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Greg Muller

Publications and source records attributed to Greg Muller.

At least 19 recordsLinked to original sources
arXiv2025

Separating dots with circles

Given a finite set of points in general position in the plane or sphere, we count the number of ways to separate those points using two types of circles: circles through three of the points, and circles through none of the points (up to an equivalence). In each case, we show the number of circles which separate the points into subsets of size k and l is independent of the configuration of points, and we provide an explicit formula in each case. We also consider how the circles change as the configuration of dots varies continuously. We show that an associated higher order Voronoi decomposition of the sphere changes by a sequence of local `moves'. As a consequence, an associated cluster algebra is independent of the configuration of dots, and only depends on the number of dots and the order of the Voronoi decomposition.

math.CO↗
arXiv2025

Valuative independence and cluster theta reciprocity

We prove that theta functions constructed from positive scattering diagrams satisfy valuative independence. That is, for certain valuations $\operatorname{val}_{v}$, we have $\operatorname{val}_v(\sum_u c_u \vartheta_u)=\min_{c_u\neq 0} \operatorname{val}_v(\vartheta_u)$. As applications, we prove linear independence of theta functions with specialized coefficients and characterize when theta functions for cluster varieties are unchanged by the unfreezing of an index. This yields a general gluing result for theta functions from moduli of local systems on marked surfaces. We then prove that theta functions for cluster varieties satisfy a symmetry property called theta reciprocity: briefly, $\operatorname{val}_v(\vartheta_u)=\operatorname{val}_u(\vartheta_v)$. For this we utilize a new framework called a "seed datum" for understanding cluster-type varieties. One may apply valuative independence and theta reciprocity together to identify theta function bases for global sections of line bundles on partial compactifications of cluster varieties.

math.AG↗
arXiv2024

Deep Points of Cluster Algebras

We initiate a systematic study of the deep points of a cluster algebra; that is, the points in the associated variety which are not in any cluster torus. We describe the deep points of cluster algebras of type A, rank 2, Markov, and unpunctured surface type.

math.AG↗
arXiv2023

A short proof of the finiteness of Dynkin friezes

We give a short and elementary proof that every Dynkin diagram admits finitely many (positive integral) friezes. This was originally proven by Gunawan-Muller using the geometry of cluster algebras. The proof here provides an explicit (albeit inefficient) bound on values.

math.CO↗
arXiv2022

Poisson geometry and Azumaya loci of cluster algebras

There are two main types of objects in the theory of cluster algebras: the upper cluster algebras ${{\boldsymbol{\mathsf U}}}$ with their Gekhtman-Shapiro-Vainshtein Poisson brackets and their root of unity quantizations ${{\boldsymbol{\mathsf U}}}_\varepsilon$. On the Poisson side, we prove that (without any assumptions) the spectrum of every finitely generated upper cluster algebra ${{\boldsymbol{\mathsf U}}}$ with its GSV Poisson structure always has a Zariski open orbit of symplectic leaves and give an explicit description of it. On the quantum side, we describe the fully Azumaya loci of the quantizations ${{\boldsymbol{\mathsf U}}}_\varepsilon$ under the assumption that ${{\boldsymbol{\mathsf A}}}_\varepsilon = {{\boldsymbol{\mathsf U}}}_\varepsilon$ and ${{\boldsymbol{\mathsf U}}}_\varepsilon$ is a finitely generated algebra. All results allow frozen variables to be either inverted or not.

math.RT↗
arXiv2022

Superunitary regions of cluster algebras

This note introduces the superunitary region of a cluster algebra, the subspace of the totally positive region on which each cluster variable is at least 1. Our main result is that the superunitary region of a finite type cluster algebra is a regular CW complex which is homeomorphic to the generalized associahedron of the cluster algebra. As an application, the compactness of the superunitary region implies that each Dynkin diagram admits finitely many positive integral friezes.

math.CO↗
arXiv2022

Juggler's friezes

This note generalizes $\mathrm{SL}(k)$-friezes to configurations of numbers in which one of the boundary rows has been replaced by a ragged edge (described by a juggling function). We provide several equivalent definitions/characterizations of these juggler's friezes, in terms of determinants, linear recurrences, and a dual juggler's frieze. We generalize classic results, such as periodicity, duality, and a parametrization by part of a Grassmannian. We also provide a method of constructing such friezes from certain $k \times n$ matrices using the twist of a matrix.

math.CO↗
arXiv2020

Algebraically-Informed Deep Networks (AIDN): A Deep Learning Approach to Represent Algebraic Structures

