Poster Session
Poster session - 25.08.2025
Presenter: Barbara Betti
Title of the poster: Proudfoot-Speyer degeneration of scattering equations
Abstract of the poster: We study scattering equations of hyperplane arrangements using a combinatorial and numerical approach. We restate the problem as linear equations on a reciprocal linear space and solve it with a homotopy algorithm. This is based on the Gröbner degeneration of the coordinate ring into the Stanley-Reisner ring of the broken circuit complex. We study the regularity of the ideal defined by the equations and apply our methods to scattering equations.
This is a joint work with Viktoriia Borovik and Simon Telen.
Presenter: Alexander Taveira Blomenhofer
Title of the poster: The contraction variety of a tensor
Abstract of the poster: We present the contraction variety of a tensor and compute it for concise tensors of small rank, Chow rank, or skew rank. As a consequence, we obtain a nearly linear time algorithm to compute low-rank Chow decompositions. The algorithm employs a pseudo-diagonalization method to recover the irreducible components of a the contraction variety.
Based on joint work with Benjamin Lovitz.
Presenter: Alex Bortolotti
Title of the poster: Efficient Solvability of the Ideal Membership Problem via Polynomial Calculus and Its Connections to the Automability of the Sum-of-Squares Proof System
Abstract of the poster: The Polynomial Ideal Membership Problem (IMP), which asks whether a polynomial f belongs to an ideal I, is a fundamental problem with wide-ranging applications. Despite its importance, it is known to be computationally intractable in the general case. A significant result is a dichotomy between “hard” (NP-hard) and “easy” (polynomial time) instances of the degree-d IMP ( IMP(d) ), where the input polynomial has a degree at most d. This dichotomy was initially established for the Boolean domain [Mastrolilli (2017)] and later partially extended to the general domain cases [Bulatov-Rafiey (2019)]. Both results are based on classifying IMP instances using functions called polymorphisms, a concept central to the study and classification of Constraint Satisfaction Problems (CSPs). In this work, we present novel insights into the efficient computability of IMP(d) for broad classes of CSPs over finite domains. We demonstrate that for CSPs whose constraint languages are closed under semilattice or dual discriminator polymorphisms – classes that generalize HORN-SAT and 2-SAT and are foundational in CSP theory – IMP(d) can be solved efficiently in polynomial time with respect to the number of variables, for a fixed degree, within the Polynomial Calculus (PC) proof system. The PC proof system can be viewed as a truncated version of the Buchberger algorithm. Furthermore, we establish a strong connection between the solvability of IMP(d) within the PC proof system and the effectiveness of computing Sum-of-Squares (SoS) hierarchies. The SoS hierarchy has emerged as a promising technique in semidefinite optimization and is inspired by Putinar’s Positivstellensatz in real algebraic geometry. It remains an open question whether fixed-degree SoS proofs can be automated [O’Donnell (2017)]. In this work, we develop a new criterion based on the IMP(d) solvability by the PC proof system for showing fixed-degree automatability, thereby broadening the class of polynomial systems for which degree-d SoS proofs can be automated.
Presenter: Clemens Brüser
Title of the poster: Determinantal Representations of Adjoint Polynomials
Abstract of the poster: Adjoint polynomials of convex polytopes have recently received attention from the field of particle physics, and the question has been raised whether they admit determinantal representations. We prove that the adjoint curve of a polygon always has a natural symmetric determinantal representation that certifies hyperbolicity. For three-dimensional polytopes we show that if the adjoint is smooth, then a determinantal representation exists. The methods to find these representations are computationally viable. There are also some negative results for higher dimensions.
The presented results are joint work with Mario Kummer and Dmitrii Pavlov (both TU Dresden).
Presenter: Erin Connelly
Title of the poster: Degenerate configurations in 3D-Image reconstruction
Abstract of the poster: Given k points (xi , yi) ∈ P2 × P2 , we use invariant theory to characterize rank deficiency of the k × 9 matrix Zk with rows x_i^T ⊗ y_i^T in terms of the geometry of the point configurations {xi} and {yi}. For k = 6 the answer is in terms of the lines on a cubic surface. For k = 7 and k = 8 the answer is in terms of Cremona transformations. This problem is motivated by a study of the numerical conditioning of classical 3D-Image reconstruction algorithms in Computer Vision.
