arXiv subjects

Felix Lotter

Publications and source records attributed to Felix Lotter.

8 recordsLinked to original sources

arXiv2026

Piecewise Symmetric Tensors

Every square matrix is uniquely the sum of a symmetric matrix and a skew-symmetric matrix. We extend this familiar fact to higher-order tensors: every cubic $k$-tensor is uniquely the sum of an $m$-piecewise symmetric tensor and a $(k-m)$-piecewise skew-symmetric tensor, for each choice of $m\leq k$. We study these tensor spaces via the combinatorics of word descents and compositions. Under Schur--Weyl duality, the two spaces naturally decompose into descent representations of the symmetric group. In fact, we prove a variant of Solomon's classical decomposition of its group algebra that explains this decomposition. Our motivation stems from signature tensors in stochastic analysis, algebraic geometry and data science: more specifically, we show that our tensor space decompositions determine the vanishing ideals of the signatures of piecewise linear paths with a fixed number of segments.

math.RT↗

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

The Universe Fan

The wavefunction of the universe, as studied in perturbative quantum field theory, is a rational function whose singularities and factorization properties encode a rich underlying combinatorial structure. We define and study a broad generalization of such wavefunctions that can be associated to any lattice. We obtain these wavefunctions as the Laplace transform of a polyhedral fan, the universe fan, whose cones are defined by positivity conditions reflecting a notion of causality in the lattice, and we describe its face lattice. In the matroid case, the universe fan projects to the nested set fan, and the wavefunctions we define recover the matroid amplitudes introduced by Lam as residues. Moreover, in the case relevant for physics, the positivity conditions give a novel way to study the wavefunction, and we show how it is related to the cosmological polytopes of Arkani-Hamed, Benincasa, Postnikov. Finally, we study refinements of the universe fan induced by piecewise linear (tropical) functions. The resulting subdivisions project to refinements of the nested set fan and correspond dually to blow-ups of matroid polytopes, generalizing the cosmohedron polytope.

math.CO↗

arXiv2026

Signature Varieties of Splines

Splines are central objects for the interpolation of discrete data via piecewise smooth paths. Their iterated-integral signature is an infinite collection of tensors which characterizes paths almost uniquely. We study truncations of this collection, which define algebraic maps from parameter space to tensor space. We prove that the images of these maps are given by orbits of a matrix-tensor action. Furthermore, taking the Zariski closure, we define and study varieties of spline signature tensors. We determine dimension and degree of these tensor varieties in a number of examples, relying on symbolic computations. With a view towards learning, constructing paths with a given signature tensor translates to studying the fibers of the signature map. We use computational methods to determine their cardinality, with a focus on its dependence on different classes of splines. We observe in explicit examples that reconstructing splines from a given signature tensor of a path yields close approximations of the original path.

math.AG↗

arXiv2025

Signature matrices of membranes

The signature of a membrane is a sequence of tensors whose entries are iterated integrals. We study algebraic properties of membrane signatures, with a focus on signature matrices of polynomial and piecewise bilinear membranes. Generalizing known results for path signatures, we show that the two families of membranes admit the same set of signature matrices and we examine the corresponding affine varieties. In particular, we prove that there are no algebraic relations on signature matrices of membranes, in contrast to the path case. We complement our results by a linear time algorithm for the computation of signature tensors for piecewise bilinear membranes.

math.AG↗

arXiv2025

Computing Path Signature Varieties in Macaulay2

The signature of a path is a non-commutative power series whose coefficients are given by certain iterated integrals over the path coordinates. This series almost uniquely characterizes the path up to translation and reparameterization. Taking only fixed degree parts of these series yields signature tensors. We introduce the Macaulay2 package $\texttt{PathSignatures}$ to simplify the study of these interesting objects for piecewise polynomial paths. It allows for the creation and manipulation of parametrized families of paths and provides methods for computing their signature tensors and their associated algebraic varieties.

math.AG↗

arXiv2025

Cyclic polytopes through the lens of iterated integrals

The volume of a cyclic polytope can be obtained by forming an iterated integral along a suitable piecewise linear path running through its edges. Different choices of such a path are related by the action of a subgroup of the combinatorial automorphisms of the polytope. Motivated by this observation, we look for other linear combinations of iterated integrals that are invariant under the subgroup action. This yields interesting polynomial attributes of the cyclic polytope. We prove that there are infinitely many of these invariants which are algebraically independent in the shuffle algebra.

math.RA↗

arXiv2024

Cohomologically smooth morphisms for étale $\mathbb F_p$-sheaves in characteristic $p$

We scrutinise the notions of cohomologically smooth morphisms and smooth objects for the six functor formalism of étale $\mathbb F_p$-sheaves on schemes in characteristic $p$. We show that only cohomologically étale morphisms are cohomologically smooth in this setting. This is complemented by a characterisation of cohomologically étale morphisms in arbitrary characteristic. In fact, we prove that such a morphism is already étale up to universal homeomorphism.

math.AG↗