arXiv subjects

Nathan Pflueger

Publications and source records attributed to Nathan Pflueger.

At least 19 recordsLinked to original sources

arXiv2026

Formalizing chip-firing and Riemann--Roch for graphs in Lean 4

The Riemann--Roch theorem for graphs, due to Baker and Norine, is a foundational result establishing a powerful analogy between finite graphs and algebraic curves. We describe a complete formal proof of this theorem implemented in the Lean 4 theorem prover. Our formalization includes the existence and uniqueness of q-reduced divisors, a modified form of Dhar's burning algorithm, the bijection between acyclic orientations with unique source and maximal superstable configurations, and Clifford's theorem. We also include several challenges for future formalization.

math.CO↗

arXiv2026

Benchmarks in Leipzig

Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the 3-day workshop *Benchmarks in Leipzig* with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100 questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive.

math.HO↗

arXiv2026

Transmission permutations and Demazure products in Hurwitz--Brill--Noether theory

A line bundle on a curve with two marked points can be special in many ways, as measured by the global sections of all of its twists by these points. All of this information is conveniently packaged into a permutation, which we call the transmission permutation. We prove that when twice-marked curves are chained together, these permutations are composed via the Demazure product; in reverse, bundles with given permutation can be enumerated via reduced decompositions of a permutation. This paper demonstrates the utility of transmission permutations by giving a short derivation of the basic dimension bounds of both classical Brill--Noether theory and Hurwitz--Brill--Noether theory in a unified framework. The difference between the two cases derives from taking permutations in either symmetric groups or affine symmetric groups.

math.AG↗

arXiv2025

chipfiring: A Python Package for Efficient Mathematical Analysis of Chip-Firing Games on Multigraphs

This paper presents `chipfiring`, a comprehensive Python package for the mathematical analysis of chip-firing games on finite graphs. The package provides a robust toolkit for defining graphs and chip configurations (divisors), performing chip-firing operations, and analyzing fundamental properties such as winnability, linear equivalence, and divisor rank. We detail the core components of the library, including its object-oriented graph and divisor implementations, integrated Laplacian matrix computations, and an efficient implementation of Dhar's algorithm for determining the solvability of the dollar game. The `chipfiring` package is designed for researchers and students in graph theory, combinatorics, and algebraic geometry, providing essential algorithms and data structures for exploring these rich mathematical models. We describe the library's architecture, illustrate its usage with comprehensive examples, and highlight its specialized contributions compared to general-purpose graph libraries.

math.CO↗

arXiv2024

Versality of Brill-Noether flags and degeneracy loci of twice-marked curves

A Brill-Noether degeneracy locus is closure in $\Pic^d(C)$ of the locus of line bundles with a specified rank function $r(a,b) = h^0(C,L(-ap-bq))$. These loci generalize the classical Brill-Noether loci $W^r_d(C)$ as well as Brill-Noether loci with imposed ramification. For general $(C,p,q)$ we determine the dimension, singular locus, and intersection class of Brill-Noether degeneracy loci, generalizing classical results about $W^r_d(C)$. The intersection class has a combinatorial interpretation in terms of the number of reduced words for a permutation associated to the rank function, or alternatively the number of saturated chains in the Bruhat order. The essential tool is a versality theorem for a certain pair of flags on $\Pic^d(C)$, conjectured by Melody Chan and the author.

math.AG↗

arXiv2023

Weierstrass semigroups from cyclic covers of hyperelliptic curves

The {\it Weierstrass semigroup} of pole orders of meromorphic functions in a point $p$ of a smooth algebraic curve $C$ is a classical object of study; a celebrated problem of Hurwitz is to characterize which semigroups ${\rm S} \subset \mathbb{N}$ with finite complement are {\it realizable} as Weierstrass semigroups ${\rm S}= {\rm S}(C,p)$. In this note, we establish realizability results for cyclic covers $π: (C,p) \rightarrow (B,q)$ of hyperelliptic targets $B$ marked in hyperelliptic Weierstrass points; and we show that realizability is dictated by the behavior under $j$-fold multiplication of certain divisor classes in hyperelliptic Jacobians naturally associated to our cyclic covers, as $j$ ranges over all natural numbers.

math.AG↗

arXiv2023

Relative Richardson Varieties

A Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties, which are defined over a base scheme with a vector bundle and two flags. To do so, we generalize transversality of flags to a relative notion, versality, that allows the flags to be non-transverse over some fibers. Relative Richardson varieties share many of the geometric properties of Richardson varieties. We generalize several geometric and cohomological facts about Richardson varieties to relative Richardson varieties. We also prove that the local geometry of a relative Richardson variety is governed, in a precise sense, by the two intersecting Schubert varieties, giving a generalization, in the flag variety case, of a theorem of Knutson-Woo-Yong; we also generalize this result to intersections of arbitrarily many relative Schubert varieties. We give an application to Brill-Noether varieties on elliptic curves, and a conjectural generalization to higher genus curves.

math.AG↗

arXiv2022

Linear series with $ρ< 0$ via thrifty lego-building

The moduli space $\mathcal{G}^r_{g,d} \to \mathcal{M}_g$ parameterizing algebraic curves with a linear series of degree $d$ and rank $r$ has expected relative dimension $ρ= g - (r+1)(g-d+r)$. Classical Brill-Noether theory concerns the case $ρ\geq 0$; we consider the non-surjective case $ρ< 0$. We prove the existence of components of this moduli space with the expected relative dimension when $0 > ρ\geq -g+3$, or $0 > ρ\geq -C_r g + \mathcal{O}(g^{5/6})$, where $C_r$ is a constant depending on the rank of the linear series such that $C_r \to 3$ as $r \to \infty$. These results are proved via a two-marked-point generalization suitable for inductive arguments, and the regeneration theorem for limit linear series.