Presenter: Sarah Eggleston
Title of the poster: Real subrank of order-three tensors
Abstract of the poster: We study the subrank of real order-three tensors and give an upper bound to the subrank of a real tensor given its complex subrank. Using similar arguments to those used by Bernardi-Blekherman-Ottaviani, we show that all subranks between the minimal typical subrank and the maximal typical subrank, which equals the generic subrank, are also typical. We then study small tensor formats with more than one typical subrank. In particular, we construct a 3 × 3 × 5-tensor with subrank 2 and show that the subrank of the 4 × 4 × 4-quaternion multiplication tensor is 2. Finally, we consider the tensor associated to componentwise complex multiplication in Cn and show that this tensor has real subrank n – informally, no more than n real scalar multiplications can be carried out using a device that does n complex scalar multiplications. We also prove a version of this result for other real division algebras.
This is joint work with Benjamin Biaggi and Jan Draisma.
Presenter: Joris Koefler
Title of the poster: Adjoints and Amplituhedra
Abstract of the poster: Semi-algebraic sets equipped with a unique differential form, have recently attracted attention in the field of positive geometry. The unique hypersurface interpolating residual intersections of the algebraic boundary of the semi-algebraic set, called adjoint hypersurfaces, play a major role for these canonical forms. We showcase an application of adjoints for polygons and the limit amplituhedron.
Presenter: Francesco Maria Mascarin
Title of the poster: Lissajous varieties
Abstract of the poster: The Kuramoto model is one of the most investigated models in the context of coupled oscillators. The primary focus of research is on understanding equilibrium states, which can be expressed as solutions to a system of algebraic equations. To investigate the equilibrium states from alternative perspectives, a novel parametrization is introduced, leading to algebraic object called Lissajous varieties.
Presenter: Elia Mazzucchelli
Title of the poster: Aspects of Positive Geometries
Abstract of the poster: Positive geometries intersect many areas of mathematics, including computational and real semialgebraic geometry and tropical geometry. We focus on the connection with the first two, in the context of positive geometries in the Grassmannian, called Amplituhedra, which arise in the context of amplitudes in a supersymmetric quantum field theory.
Presenter: Riccardo Ontani
Title of the poster: The Vafa-Intriligator formula for quotients of linear spaces
Abstract of the poster: The Vafa-Intriligator formula conjecturally expresses generating functions of quasimap invariants for GIT targets as finite sums of evaluations of a rational function at certain points. In the case of positive targets of the form V//G, where G is a reductive group acting on a vector space V with a suitably nice linearisation, the formula is proven in genus zero. Here, the relevant points arise as explicit solutions to polynomial equations in an ambient affine space. Solving these equations is a combinatorially interesting task, and doing so yields concrete expressions for intersection numbers on moduli spaces of quasimaps. I will present this formula and the associated equations, with examples obtained by specialising to particular targets.
Presenter: Dmitrii Pavlov
Title of the poster: Santalo Geometry of Convex Polytopes
Abstract of the poster: The Santalo point of a convex polytope is the interior point which leads to a polar dual of minimal volume. This minimization problem is relevant in interior point methods for convex optimization, where the logarithm of the dual volume is known as the universal barrier function. When translating the facet hyperplanes, the Santalo point traces out a semi-algebraic set. We describe and compute this geometry using algebraic and numerical techniques. We exploit connections with statistics, optimization and physics.
Presenter: Gianni Petrella
Title of the poster: Advancing quiver moduli research with AI
Abstract of the poster: We apply machine learning techniques to representations of quivers to estimate various numerical invariants of their moduli spaces and to generate extremal examples optimising known or conjectured bounds. We conclude on learnability of various geometric properties of the corresponding moduli problems, and generate interesting examples to test conjectures. The workflow we propose can be adapted to other areas of algebraic geometry.
Presenter: Anaelle Pfister
Title of the poster: Coaction of the Motivic Galois Group on Cosmological Correlators
Abstract of the poster: Cosmological correlators are physical quantities measuring the strength of particle interactions in the Early Universe. Following Cartier’s idea that physical quantities are closed under the action of the motivic Galois group, we study in this work the action of the motivic Galois group on the vector space spanned by cosmological correlators. This gives a recursive structure to these physical quantities. This subject sits at the intersection of several fields, including cosmology, cohomology theory and hyperplane arrangements.
Presenter: Max Wiesmann
Title of the poster: Arrangements and Likelihood
Abstract of the poster: We establish connections between hypersurface arrangements and likelihood geometry. Thereby arises a new description of the prime ideal of the likelihood correspondence of a parametrised statistical model. The description rests on the Rees algebra of the likelihood module of the arrangement, a module that is closely related to the module of logarithmic derivations introduced by Saito for a general hypersurface. Our new description is often computationally advantageous. A particular focus is put on generic hypersurface arrangements.
Based on joint works with T. Kahle, L. Kühne, L. Mühlherr, H. Schenck and B. Sturmfels