This workshop will highlight new results and recent developments in the theory and applications of computational algebraic geometry. Speakers will present results on a range of techniques in the field including symbolic and numerical methods. Tools from algebraic geometry have been applied to a broad range of fields including: biology, chemistry, economics, optimization, physics, robotics, and statistics.
Workshops with close interactions:
I.4: Computational Geometry and Topology
I.5: Real Number Complexity
II.7: Computational Number Theory
III.1: Continuous Optimization
III.5: Symbolic Analysis
Organizers
MPI CBG and Center for Systems Biology Dresden
Texas A&M University
North Carolina State University
Speakers
Semi-plenary speakers
TU Dresden
Harvard University
Invited speakers
TU Berlin
MPI CBG Dresden
Max Planck Institute for Molecular Cell Biology and Genetics
MPI Leipzig
Osnabrück University
Western University
Worcester Polytechnic Institute
North Carolina State University
University of Barcelona
University of Leeds
University of Bern
Cunef University
University of Hawai’i
University of Notre Dame
OvGU Magdeburg
University of Waterloo
University of Michigan
MPI Leipzig
Leiden University
University of Barcelona
Monday, 13 July
![[HS 09 – 1st Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS09.png){kind=link}
14:00-14:30
Carlos Améndola (TU Berlin)
The expected signature of a family of paths need not be itself a signature of a path. Motivated by this, we consider the notion of a Lie group barycenter introduced by Buser and Karcher to propose a barycenter on path signatures. We show that the barycenter is an epimorphism of algebraic varieties that is represented by a non-commutative polynomial. In the case of piecewise linear paths, we study the problem of recovering an underlying path corresponding to the barycenter of signatures. This is joint work with Leonard Schmitz.
14:30-15:00
Jane Coons (Worcester Polytechnic Institute)
Given a fixed graph G, we are interested in finding the set of all graphs with the same discrete curvature sequence as G. Roost et al. recently described a Markov bases method for exploring this set of graphs. In this talk, we give a lattice basis for this problem which allows us to employ new machine learning methods for finding graphs with a fixed discrete curvature. We also show that the degree of the unique minimal Markov basis grows at least quadratically in the maximum degree of the graph. This is joint work with Giulio Zucal.
15:00-16:00 semi-plenary talk
Anna Seigal (Harvard University)
Many methods to describe systems rely on the assumption that unobserved variables in the system are mutually independent. Examples include independent component analysis and causal discovery procedures such as LiNGAM. Independence is a strong assumption. We show that it can be replaced by weaker restrictions, which say that the mean of a variable (rather than the whole distribution) is unchanged by conditioning on another variable. We demonstrate the identifiability of component analysis and causal discovery under these weaker mean-independence restrictions.
Just as independence-based procedures correspond to low-rank tensor decomposition problems, mean-independence is a different structured tensor decomposition. Parameter estimation methods in the mean-independent models compute eigenvectors of tensors and test the vanishing of polynomials. Based on joint work with Geert Mesters, Alvaro Ribot, and Piotr Zwiernik.
16:00-16:30 Coffee Break
16:30-17:00
Thomas Kahle (OvGU Magdeburg)
Given an arrangement of hypersurfaces, the critical locus of a Laurent monomial in their defining equations is an irreducible variety in a product
of two projective spaces. In algebraic statistics it is known as the likelihood correspondence and in particle physics as the scattering correspondence. We give an explicit determinantal representation of its bihomogeneous prime ideal whenever the arrangement satisfies the
strict normal crossing (SNC) condition. We show that this is also characterized by the non-vanishing of the Euler discriminant. The proof connects hyperplane arrangement theory with the commutative algebra of Rees algebras and Buchsbaum–Rim complexes. Joint work with Lukas Kühne, Leonie Mühlherr, Hal Schenck, Bernd Sturmfels, and Maximilian Wiesmann.