math.AG↗

arXiv2022

Twice-Marked Banana Graphs & Brill-Noether Generality

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 are the first such examples since the analysis of chains of loops by Cools, Draisma, Payne and Robeva. The graphs constructed are chains of loops and "theta graphs," which are banana graphs of genus 2. We also demonstrate that almost all banana graphs of genus at least 3 cannot be used for this purpose, due either to failure of a submodularity condition or to the presence of far too many inversions in certain permutations associated to divisors called transmission permutations.

math.CO↗

arXiv2022

An extended Demazure product on integer permutations via min-plus matrix multiplication

Coxeter groups possess an associative operation, called variously the Demazure, greedy, or 0-Hecke product. For symmetric groups, this product has an amusing formulation, due to Tiskin, as matrix multiplication in the min-plus (tropical) semiring of two matrices associated to the permutations. We prove that this min-plus formulation extends to furnish a Demazure product on a much larger group of integer permutations, consisting of all permutations that change the sign of finitely many integers. We prove several alternative descriptions of this product and some useful properties. These results were developed in service of Brill--Noether theory of algebraic and tropical curves; the connection is surveyed in an appendix. The main theorems of this paper have been fully formalized in Lean 4.

math.CO↗

arXiv2022

(Hurwitz-)Brill-Noether general marked graphs via the Demazure product

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 loops of torsion order $k$ is Hurwitz-Brill-Noether general in the sense of splitting loci (a theorem of Cook-Powell-Jensen). In fact, we prove a generalization to (metric) graphs with two marked points, that behaves well under vertex gluing. The key construction is a way to associate permutations to divisors on twice-marked graphs, simultaneously encoding the ranks of every twist of the divisor by the marked points. Vertex gluing corresponds to the Demazure product, which can be formulated via tropical matrix multiplication.

math.CO↗

arXiv2021

Euler characteristics of Brill-Noether varieties

We prove an enumerative formula for the algebraic Euler characteristic of Brill-Noether varieties, parametrizing degree d and rank r linear series on a general genus g curve, with ramification profiles specified at up to two general points. Up to sign, this Euler characteristic is the number of standard set-valued tableaux of a certain skew shape with g labels. We use a flat degeneration via the Eisenbud-Harris theory of limit linear series, relying on moduli-theoretic advances of Osserman and Murray-Osserman; the count of set-valued tableaux is an explicit enumeration of strata of this degeneration.

math.AG↗

arXiv2021

Weierstrass semigroups on Castelnuovo curves

We define a class of numerical semigroups S, which we call Castelnuovo semigroups, and study the subvariety $M^S_{g,1}$ of $M_{g,1}$ consisting of marked smooth curves with Weierstrass semigroup S. We determine the number of irreducible components of these loci and determine their dimensions. Curves with these Weierstrass semigroups are always Castelnuovo curves, which provides the basic tool for our argument. This analysis provides examples of numerical semigroups for which $M^S_{g,1}$ is reducible and non-equidimensional.

math.AG↗

arXiv2021

On non-primitive Weierstrass points

We give an upper bound on the codimension in $M_{g,1}$ of the variety $M^S_{g,1}$ of marked curves $(C,p)$ with a given Weierstrass semigroup. The bound is a combinatorial quantity which we call the effective weight of the semigroup; it is a refinement of the weight of the semigroup, and differs from it precisely when the semigroup is not primitive. We prove that whenever the effective weight is less than g, the variety $M^S_{g,1}$ is nonempty and has a component of the predicted codimension. These results extend previous results of Eisenbud, Harris, and Komeda to the case of non-primitive semigroups. We also survey other cases where the codimension of $M^S_{g,1}$ is known, as evidence that the effective weight estimate is correct in much wider circumstances.

math.AG↗

arXiv2020

The Gieseker-Petri theorem and imposed ramification

We prove a smoothness result for spaces of linear series with prescribed ramification on twice-marked elliptic curves. In characteristic 0, we then apply the Eisenbud-Harris theory of limit linear series to deduce a new proof of the Gieseker-Petri theorem, along with a generalization to spaces of linear series with prescribed ramification at up to two points. Our main calculation involves the intersection of two Schubert cycles in a Grassmannian associated to almost-transverse flags.

math.AG↗

arXiv2020

Combinatorial relations on skew Schur and skew stable Grothendieck polynomials

We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in earlier joint work with López Martín and Teixidor i Bigas concerning Brill-Noether curves, and it generalizes a 2000 formula of Lenart and a recent result of Reiner-Tenner-Yong to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms of skew Grothendieck polynomials.

math.CO↗

arXiv2017

Special divisors on marked chains of cycles

We completely describe all Brill-Noether loci on metric graphs consisting of a chain of g cycles with arbitrary edge lengths, generalizing work of Cools, Draisma, Payne, and Robeva. The structure of these loci is determined by displacement tableaux on rectangular partitions, which we define. More generally, we fix a marked point on the rightmost cycle, and completely analyze the loci of divisor classes with specified ramification at the marked point, classifying them using displacement tableaux. Our results give a tropical proof of the generalized Brill-Noether theorem for general marked curves, and serve as a foundation for the analysis of general algebraic curves of fixed gonality.

math.CO↗

arXiv2017

Brill-Noether varieties of k-gonal curves

We consider a general curve of fixed gonality k and genus g. We propose an estimate for the dimension of the variety $W^r_d(C)$ of special linear series on C, by solving an analogous problem in tropical geometry. Using work of Coppens and Martens, we prove that this estimate is exactly correct if k is at least g/5 + 2, and is an upper bound in all other cases. We also completely characterize the cases in which $W^r_d(C)$ has the same dimension as for a general curve of genus g.

math.AG↗