Related papers: Transmission permutations and Demazure products in…
This paper gives a novel and compact proof that a metric graph consisting of a chain of loops of torsion order $0$ is Brill-Noether general (a theorem of Cools-Draisma-Payne-Robeva), and a finite or metric graph consisting of a chain of…
Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main…
We analyze a family of graphs known as banana graphs, with two marked vertices, through the lens of Hurwitz-Brill-Noether theory. As an application, we construct explicit new examples of finite graphs which are Brill-Noether general. These…
Given a curve $C$ that is a degree $k$ cover $C \to \mathbb{P}^1$ totally ramified at two points $p$ and $q$, we can seek to understand the space of degree $d$ line bundles on $C$ with prescribed ramification at $p$ and $q$. The…
In this paper, we describe the Brill--Noether theory of a general smooth plane curve and a general curve $C$ on a Hirzebruch surface of fixed class. It is natural to study the line bundles on such curves according to the splitting type of…
In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work…
In this paper we study the Brill-Noether theory of sub-line bundles of a general, stable rank-two vector bundle on a curve C with general moduli. We relate this theory to the geometry of unisecant curves on smooth, non-special scrolls with…
For each endotrivial complex arising from Bredon homology of a representation sphere, we construct $p$-local quasi-isomorphisms, called forerunners, enabling us to extend Balmer--Gallauer's results in arXiv:2307.04398 Part II concerning the…
For a $p$-permutation equivalence between two block algebras of finite groups, we introduce new square diagrams that link the $p$-permutation equivalence via the Brauer construction to local equivalences between stabilizers of corresponding…
In this (mostly) survey article, we give a synopsis of a number of results relating to Brill--Noether theory on curves and metric graphs, together with some speculations about the behavior of one-dimensional linear series on a class of…
A duality theory of bundles of C$^*$-algebras whose fibres are twisted transformation group algebras is established. Classical T-duality is obtained as a special case, where all fibres are commutative tori, i.e. untwisted group algebras for…
We generalize the Embedding Theorem of Eisenbud-Harris from classical Brill-Noether theory to the setting of Hurwitz-Brill-Noether theory. More precisely, in classical Brill-Noether theory, the embedding theorem states that a general linear…
Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle…
We present in this paper the algebra of fused permutations and its deformation the fused Hecke algebra. The first one is defined on a set of combinatorial objects that we call fused permutations, and its deformation is defined on a set of…
Let $f\colon C \rightarrow \mathbb{P}^1$ be a degree $k$ genus $g$ cover. The stratification of line bundles $L \in \mathrm{Pic}^d(C)$ by the splitting type of $f_*L$ is a refinement of the stratification by Brill-Noether loci $W^r_d(C)$.…
We study two-dimensional topological gauge theories with gauge group equal to the symmetric group $S_n$ and their string theory duals. The simplest such theory is the topological quantum field theory of principal $S_n$ fiber bundles. Its…
We introduce a class of permutation centralizer algebras which underly the combinatorics of multi-matrix gauge invariant observables. One family of such non-commutative algebras is parametrised by two integers. Its Wedderburn-Artin…
Double Hurwitz numbers have at least four equivalent definitions. Most naturally, they count covers of the Riemann sphere by genus g curves with certain specified ramification data. This is classically equivalent to counting certain…
Coxeter groups possess an associative operation, called variously the Demazure, greedy, or $0$-Hecke product. For symmetric groups, this product has an amusing formulation as matrix multiplication in the min-plus (tropical) semiring of two…
Trigonal curves provide an example of Brill-Noether special curves. Theorem 1.3 of [9] characterizes the Brill-Noether theory of general trigonal curves and the refined stratification by Brill-Noether splitting loci, which parametrize line…