17:00-17:30
Viktoriia Borovik (MPI Leipzig)
In this talk, I will discuss how discriminants behave under flat degenerations. The key object is the relative conormal space, whose special fiber controls the limit of conormal varieties and, after projection, the limit of dual varieties. For Gröbner degenerations, the relative conormal space is itself a Gröbner degeneration, with the opposite weight on the dual coordinates. Extra components in the limit arise from Whitney strata of the family lying in the special fiber, and their multiplicities can be computed from Euler obstructions and local Milnor fibers via Sabbah’s formula. I will illustrate the results with examples from singular hypersurfaces, generic complete intersections, reciprocal linear spaces, and mixed discriminants of polynomial systems. This is joint work with Clara Briand.
17:30-18:00
Irem Portakal (MPI Leipzig)
In this talk, we explore the geometry underlying totally mixed Nash equilibria for n-player games. We model these equilibria using vector bundles and introduce the Nash equilibrium scheme, which parametrizes all totally mixed equilibria of a given game. Depending on the format of the game, we define two key geometric objects: the Nash discriminant and the Nash resultant varieties. These capture games whose Nash equilibrium schemes exhibit unexpected behavior – such as being nonreduced or having positive-dimensional components. No prior background in game theory will be assumed, and we will illustrate the main ideas with examples.
18:00-18:30
Elizabeth Gross (University of Hawaiʻi)
An important consideration for a model-based method of phylogenetic network inference is the identifiability of the network parameter of the model. A recurring theme in the works exploring this issue is that it is often difficult to identify the orientation of edges in a triangle of the network. In fact, it has been shown that for some models it is impossible to determine the orientation of triangle edges utilizing the standard algebraic technique of phylogenetic invariants. In this talks, we consider one such model with a Jukes-Cantor site-substitution process and no coalescence. We give a complete semialgebraic description of three, 3-leaf Jukes-Cantor phylogenetic network models with embedded triangles. By describing these base cases, we will resolve several questions about the identifiability of networks with embedded triangles. In particular, we will show that for any pair of models, the intersection and set differences of the models are full-dimensional regions of the space of site-pattern probability distributions. Thus, despite being algebraically indistinguishable, these network models are not identical, nor are they identifiable (or generically identifiable). Our results also yield a straightforward biological interpretation–that the signal from a hybridization event is immediately detectable but decays over time until it is impossible to identify the orientation of edges in the triangle of a network. This is joint work with Bryan Currie, Aviva Englander, Jose Esparza-Lozano, Max Hill, Colby Long, Devon Olds, Kawika O’Connor, Udani Ranasinghe, and Christin Sum.
Tuesday, 14 July
14:00-14:30
Paul Breiding (Osnabrück University)
HomotopyContinuation.jl is a software for solving systems of nonlinear polynomial equations by numerical homotopy continuation. I will discuss the recent developments and features of the software, including numerical irreducible decomposition, and sketch the roadmap towards v.3.0.
14:30-15:00
Silviana Amethyst (MPI CBG Dresden)
I started working on Bertini 2 in 2014, as the re-implementation of Bertini 1 into C++, providing a native Python interface for numerical algebraic geometry, and clean C++ interface for ready use as a library. Development continues today, and with a lot of help and hard work the package is getting closer to what I want it to be. In this talk, I would like to share some lessons I learned from the worlds of mathematical software, research with undergraduates, research software engineering, and adapting to an ever-changing world.
15:00-15:30
Jon Hauenstein (University of Notre Dame)
Given a polynomial system with real coefficients, a standard question in computational algebraic geometry is to compute its set of real solutions. Some examples include synthesizing a linkage given a list of constraints, analyzing the stability of steady states of a dynamical system, and computing nesting structure of curves in the real projective plane. This talk will explore a computational approached based on routing functions to numerically compute and analyze real solution sets with applications arising in mathematics, science, and engineering.
15:30-16:00
Taylor Brysiewicz (Western University)
Many algorithms in numerical algebraic geometry depend heavily on choices made by the user: numerical tolerances, step sizes, path-tracking order, and the distribution of monodromy loops, among others. Traditionally, this freedom is treated as a source of instability to be controlled, or even ignored. In this talk, I will discuss a different perspective, which seeks to understand the statistical nature of this freedom. Rather than eliminating randomness, we use it to study the behavior of the algorithms themselves. I will describe several stochastic models for the output spaces of core algorithms in numerical algebraic geometry. The focus will be on monodromy-based models and path-jumping models.