One of the central problems in the interface of deep learning and mathematics is that of building learning systems that can automatically uncover underlying mathematical laws from observed data. In this work, we make one step towards building a bridge between algebraic structures and deep learning, and introduce \textbf{AIDN}, \textit{Algebraically-Informed Deep Networks}. \textbf{AIDN} is a deep learning algorithm to represent any finitely-presented algebraic object with a set of deep neural networks. The deep networks obtained via \textbf{AIDN} are \textit{algebraically-informed} in the sense that they satisfy the algebraic relations of the presentation of the algebraic structure that serves as the input to the algorithm. Our proposed network can robustly compute linear and non-linear representations of most finitely-presented algebraic structures such as groups, associative algebras, and Lie algebras. We evaluate our proposed approach and demonstrate its applicability to algebraic and geometric objects that are significant in low-dimensional topology. In particular, we study solutions for the Yang-Baxter equations and their applications on braid groups. Further, we study the representations of the Temperley-Lieb algebra. Finally, we show, using the Reshetikhin-Turaev construction, how our proposed deep learning approach can be utilized to construct new link invariants. We believe the proposed approach would tread a path toward a promising future research in deep learning applied to algebraic and geometric structures.

cs.LG↗
arXiv2019

Linear recurrences indexed by $\mathbb{Z}$

This note considers linear recurrences (also called linear difference equations) in unknowns indexed by the integers. We characterize a unique \emph{reduced} linear recurrence with the same solutions as a given linear recurrence, and construct a \emph{solution matrix} which parametrizes the space of solutions. Several properties of solution matrices are shown, including a combinatorial characterization of bases and dimension of the space of solutions.

math.CO↗
arXiv2018

Non-Commutative Resolutions of Toric Varieties

Let $R$ be the coordinate ring of an affine toric variety. We show that the endomorphism ring $End_R(\mathbb A),$ where $\mathbb A$ is the (finite) direct sum of all (isomorphism classes of) conic $R$-modules, has finite global dimension. Furthermore, we show that $End_R(\mathbb A)$ is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field $k$ of prime characteristic, we show that the ring of differential operators $D_\mathsf{k}(R)$ has finite global dimension.

math.AC↗
arXiv2016

The twist for positroid varieties

The purpose of this note is to connect two maps related to certain graphs embedded in the disc. The first is Postnikov's boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain Pl\"ucker coordinates which are expected to be a cluster in a cluster structure. This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of Pl\"ucker coordinates.

math.CO↗
arXiv2015

Lower bound cluster algebras: presentations, Cohen-Macaulayness, and normality

We give an explicit presentation for each lower bound cluster algebra. Using this presentation, we show that each lower bound algebra Grobner degenerates to the Stanley-Reisner scheme of a vertex-decomposable ball or sphere, and is thus Cohen-Macaulay. Finally, we use Stanley-Reisner combinatorics and a result of Knutson-Lam-Speyer to show that all lower bound algebras are normal.

math.AC↗
arXiv2015

The greedy basis equals the theta basis

We prove the equality of two canonical bases of a rank 2 cluster algebra, the greedy basis of Lee-Li-Zelevinsky and the theta basis of Gross-Hacking-Keel-Kontsevich.

math.QA↗
arXiv2015

The existence of a maximal green sequence is not invariant under quiver mutation

This note provides a quiver which does not admit a maximal green sequence, but which is mutation-equivalent to a quiver which does admit a maximal green sequence. The proof uses the `scattering diagrams' of Gross-Hacking-Keel-Kontsevich to show that a maximal green sequence for a quiver determines a maximal green sequence for any induced subquiver.

math.QA↗
arXiv2014

Singularities of locally acyclic cluster algebras

We show that locally acyclic cluster algebras have (at worst) canonical singularities. In fact, we prove that locally acyclic cluster algebras of positive characteristic are strongly F-regular. In addition, we show that upper cluster algebras are always Frobenius split by a canonically defined splitting, and that they have a free canonical module of rank one. We also give examples to show that not all upper cluster algebras are F-regular if the local acyclicity is dropped.

math.AC↗
arXiv2014

Cluster Algebras of Grassmannians are Locally Acyclic

Considered as commutative algebras, cluster algebras can be very unpleasant objects. However, the first author introduced a condition known as "local acyclicity" which implies that cluster algebras behave reasonably. One of the earliest and most fundamental examples of a cluster algebra is the homogenous coordinate ring of the Grassmannian. We show that the Grassmannian is locally acyclic. Morally, we are in fact showing the stronger result that all positroid varieties are locally acyclic. However, it has not been shown that all positroid varieties have cluster structure, so what we actually prove is that certain cluster varieties associated to Postnikov's alternating strand diagrams are locally acylic. Moreover, we actually establish a slightly stronger property than local acyclicity, which we term the Louise property, that is designed to facilitate proofs involving the Mayer-Vietores sequence.

math.CO↗
arXiv2013

Computing upper cluster algebras

This paper develops techniques for producing presentations of upper cluster algebras. These techniques are suited to computer implementation, and will always succeed when the upper cluster algebra is totally coprime and finitely generated. We include several examples of presentations produced by these methods.

math.AC↗