Searcharxiv⌕ Search

arXiv subjects

Christian Stump

Publications and source records attributed to Christian Stump.

At least 19 recordsLinked to original sources
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

Soohak: A Mathematician-Curated Benchmark for Evaluating Research-level Math Capabilities of LLMs

Following the recent achievement of gold-medal performance on the IMO by frontier LLMs, the community is searching for the next meaningful and challenging target for measuring LLM reasoning. Whereas olympiad-style problems measure step-by-step reasoning alone, research-level problems use such reasoning to advance the frontier of mathematical knowledge itself, emerging as a compelling alternative. Yet research-level math benchmarks remain scarce because such problems are difficult to source (e.g., Riemann Bench and FrontierMath-Tier 4 contain 25 and 50 problems, respectively). To support reliable evaluation of next-generation frontier models, we introduce Soohak, a 439-problem benchmark newly authored from scratch by 64 mathematicians. Soohak comprises two subsets. On the Challenge subset, frontier models including Gemini-3-Pro, GPT-5, and Claude-Opus-4.5 reach 30.4%, 26.4%, and 10.4% respectively, leaving substantial headroom, while leading open-weight models such as Qwen3-235B, GPT-OSS-120B, and Kimi-2.5 remain below 15%. Notably, beyond standard problem solving, Soohak introduces a refusal subset that probes a capability intrinsic to research mathematics: recognizing ill-posed problems and pausing rather than producing confident but unjustified answers. On this subset, no model exceeds 50%, identifying refusal as a new optimization target that current models do not directly address. To prevent contamination, the dataset will be publicly released in late 2026, with model evaluations available upon request in the interim.

cs.CL↗
arXiv2025

Extending the ab-index

We prove for finite, graded, bounded posets, that the Poincar\'e-extended ab-index is obtained from the ab-index via the omega-transformation. This proves a conjecture by Dorpalen-Barry, Maglione, and the second author, and provides a more conceptual approach to ab-indices and Chow polynomials beyond R-labeled posets.

math.CO↗
arXiv2025

Positive $m$-divisible non-crossing partitions and their Kreweras maps

We study positive $m$-divisible non-crossing partitions and their positive Kreweras maps. In classical types, we describe their combinatorial realisations as certain non-crossing set partitions. We also realise these positive Kreweras maps as pseudo-rotations on a circle, respectively on an annulus. We enumerate positive $m$-divisible non-crossing partitions in classical types that are invariant under powers of the positive Kreweras maps with respect to several parameters. In order to cope with the exceptional types, we develop a different combinatorial model in general type describing positive $m$-divisible non-crossing partitions that are invariant under powers of the positive Kreweras maps. We finally show that altogether these results establish several cyclic sieving phenomena.

math.CO↗
arXiv2025

On the global dimension of Nakayama algebras

We study the global dimension of Nakayama algebras. In the case of linear Nakayama algebras, which are in canonical bijection to Dyck paths, we show that the global dimension has the same distribution as the height of Dyck paths. For cyclic Nakayama algebras an explicit classification of finite global dimension is not known. However, we show that in certain special cases cyclic Nakayama algebras with finite global dimension can again be interpreted as Dyck paths. In particular, we show that there is a natural bijection between sincere Nakayama algebras and Dyck paths. In this case, we find that the global dimension is in fact twice the bounce count of the corresponding Dyck path.

math.CO↗
arXiv2024

Chow and augmented Chow polynomials as evaluations of Poincar\'e-extended ab-indices

We show that Chow polynomials and augmented Chow polynomials of matroids, and more generally of finite graded posets admitting R-labelings, are obtained as evaluations of their Poincar\'e-extended ab-indices. This implies in particular explicit combinatorial $\gamma$-positive expansions for both, providing the first proof of the $\gamma$-positivity not relying on the K\"ahler package for the Chow ring. We then evaluate this expansion to obtain an explicit closed formula for the braid arrangement.

math.CO↗
arXiv2024

The coarse flag Hilbert-Poincar\'e series of the braid arrangement

The paper concerns the coarse flag Hilbert-Poincar\'e series of Maglione-Voll in the case of the braid arrangement associated to the symmetric group. We explicitly construct a companion statistic $\operatorname{ino} : \mathfrak{S}_{n+1} \times \operatorname{Sym}(n) \rightarrow \mathbb{N}$ for the descent statistic on $\operatorname{Sym}(n)$ using reverse $(P,\omega)$-partitions and quasisymmetric functions.

math.CO↗
arXiv2023

The Poincar\'e-extended ab-index

Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincar\'e-extended ab-index, which generalizes both the ab-index and the Poincar\'e polynomial. For posets admitting R-labelings, we give a combinatorial description of the coefficients of the extended ab-index, proving their nonnegativity. In the case of intersection posets of hyperplane arrangements, we prove the above conjecture of the second author and Voll as well as another conjecture of the second author and K\"uhne. We also define the pullback ab-index generalizing the cd-index of face posets for oriented matroids. Our results recover, generalize and unify results from Billera-Ehrenborg-Readdy, Bergeron-Mykytiuk-Sottile-van Willigenburg, Saliola-Thomas, and Ehrenborg. This connection allows us to translate our results into the language of quasisymmetric functions, and-in the special case of symmetric functions-make a conjecture about Schur positivity. A proof of this conjecture now appears an appendix by Ricky Ini Liu.

