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Matthew Faust

Publications and source records attributed to Matthew Faust.

18 recordsLinked to original sources
arXiv2026

Generic Irreducibility of Bloch Varieties for Periodic Graph Operators

We give a complete characterization of generic irreducibility for dispersion polynomials and Bloch varieties of periodic graph operators. More precisely, we prove that for a generic choice of edge weights and potentials, the dispersion polynomial/Bloch variety of a nontrivial periodic graph is irreducible if and only if the quotient graph is connected. Our proof uses a strong dichotomy for parameterized Laurent polynomials: reducibility either occurs for every parameter or fails on a nonempty Zariski-open set. After establishing this dichotomy, we reduce the problem to minimally connected periodic graphs.

math.SP↗
arXiv2026

Common Real Secants to Pairs of Real Twisted Cubic Curves

It is well established that a general pair of twisted cubic curves in complex projective space has ten common secant lines. As an initial investigation, we show that the monodromy group of the ten common secant lines over the complex numbers is the full symmetric group demonstrating that the common secant lines have no special structure over the complex numbers. We then investigate a novel question in real algebraic geometry: describe the possible collections of ten common secant lines to a pair of real projective twisted cubic curves. In addition to distinguishing between real and nonreal secant lines, we introduce a refinement of this classification which takes intersection points into account yielding totally real, partially real, and minimally real secant lines. Using computational algebraic geometry as well as combinatorics, we show that for each $k$ between 0 and 10, there exist pairs of real twisted cubic curves with exactly $k$ common totally real secant lines. We also obtain examples of real twisted cubics whose sets of common real secants cover a wide range of possibilities within our admissible classification of common real secant lines.

math.AG↗
arXiv2026

A counterexample to Fermi isospectral rigidity for two dimensional discrete periodic Schr\"odinger operators

Using numerical certification, we prove the existence of a nontrivial real-valued two dimensional periodic potential whose associated discrete Schr\"odinger operator is Fermi isospectral to the zero potential. This provides a negative answer to a question posed by the third author concerning the rigidity of Fermi isospectrality in dimension two. This example also disproves a conjecture of Gieseker, Kn\"orrer, and Trubowitz in the 1990s stating that for any nontrivial real-valued periodic potential in dimension two, the Fermi variety is irreducible at all energy levels.

math.SP↗
arXiv2026

Algebraic Properties of the Ideal of Spectral Invariants for the Discrete Laplacian

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, with $q_j\in \mathbb{Z}^+$ for each $j\in \{1,\ldots,d\}$, and denote by $\Delta$ the discrete Laplacian on $\ell^2\left( \mathbb{Z}^d\right)$. We describe various algebraic properties of the ideal of spectral invariants for the discrete Laplacian when $d=1$, including a construction of a Gr\"obner basis. We also present various collections of complex $\Gamma$-periodic potentials $V$ that are such that $\Delta$ and $\Delta + V$ are Floquet isospectral. We end with a discussion of the general setting, where the $q_i$ are taken to be vectors in $\mathbb{Z}^d$.

math.SP↗
arXiv2026

Inverse Eigenvalue Problems, Floquet Isospectrality and the Hilbert--Chow Morphism

When can one change the diagonal of a matrix without changing its spectrum? We completely answer this question over an algebraically closed field of characteristic zero or larger than the size of the matrix: An $n \times n$ matrix $A$ admits a nonzero diagonal matrix $D$ such that $A$ and $A+D$ have the same spectrum if and only if, for some size $k$, the $k \times k$ principal minors of $A$ are not all equal. This relates to the classical additive inverse eigenvalue problem in numerical analysis and has implications for existence and rigidity results in the theory of Floquet isospectrality of discrete periodic operators in solid state physics. The proof employs new techniques involving Hilbert schemes of points and the infinitesimal structure of the Hilbert--Chow morphism.

math.AG↗
arXiv2025

The Critical Point Degree of a Periodic Graph

The critical point degree of a periodic graph operator is the number of critical points of its complex Bloch variety. Determining it is a step towards the spectral edges conjecture and more generally understanding Bloch varieties. Previous work showed that it is bounded above by the volume of the Newton polytope of the graph, and that the inequality is strict when there are asymptotic critical points. We identify contributions from asymptotic critical points that arise from the structure of the graph, and show that the critical point degree is bounded above by the difference of the volume of the Newton polytope and these contributions. These results have implications for nonlinear optimization.

math.SP↗
arXiv2025

The Spectral Edges Conjecture via Corners

The Spectral Edges Conjecture is a well-known and widely believed conjecture in the theory of discrete periodic operators. It states that the extrema of the dispersion relation are isolated, non-degenerate, and occur in a single band. We present two infinite families of periodic graphs which satisfy the Spectral Edges Conjecture. For each, every extremum of the dispersion relation is a corner point (point of symmetry). In fact, each spectral band function is a perfect Morse function. We also give a construction that increases dimension, while preserving that each spectral band function is a perfect Morse function.