16:00-16:30 Coffee Break
16:30-17:00
Lorenzo Baldi (Max Planck Institute for Molecular Cell Biology and Genetics)
The Archimedean Positivstellensatz is a key result in real algebra, providing denominator-free representations of positive functions. This result, which has been rediscovered many times in the literature, serves as a bridge between real algebra and analysis, and has many applications. Motivated by the study of the convergence of Feynman integrals in particle physic, we show how to use the Archimedean Positivstellensatz to prove a generalization of Pólya’s theorem in the setting of positive toric varieties. Unlike the classical result, our generalizations apply to sparse and non-homogeneous polynomials. Based on a joint work with Rainer Sinn, Máté L. Telek, and Julian Weigert.
17:00-17:30
Carlos D’Andrea (University of Barcelona)
It has been proven by Y. Pourchet in 1971 that any positive univariate polynomial with rational coefficients can be expressed as a sum of up to five squares of polynomials in Q[x], and that the number five is optimal in the sense that there are positive polynomials which cannot be written as a sum of four or less squares. In 2023 an algorithm was proposed by Koprowski, Magron, and Vaccon with a conjecture that it may always produce an optimal decomposition as a sum of five squares.
In this presentation, we will review the problem and its algorithmic approach, and show that the aforementioned conjecture is not correct. We also explore further generalizations of this approach. This is joint work with Teresa Cortadellas, Ana de Felipe, Joel Hurtado, and Eulalia Montoro.
17:30-18:00
Martin Sombra (University of Barcelona)
Submanifolds of the projective space equipped with the Fubini-Study metric are homogeneous spaces. I will explain an approach aiming to test this conjecture for toric manifolds of arbitrary codimension. This is joint work with Antonio Di Scala (Politecnico di Torino).
18:00-18:30
Elana Kalashnikov (University of Waterloo)
The classification of four dimensional Fano varieties is an important open question, and it has been conjectured that mirror symmetry may make this a computationally (more) accessible problem. In this talk, I’ll survey various Fano searches and mirror symmetry constructions that have contributed to what is currently known.
Wednesday, 15 July
14:00-14:30
Claudia Fevola (Cunef University)
The Kadomtsev–Petviashvili (KP) equation is a central object in the theory of integrable systems, whose solutions reveal deep connections with algebraic geometry and combinatorics. KP solutions can be constructed from algebraic curves or from Grassmannians points.
In this talk, I will discuss how these constructions behave under degenerations. We focus on banana curves—reducible rational nodal curves that arise as degenerations of hyperelliptic curves. I will describe the geometry and combinatorics of the tropical theta divisor, showing how it canonically encodes the matroid and Grassmannian structures underlying the associated KP multi-soliton solutions.
This is joint work with Simonetta Abenda, Türku Özlum Çelik, and Yelena Mandelshtam.
14:30-15:00
Yelena Mandelshtam (University of Michigan)
A metric graph determines a positive semidefinite quadratic form on its cycle space, equivalently the quadratic form induced by its graph Laplacian. The associated Voronoi cell is a polytope whose combinatorics is closely related to the geometry of the graph and to tropical limits of principally polarized abelian varieties. While previous work of Caporaso-Viviani and Amini describes the combinatorial type of these Voronoi cells, effective computation requires explicit inequalities and normal vectors. I will describe an algorithm for computing the Voronoi cell associated to a metric graph. The input is a graph together with edge lengths, and the output is a system of inequalities defining the Voronoi polytope, expressed in terms of cycles in the graph. The construction makes Amini’s graph-theoretic description computationally explicit and is designed for implementation in polyhedral software. These explicit computations are important in applications involving tropical degenerations of Riemann theta functions, including tropical KP theory and tropical limits of string-theoretic worldsheet integrals. This is joint work with Hadleigh Frost.
