneson posits equality between an algebraic invariant and a geometric invariant of links in S^3. I will describe a diagram coloring technique - which amounts to finding a Coxeter quotient of the fundamental group of the link complement - by which we can establish the conjecture for infinite classes of links. As a corollary, we derive formulas for the bridge numbers for the links in question. This talk should check the "T" and "A" boxes in "TACO" and lightly flirt with "C". Based on joint works with Blair, Baader, Misev.

The history of Lie bialgebras began with the paper where the Lie bialgebras were defined: V. G. Drinfeld, "Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations", Dokl. Akad. Nauk SSSR, 268:2 (1983) Presented: L.D. Faddeev. Received: 04.06.1982. The aim of my talk is to celebrate 40 years of Lie bialgebras in mathematics and to explain how these important algebraic structures can be classified. This classification goes "hand in hand" with the classification of the so-called Manin triples and Drinfeld doubles also introduced in Drinfeld's paper cited above. The ingenious idea how to classify Drinfeld doubles associated with Lie algebras possessing a root system is due to F. Montaner and E. Zelmanov. In particular, using their approach the speaker classified Lie bialgebras, Manin triples and Drinfeld doubles associated with a simple finite dimensional Lie algebra g (the paper was based on a private communication by E. Zelmanov and it was published in Comm. Alg. in 1999). Further, in 2010, F. Montaner, E. Zelmanov and the speaker published a paper in Selecta Math., where they classified Drinfeld doubles on the Lie algebra of the formal Taylor power series g[[u]] and all Lie bialgebra structures on the polynomial Lie algebra g[u]. Finally, in March 2022 S. Maximov, E. Zelmanov and the speaker published an Arxiv preprint, where they made a crucial progress towards a complete classification of Manin triples and Lie bialgebra structures on g[[u]]. Of course, it is impossible to compress a 40 years history of the subject in one talk but the speaker will try his best to do this.