Algebraic and geometric aspects of signatures and rough analysis
Weierstrass Institute Berlin, February 4—6, 2026
Joscha Diehl
Tensor-to-Tensor Models with Fast Iterated Sum Features
We present a new class of tensor-to-tensor (in particular: image- to-image) models based on iterated sums. Their efficient computation is inspired by recent breakthrough results in the field of permutation pattern counting. Work in progress with R. Ibraheem, L. Schmitz and Y. Wue.
Pardis Semnani
Path-Dependent SDEs: Solutions and Parameter Estimation
We develop a consistent method for estimating the parameters of a rich class of path-dependent SDEs, called signature SDEs, which can model general path-dependent phenomena. Path signatures are iterated integrals of a given path with the property that any sufficiently nice function of the path can be approximated by a linear functional of its signatures. This is why we model the drift and diffusion of our signature SDE as linear functions of path signatures. We provide conditions that ensure the existence and uniqueness of solutions to a general signature SDE. We then introduce the Expected Signature Matching Method (ESMM) for linear signature SDEs, which enables inference of the signature-dependent drift and diffusion coefficients from observed trajectories. Furthermore, we show that ESMM is consistent: given sufficiently many samples and Picard iterations used by the method, the parameters estimated by the ESMM approach the true parameter with arbitrary precision. Finally, we demonstrate on a variety of empirical simulations that the ESMM accurately infers the drift and diffusion parameters from observed trajectories.
This is joint work with Vincent Guan, Elina Robeva, and Darrick Lee.
Annika Burmester
Algebraic structures of multiple (q-)zeta values
We give an introductory overview of the theory of multiple zeta values and indicate how to obtain a Hopf algebra structure on a suitable quotient of the algebra of (formal) multiple zeta values. Multiple q-zeta values are certain q-series that degenerate to multiple zeta values in the limit q to 1. We discuss how the algebraic structures associated with multiple zeta values extend to the setting of multiple q-zeta values.
Georg Regensburger
Integro-differential rings with generalized evaluation and shuffle relations
In this talk, we discuss the fundamental theorem of calculus and its algebraic implications for differential rings, allowing for functions with singularities and a generalized notion of evaluation. We give an overview of integro-differential rings and present several examples. This framework generalizes classical results such as shuffle relations for nested integrals and the Taylor formula, incorporating additional terms that account for singularities.
In general, not every element of a differential ring has an antiderivative within the same ring. Starting from a commutative differential ring together with a direct decomposition into integrable and non-integrable elements, we outline aspects of the construction of the free integro-differential ring. This integro-differential closure contains, in particular, all nested integrals of elements of the original differential ring.
This is joint work with Clemens Raab.
Fride Straum
Iterated Tensor Algebras
In this talk, we will present the algebraic skeleton of the theory related to iterated signatures, setting aside the topological, analytical, and applied considerations. Iterated signatures can model 2D and 3D data, hence may prove useful in applications. The iterated tensor algebra is surprisingly well-behaved, despite being an iteration of infinite-dimensional objects. However, if one wants a more detailed understanding of how elements of an n-iterated tensor algebra look like, one needs to zoom in on all n iterations down to the underlying vector space. Therefore, we end this talk with an investigation of the double tensor algebra. This talk is a section of ongoing work together with Fred Espen Benth, Kurusch Ebrahimi-Fard, and Fabian Andsem Harang.
Darrick Lee
Iterated Signatures
In this talk, we discuss an approach to use iterated path signatures to encode information about higher dimensional maps . We introduce the iterated signature from the point of view of half shuffles, and discuss the corresponding algebraic structures. Then, we will discuss some examples which demonstrate how iteration captures additional information. This is based on joint work in progress with Rosa Preiss.