15:00-15:30
Emre Sertöz (Leiden University)
CANCELLED
I will present a method, developed with Edgar Costa (MIT), that aspires to compute the Picard lattice of a smooth quartic surface in P3 as a Galois module, starting from its defining equation. Although there are constraints to guarantee the success of the method, the method always produces rigorous results despite relying on numerical approximations of periods. The method builds on the ideas that seeded IVHS in the work of Griffiths–Harris (1983) and was further developed by Movasati–Sertöz (2021) and Cifani–Pirola–Schlesinger (2022). With Edgar Costa, we further engineered the method to handle mathematically hazardous environments, chief among them the Mukai–Klein quartic. This surface contains no lines, conics, or twisted cubics, but 133056 quartic rational curves, making Galois reconstruction especially challenging. The problem was originally posed by Noam Elkies, and we incorporate many of his insights.
15:30-16:00
Linden Disney-Hogg (University of Leeds)
I will describe recent work determining the orbit structure of theta characteristics (equivalently spin structures) on Riemann surfaces under the action of the automorphism group, focusing in particular on invariant characteristics. Theoretical results, practical computational tools developed in Sagemath, and explorative applications of machine learning will all be featured.
16:00-16:30 Coffee Break
16:30-17:30 semi-plenary talk
Mario Kummer (TU Dresden)
A quasicrystal is a structure that is not periodic yet still exhibits long-range order. One of the defining features of quasicrystals is their X-ray diffraction pattern, which shows isolated peaks — a hallmark of order previously associated only with periodic structures. Mathematically, quasicrystals are often modeled as discrete point sets, with their diffraction patterns represented by the absolute square of their Fourier transform, known as the diffraction measure. Interestingly, the support of this diffraction measure is typically dense, and only by ignoring contributions below a certain threshold (corresponding to experimental resolution) does it appear discrete. Fourier quasicrystals, however, are discrete point sets whose diffraction patterns are genuinely discrete, independent of experimental precision. To what extent aperiodic Fourier quasicrystals can exist, at least mathematically, has been a very active research topic over the last years. In this talk, we explain how Fourier quasicrystals can be constructed using certain real algebraic varieties with remarkable geometric, algebraic, and analytic properties. This is based on joint works with Alon, Kurasov, and Vinzant.
17:30-18:00
Yairon Cid-Ruiz (North Carolina State University)
A fundamental property of Segre classes is their birational invariance. This invariance implies that the Segre class of a closed subscheme only depends on the integral closure of the defining ideal sheaf.
We show that, conversely, the Segre class of a closed subscheme encodes an integral dependence criterion for its defining ideal sheaf. As an application, we prove that Aluffi’s Segre zeta function provides an integral dependence criterion for homogeneous ideals in polynomial rings.
18:00-18:30
Jan Draisma (University of Bern)
We show that there exists an algorithm that, on input a finite boolean combination of inequalities such as u^2*v.
Posters
![[PZ09 – 1st Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/poster_zones_1-stock.png){kind=link}
Lorenzo Cecchi
(Sissa)
In numerical algebraic geometry, the efficiency of homotopy continuation algorithms is strictly controlled by the distance of the homotopy path from the discriminant locus, the set of ill-posed systems with multiple roots. While classical approaches quantify this via the condition number and conformal “condition metrics”, we propose a new intrinsic geometric framework based on optimal transport.
We view the space of polynomial systems as the base of a (singular) holomorphic fibration, where each fiber consists of the roots of the system, seen as atomic probability measures. By endowing the space of systems with the intrinsic Wasserstein distance between these measures, we construct a metric structure that naturally encodes the geometry of the roots.
We prove that for pencils of polynomial systems, the resulting metric space is the completion of a specific Kahler metric defined on the smooth locus. Crucially, we establish that the set of well-posed systems is geodesically convex within this completion. This geometric property suggests that geodesics in the Wasserstein metric inherently repel from the discriminant, providing a “constrained” analogue of the Monge–Mather shortening principle. This framework offers a theoretical guarantee for constructing well-conditioned homotopies that avoid singularities by design.