math.CO↗
arXiv2022

Shi arrangements restricted to Weyl cones

We consider the restrictions of Shi arrangements to Weyl cones, their relations to antichains in the root poset, and their intersection posets. For any Weyl cone, we provide bijections between regions, flats intersecting the cone, and antichains of a naturally-defined subposet of the root poset. This gives a refinement of the parking function numbers via the Poincar\'e polynomials of the intersection posets of all Weyl cones. Finally, we interpret these Poincar\'e polynomials as the Hilbert series of three isomorphic graded rings. One of these rings arises from the Varchenko-Gel'fand ring, another is the coordinate ring of the vertices of the order polytope of a subposet of the root poset, and the third is purely combinatorial.

math.CO↗
arXiv2022

Central limit theorems for generalized descents and generalized inversions in finite root systems

We consider generalized inversions and descents in finite Weyl groups. We establish Coxeter-theoretic properties of indicator random variables of positive roots such as the covariance of two such indicator random variables. We then compute the variances of generalized inversions and descents in classical types. We finally use the dependency graph method to prove central limit theorems for general antichains in root posets and in particular for generalized descents, and then for generalized inversions.

math.PR↗
arXiv2021

Bruhat intervals, subword complexes and brick polyhedra for finite Coxeter groups

We study the interplay between the discrete geometry of Bruhat poset intervals and subword complexes of finite Coxeter systems. We establish connections between the cones generated by cover labels for Bruhat intervals and of root configurations for subword complexes, culminating in the notion of brick polyhedra for general subword complexes.

math.CO↗
arXiv2020

A combinatorial classification of 2-regular simple modules for Nakayama algebras

Enomoto showed for finite dimensional algebras that the classification of exact structures on the category of finitely generated projective modules can be reduced to the classification of 2-regular simple modules. In this article, we give a combinatorial classification of 2-regular simple modules for Nakayama algebras and we use this classification to answer several natural questions such as when there is a unique exact structure on the category of finitely generated projective modules for Nakayama algebras. We also classify 1-regular simple modules, quasi-hereditary Nakayama algebras and Nakayama algebras of global dimension at most two. It turns out that most classes are enumerated by well-known combinatorial sequences, such as Fibonacci, Riordan and Narayana numbers. We first obtain interpretations in terms of the Auslander-Reiten quiver of the algebra using homological algebra, and then apply suitable bijections to relate these to combinatorial statistics on Dyck paths.

math.CO↗
arXiv2020

Minkowski decompositions for generalized associahedra of acyclic type

We give an explicit subword complex description of the generators of the type cone of the g-vector fan of a finite type cluster algebra with acyclic initial seed. This yields in particular a description of the Newton polytopes of the F-polynomials in terms of subword complexes as conjectured by S. Brodsky and the third author. We then show that the cluster complex is combinatorially isomorphic to the totally positive part of the tropicalization of the cluster variety as conjectured by D. Speyer and L. Williams.

math.CO↗
arXiv2019

Standard complexes of matroids and lattice paths

Motivated by Gröbner basis theory for finite point configurations, we define and study the class of "standard complexes" associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant under matroid duality. For the lexicographic term order, the standard complexes satisfy a deletion-contraction-type recurrence. We explicitly determine the lexicographic standard complexes for lattice path matroids using classical bijective combinatorics.

math.CO↗
arXiv2019

Counting inversions and descents of random elements in finite Coxeter groups

We investigate Mahonian and Eulerian probability distributions given by inversions and descents in general finite Coxeter groups. We provide uniform formulas for the means and variances in terms of Coxeter group data in both cases. We also provide uniform formulas for the double-Eulerian probability distribution of the sum of descents and inverse descents. We finally establish necessary and sufficient conditions for general sequences of Coxeter groups of increasing rank under which Mahonian and Eulerian probability distributions satisfy central and local limit theorems.

math.CO↗
arXiv2019

A new face iterator for polyhedra and more general finite locally branched lattices

We discuss a new memory-efficient depth-first algorithm and its implementation that iterates over all elements of a finite locally branched lattice. This algorithm can be applied to face lattices of polyhedra and to various generalizations such as finite polyhedral complexes and subdivisions of manifolds, extended tight spans and closed sets of matroids. Its practical implementation is very fast compared to state-of-the-art implementations of previously considered algorithms. Based on recent work of Bruns, Garc\'ia-S\'anchez, O'Neill and Wilburne, we apply this algorithm to prove Wilf's conjecture for all numerical semigroups of multiplicity 19 by iterating through the faces of the Kunz cone and identifying the possible bad faces and then checking that these do not yield counterexamples to Wilf's conjecture.

math.CO↗
arXiv2019

Freeness of multi-reflection arrangements via primitive vector fields

In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. We then extend our results further to all imprimitive irreducible unitary reflection groups. In this case the exponents turn out to depend on the exponents of a certain Galois twist of the dual reflection representation that comes from a Beynon-Lusztig type semi-palindromicity of the fake degrees.

math.GR↗