Horatio Boedihardjo
Decay rate of logarithmic signature
It is well-known that the coefficients of path signatures decay at least factorially fast, while the decay rate of the coefficients of the logarithmic signature is generally geometric. It was conjectured by T. Lyons and N. Sidorova that, with the exceptions of straight lines, the logarithmic signatures of tree-reduced bounded variation paths have infinite radius of convergence. This conjecture was confirmed in the same work for certain types of paths and the general BV case remains unsolved.
In this talk, we develop a deeper understanding towards the Lyons-Sidorova conjecture. We prove that, if the logarithmic signature has infinite radius of convergence, the signature coefficients must satisfy an infinite system of rigid algebraic identities defined in terms of iterated integrals along complex exponential one-forms. These iterated integral identities impose strong geometric constraints on the underlying path, and in some special situations, confirm the conjecture. As a non-trivial application of our integral identities, we prove a weak version of the conjecture, which asserts that if the logarithmic signature of a BV path has infinite radius of convergence over all sub-intervals of time, the underlying path must be a straight line.
Joint work with Xi Geng and Sheng Wang.
Tim Seynnaeve
Decomposing tensor spaces via path signatures
The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. The set of all signature tensors at a particular level gives rise to the universal signature variety. We show that the parametrization of this variety induces a natural decomposition of the tensor space via representation theory, and connect this to the study of path invariants. We also reveal certain constraints that apply to the rank and symmetry of a signature tensor. This talk is based on joint work with Carlos Améndola, Francesco Galuppi, Ángel David Ríos Ortiz, and Pierpaola Santarsiero.
Pierpaola Santarsiero
Algebraic aspects of signature tensors
In this talk, we present a systematic study of basic algebraic properties of signature tensors, with a particular focus on rank, symmetry and conciseness. We will see a sharp upper bound on the rank of signature tensors of piecewise linear paths; we will analyze several types of symmetry, and provide a characterization of conciseness of signature tensors in terms of the corresponding path. Time permitting, we will also explore how to algebraically characterize paths lying on an algebraic variety by giving conditions on the entries of the signatures.
Vincenzo Galgano
(Discrete) Signature tensors for persistence barcodes
Signature tensors of paths were applied to persistence barcodes in topological data analysis (Chevyrev, Nanda and Oberhauser, 2020). We further investigate this direction in both an applied and theoretical perspective. On the applied side we consider discrete signature tensors (Diehl, Ebrahimi-Fard and Tapia - 2020) to define the discrete landscape feature map: to any persistence landscape we associate the discrete signature tensor of the time-series of the critical points. We show that this map is stable and apply it to a protein dataset. On the theoretical side, we give results on algebraic varieties of signature matrices for naive paths of barcodes.
Yannic Vargas
Schröder Trees and Anshelevich's Free Wick Polynomials
A non-commutative analogue of classical probability notion in free probability is given by the concept of free Wick polynomials, introduced by Anshelevich. In both classical and quantum stochastic analysis, the classical Wick polynomials play a fundamental role, particularly in the study of orthogonal polynomials and operator product renormalisation. In classical probability, they normalize random variable products to create orthogonal systems, acting as multi-dimensional analogues of Hermite polynomials linked to Wiener–Itô integrals. In free probability, free Wick polynomials extend these ideas in the context of free independence. Anshelevich's work connects free Wick products to stochastic calculus, using free Brownian motion. In this talk, we present a new combinatorial formula that writes free Wick polynomials in terms of Schröder trees and non-crossing partitions, using the notion of antipode of a Hopf algebra. This is joint work with Adrián Celestino.
Loïc Foissy
Strange pre- and post-Lie structures on rooted trees
We present a construction of pre-Lie on rooted trees whose edges and vertices are decorated, with a grafting product twisted by an action of a map acting on both edges and vertices. We show that this construction indeed gives a pre-Lie algebra if, and only if, a certain commutation relation is satisfied. Then, this pre-Lie algebra can be extended as a post-Lie algebra through a semi-direct product.
A particular example is used for normal forms in the study of stochastics PDEs. Here, the set of decorations of edges and vertices is and the acting map is the exponentiation of a simpler map.