math.SP↗
arXiv2025

Rare Flat Bands for Periodic Graph Operators

As a corollary of our main results, we prove that for any connected $\mathbb{Z}^d$-periodic graph, when edge weights and potentials are treated as variables, the corresponding periodic graph operators generically (i.e., outside a proper algebraic subset of the variable space) do not have flat bands.

math.SP↗
arXiv2024

LikelihoodGeometry: Macaulay2 Package

This note introduces the $\texttt{LikelihoodGeometry}$ package for the computer algebra system $\textit{Macaulay2}$. This package gives tools to construct the likelihood correspondence of a discrete algebraic statistical model, a variety that that ties together data and their maximum likelihood estimators. This includes methods for constructing and combining popular statistical models and calculating their ML-degree.

stat.CO↗
arXiv2024

Floquet Isospectrality of the Zero Potential for Discrete Periodic Schr\"odinger Operators

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, with $q_j\in (\mathbb{Z}^+)^d$ for each $j\in \{1,\ldots,d\}$, and denote by $\Delta$ the discrete Laplacian on $\ell^2\left( \mathbb{Z}^d\right)$. Using Macaulay2, we first numerically find complex-valued $\Gamma$-periodic potentials $V:\mathbb{Z}^d\to \mathbb{C}$ such that the operators $\Delta+V$ and $\Delta$ are Floquet isospectral. We then use combinatorial methods to validate these numerical solutions.

math.SP↗
arXiv2023

Likelihood Correspondence of Toric Statistical Models

Maximum likelihood estimation (MLE) is a fundamental problem in statistics. Characteristics of the MLE problem for discrete algebraic statistical models are reflected in the geometry of the $\textit{likelihood correspondence}$, a variety that ties together data and their maximum likelihood estimators. We construct this ideal for the large class of toric models and find a Gr\"{o}bner basis in the case of complete and joint independence models arising from multi-way contingency tables. All of our constructions are implemented in $\textit{Macaulay2}$ in a package $\texttt{LikelihoodGeometry}$ along with other tools of use in algebraic statistics. We end with an experimental section using these implementations on several interesting examples.

math.ST↗
arXiv2023

Irreducibility of the Dispersion Polynomial for Periodic Graphs

We use methods from algebra and discrete geometry to study the irreducibility of the dispersion polynomial of a discrete periodic operator associated to a periodic graph after changing the period lattice. We provide numerous applications of these results to discrete periodic operators associated to families of graphs which include dense periodic graphs, and a family containing the hexagonal and diamond lattices.

math.AG↗
arXiv2022

Critical points of discrete periodic operators

We study the spectra of operators on periodic graphs using methods from combinatorial algebraic geometry. Our main result is a bound on the number of complex critical points of the Bloch variety, together with an effective criterion for when this bound is attained. We show that this criterion holds for Z^2- and Z^3-periodic graphs with sufficiently many edges and use our results to establish the spectral edges conjecture for some Z^2-periodic graphs.

math-ph↗
arXiv2021

The Dry Ten Martini Problem at Criticality

We prove (actually this time) that every allowed gap label of the almost Mathieu operator is realized by an open gap at critical coupling for every irrational frequency. The main step is to show that the maximum Lyapunov exponent in each labeled gap is nondecreasing as the subcritical coupling increases. In the Landau representation, an antiunitary conjugation relates commuting left and right magnetic actions that agree on the rooted resolvent. A boundary balance on the lattice diagonals yields a self-adjoint compression and expresses $\lambda h_n'(\lambda)$ as a nonnegative weighted norm ratio of Green-function entries. Together with noncritical gap opening and upper semicontinuity of the Lyapunov exponent, this establishes critical gap opening.

math-ph↗
arXiv2019

The Surprising Accuracy of Benford's Law in Mathematics

Benford's law is an empirical ``law'' governing the frequency of leading digits in numerical data sets. Surprisingly, for mathematical sequences the predictions derived from it can be uncannily accurate. For example, among the first billion powers of $2$, exactly $301029995$ begin with digit 1, while the Benford prediction for this count is $10^9\log_{10}2=301029995.66\dots$. Similar ``perfect hits'' can be observed in other instances, such as the digit $1$ and $2$ counts for the first billion powers of $3$. We prove results that explain many, but not all, of these surprising accuracies, and we relate the observed behavior to classical results in Diophantine approximation as well as recent deep conjectures in this area.

math.PR↗
arXiv2017

Leading Digits of Mersenne Numbers

It has long been known that sequences such as the powers of $2$ and the factorials satisfy Benford's Law; that is, leading digits in these sequences occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In this paper, we consider the leading digits of the Mersenne numbers $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime. In light of known irregularities in the distribution of primes, one might expect that the leading digit sequence of $\{M_n\}$ has \emph{worse} distribution properties than "smooth" sequences with similar rates of growth, such as $\{2^{n\log n}\}$. Surprisingly, the opposite seems to be the true; indeed, we present data, based on the first billion terms of the sequence $\{M_n\}$, showing that leading digits of Mersenne numbers behave in many respects \emph{more regularly} than those in the above smooth sequences. We state several conjectures to this effect, and we provide an heuristic explanation for the observed phenomena based on classic models for the distribution of primes.

math.NT↗