Jonas Ellwanger
(Goethe Universität Frankfurt)
We study a class of signomials whose positive support is the set of vertices of a simplex and which may have several negative support points in the simplex. Various groups of authors have provided an exact characterization for the global nonnegativity of a signomial in this class in terms of circuit signomials and that characterization provides a tractable nonnegativity test. We generalize this characterization to the constrained nonnegativity over a set $X$ under an additional convexity precondition in the exponential moment space. This provides a tractable nonnegativity test over $X$ for the class in terms of a power cone program. Our proof methods rely on a variant of of the convex cone of constrained SAGE signomials (sums of arithmetic-geometric exponentials) and the duality theory.
Constantin Ickstadt
(Goethe University Frankfurt)
We consider an algorithmic framework for two-player non-zero-sum semidefinite games, where each player’s strategy is a positive semidefinite matrix with trace one. We formulate the computation of Nash equilibria in such games as semidefinite complementarity problems and develop symbolic-numerical techniques to trace generalized Lemke-Howson paths. These paths generalize the piecewise affine-linear trajectories of the classical Lemke-Howson algorithm for bimatrix games, replacing them with nonlinear curve branches governed by eigenvalue complementarity conditions.
A key feature of our framework is the introduction of event points, which correspond to curve singularities. We analyze the local behavior near these points using Puiseux series expansions. We prove the smoothness of the curve branches under suitable non-degeneracy conditions and establish connections between our approach and both the classical combinatorial and homotopy-theoretic interpretations of the Lemke-Howson algorithm.
Jeong-Hoon Ju
(University Of Copenhagen)
Ye and Lim (2016) proved that every (resp. a generic) complex $n \times n$ matrix can be expressed as a product of $2n+5$ (resp. $\lfloor n/2 \rfloor +1$) Toeplitz matrices. Motivated by this result, it is natural to ask the following question: what is the minimum number of Toeplitz matrices required to factor a given matrix? We generalize this question from Toeplitz structure to more general structures. In this talk, we introduce the notion of structured matrix factorization length when the set of matrices with a given structure is an affine variety $X \subseteq \mathbb{C}^{n \times n}$. Then we define its algebro-geometric analogue, the border structured matrix factorization length via the $r$-th $X$-factorization variety. We calculate the dimension and degree of the variety for some $X$. Finally, we propose methods for deriving lower and upper bounds for (border) structured matrix factorization length. For lower bound, we suggest a method based on displacement rank, which also can be used to get some defining equations of the $r$-th $X$-factorization variety; for upper bound, we suggest an approach using alternating minimization. This is joint work with Taehyeong Kim.
Etna Lindy
(Aalto University)
In bivariate system solving, resultant methods can be used to associate the coordinates of the roots to the eigenvalues of a polynomial matrix, where the entries of the matrix are determined by the coefficients of the polynomials. We pay closer attention to two of these matrices: Sylvester matrix and Bézout matrix.
For a zero-dimensional ideal I = , these matrices are regular, and the eigenvalues correspond to the coordinates that can be extended to finite or infinite roots of the system. The algebraic multiplicity of the eigenvalue y0 is equal to the sum of intersection multiplicities of the ideal at the roots (x0,y0), including infinite roots (infty,y0), where we define the intersection multiplicity via the finite root (0,y0) of the reversals of f and g. Not only are the multiplicities of the roots preserved, but the partial multiplicities of the eigenvalue — that can be recovered from the Smith normal form of the matrix — describe precisely the leading monomials present in the dual space.
In the case of a positive-dimensional ideal, the matrices are no longer regular, and so we characterize a polynomial basis for the kernel of the matrix. The dual spaces may now be infinite dimensional at the points of interest, but the structure of the leading terms from the zero-dimensional case is still present, and is given by the partial multiplicities. In practice, this structure may be recovered from the dual basis of < f/h, g/h>, where h = gcd(f,g).
