Welcome to Seminars on Number Theory and Algebraic Geometry at KU Leuven, Algebra Section. Link to the official seminar page.
Seminar room: B.02.18 (Maths building).
Organizer (2025-2026): Fatemeh Mohammadi
Winter Semester 2025
September 19, 2025: 14:00-15:00: Hugh Thomas (Université du Québec à Montréal)
A Grassmannian generalization of triangulations of a polygon and CEGM scattering
The non-crossing complex of lattice paths in a $k \times (n-k)$ rectangle forms a Grassmannian generalization of the complex of triangulations of an $n$-gon (which arises as the $k=2$ case). I will explain how the non-crossing complex can be understood using the representation theory of gentle algebras, and how ongoing joint work with Arkani-Hamed, Frost, Plamondon, and Salvatori then applies to produce a corresponding set of $u$-equations which produce a novel geometrical model for it. In the $k=2$ case, this recovers the $u$-equations of Koba--Nielsen, used to define the tree string amplitude. The non-crossing complex has applications to Cachazo--Early--Guevara--Mizera scattering, which I will try to say something about. This talk is based on joint work with Arkani-Hamed, Frost, Plamondon, and Salvatori, and with Early and Plamondon.
September 24, 2025: 14:00-15:00: Raphaël Widdershoven (KU Leuven, STADIUS)
Fast Macaulay Null Space for Solving Multivariate Polynomial Systems
Solving systems of multivariate polynomial equations in finite precision often leads to large, structured problems in numerical linear algebra. A widely used approach involves computing the null space of the Macaulay matrix, which encodes the system in linear form and enables root separation. Despite its sparsity and rich structure, this matrix is typically treated as generic, resulting in high computational cost—especially for systems with many variables. In this talk, we present methods to accelerate null space computation by exploiting the algebraic properties of the Macaulay matrix. The approaches are iterative, constructing null spaces for matrices of increasing degree to reduce overall cost. We also incorporate the classical idea of elimination to further improve efficiency. These techniques yield improvements in both asymptotic complexity and practical performance.
8, 2025: 14:30-15:30: Gonzalo Esteban Manzano Flores (Universidad de Chile)
On the Kaplansky radical and reduction of arithmetic curves
In this talk, I will discuss the Kaplansky radical of a field, an object introduced by Irving Kaplansky in the 1960s to characterize fields with a unique non-split quaternion algebra. I will focus on the case of function fields of curves, and in particular on arithmetic curves, where I will show how the Kaplansky radical can be connected to the reduction of the curve.
October 15, 2025: 14:00-15:00: Raffael Mohr (KU Leuven, NUMA)
Polyhedral Elimination Technique
The elimination problem for a polynomial system of equations F asks to compute a set of defining equations for the projection of the zeros V(F) of F to a coordinate subspace. When this projection is defined by a single polynomial g, we present two algorithms that compute the Newton polytope of g from F. This furnishes an evaluation–interpolation approach for elimination, which is potentially advantageous compared to classical algorithms. We also discuss an application to differential elimination.
October 22, 2025: 14:30-15:30: Kaie Kubjas (Aalto University)
Uniqueness of size-2 positive semidefinite matrix factorizations
We study uniqueness of size-2 positive semidefinitite (psd) factorizations using tools from rigidity theory. In a size-k psd factorization the rows or columns of factors are vectorizations of size-k psd matrices. Psd factorizations play an important role in applications, for example in the computational complexity of semidefinite programs and quantum information theory. The main goal of rigidity theory is to determine whether there exists a unique configuration of n points in R^d up to rigid transformations with a fixed partial set of pairwise distances between the points. We transfer ideas from rigidity theory to study uniqueness of psd factorizations and give a complete characterization of unique size-2 psd factorizations of positive matrices of rank three. This talk is based on joint work with Kristen Dawson, Serkan Hosten, and Lilja Metsälampi.
October 29, 2025: 14:30-15:30: Daniel Windisch (KU Leuven, NUMA)
Real log canonical thresholds of hyperplane arrangements and statistical model selection
The log canonical threshold (lct) is a fundamental invariant in birational geometry, crucial for understanding the complexity of singularities in algebraic varieties. Its real counterpart, the real log canonical threshold (rlct) — also known as learning coefficient — has gained significance in statistics and machine learning, where it plays a key role in model selection and error estimation for statistical models. This talk presents new results on the rlct and its multiplicity for real (not necessarily reduced) hyperplane arrangements. We derive explicit formulas for these invariants based solely on the combinatorics and linear algebra of the arrangement. Moreover, we sketch explicit possible statistical applications of the theory. The talk is based on joint work with Dimitra Kosta.
November 5, 2025: 14:30-15:30: Matthias Orth (KU Leuven)
Gröbner basis for powers of a general linear form in a monomial complete intersection
We study almost complete intersection ideals in a polynomial ring over a field of characteristic zero, generated by powers of all the variables together with a power of their sum. Exploiting the fact that the Hilbert series of the corresponding quotient rings are thin, we determine all reduced Gröbner bases for such ideals. Our approach is primarily combinatorial, focusing on the structure of the initial ideal. To each monomial in the vector space basis of an Artinian monomial complete intersection, we associate a lattice path and introduce a reflection operation on these paths, which enables a key counting argument. As a consequence, our method provides a new proof that Artinian monomial complete intersections have the strong Lefschetz property over fields of characteristic zero. If time permits, I will talk about several applications of our results in special cases. If the monomial complete intersection is generated by the cubes of the variables, the Gröbner bases have degree structures governed by Motzkin numbers, Riordan numbers, and their convolutions. If the monomial complete intersection is equigenerated and the first power of the variable sum is considered, then we identify generalized Catalan numbers as a subsequence of the sequence of Gröbner basis element counts by degree. These subsequences also appear in quantum physics. The talk is based on joint work with Filip Jonsson Kling and Samuel Lundqvist (both Stockholm University) and Fatemeh Mohammadi (KU Leuven).
November 12, 2025: 14:30-15:30: Daisie Rock (KU Leuven)
November 19, 2025: 14:30-15:30:
November 26, 2025: 14:30-15:30: Signe Lundqvist (KU Leuven, NUMA)