Niels Lubbes
(Johannes Kepler University)
In order to address a classification problem for the topological types of real algebraic surfaces with respect to the Euclidean topology, we define a “cycle” and “torus” as a topological space homeomorphic to the unit circle S^1 and the product S^1xS^1, respectively. We call a cycle in S^1xS^1 “trivial” iff it spans a disc. By a “circle” in the unit sphere S^3, we mean a circle in the Euclidean sense, and thus an irreducible conic with infinitely many real points. We call an algebraic surface in S^3 “celestial” iff it contains at least two circles through a general point. We address the following problem:
Determine a set of celestial surfaces such that any celestial surface in S^3 is homeomorphic to exactly one element in this set.
This problem is of interest, for example, in architectural geometry and kinematics. The most challenging case concerns celestial surfaces that are the pointwise product of two circles in S^3, where S^3 is identified with the unit-quaternions.
Now suppose that X is the pointwise product of general circles in S^3. In this case, X is the image of a birational regular map f: S^1xS^1 -> X such that either f(S^1xS^1) is a torus, or there exists an elliptic curve W in S^1\times S^1 such that the restriced map f|W is a branched 2:1 covering and f is a homeomorphism outside W. It is known that the elliptic curve W consists of either two non-trivial cycles, one trivial cycle, or two trivial cycles. We conjecture that the latter case does not occur. Consequently, X would be homeomorphic to one of four celestial surfaces, where either two cycles in W are sent 2:1 to a single cycle in Sing(X), or each cycle in W is send to an arc in Sing(X).
For surfaces that are the pointwise product of a great and a little circle in S^3, we also address an ambient-isotopic version of the above problem.
Cecilie Olesen Recke
(University of Copenhagen)
We study discrete Lyapunov models, which consist of steady-state distributions of first-order vector autoregressive models.The parameter matrix of such a model encodes a directed graph whose vertices correspond to the components of the random vector. This combinatorial framework naturally allows for cycles in the graph structure. We focus on the fundamental problem of identifying the entries of the parameter matrix. In contrast to the classical setting, we assume non-Gaussian error terms, which allows us to use the higher-order cumulants of the model. We prove that the cumulants of the model satisfy tensor equations, which arise as generalizations of the second order discrete Lyapunov equation for the covariance matrix.
In this setup, we show generic identifiability for directed acyclic graphs with self-loops at each vertex and show how to express the parameters as a rational function of the cumulants. Furthermore, we establish local identifiability for all directed graphs containing self-loops at each vertex and no isolated vertices. The proofs rely on a combinatorial description of the entries of the cumulants called trek formulas, which we also provide. Finally, we provide first results on the defining polynomial equations of the models, showing model equivalence for certain graphs and paving the way towards structure learning.
Matija Tomic
(University Of British Columbia)
Sparse recovery is one of the fundamental problems in applied mathematics and statistics. The main objective is to recover sparse solutions of an underdetermined linear system Ax = b. Standard approaches include greedy algorithms and convex relaxation methods.
In this poster, we propose a fundamentally different perspective: sparse recovery is reformulated as the solution of a structured system of polynomial equations. The key observation is that an s-sparse vector x is characterized by the vanishing of all products of s+1 distinct coordinates. This transforms sparse recovery into a polynomial system whose solution set is a union of coordinate subspaces intersected with the affine space Ax = b. We do not solve this system directly. Instead, we exploit its structure to reduce the computational complexity and recover the solutions efficiently.
In particular, we develop two optimization-based algorithms that rely on decomposing a tensor that encodes the polynomial system. Both algorithms avoid the explicit construction of the tensor by leveraging a connection to elementary symmetric polynomials, which are computed efficiently via recursion relations. This reduces the computational complexity of solving the system from O(n^{s+1}) to O(n^3), yielding computationally tractable algorithms.
The algorithms provide explicit control over the desired sparsity level of the solution and enable recovery of multiple sparse solutions when they exist. Standard methods such as Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP)/LASSO do not offer these capabilities. Numerical experiments demonstrate competitive performance compared with OMP and BP/LASSO and highlight the advantages of the proposed approach. These results illustrate the potential of combining tensor decompositions and polynomial systems to advance sparse recovery, offering new tools and perspectives beyond traditional methods.