Schedule:
September 3, 2026 (1:30pm - 4:50 pm)
1:30 pm - Maria Torras Perez
2:00 pm - Cecilie Olesen Recke
2:30 pm - Aviva Szpirglas
3:00 pm - Coffee Break
3:30 pm - Nelly Villamizar
4:00 pm - Discussion on the interactions of AI with real algebraic geometry and topology
Title and Abstracts:
Maria Torras Perez:
Title: The fiber of multiparameter persistent homology for simplicial complexes
Abstract: Multiparameter persistent homology (MPH) associates to a vector-valued filter function on a simplicial complex a multiparameter persistence module. In this talk, I will present recent joint work with Heather Harrington and Ulrike Tillmann on the inverse problem for MPH. To study the geometry of its fibers, we equip both the space of filters and the moduli space of essentially finite persistence modules with natural stratifications, and show that MPH is a strongly stratified map. Over each stratum in its image, MPH restricts to a trivial fiber bundle whose fiber is a polyhedral complex. We also bound the dimension of MPH fibers in terms of multigraded Betti numbers, recovering the one-parameter bound of Leygonie and Tillmann as a special case.
Cecilie Olesen Recke:
Title: Completions to discrete probability distributions in Log-linear models.
Abstract: Completion problems, of recovering points from a set of observed coordinates, are abundant in applications to image reconstruction, phylogenetics, and data science. We consider a completion problem of determining those observed probabilities which can be finitely completed to a probability distribution in a given log-linear model. These observed probabilities either have a unique completion or two completions to the log–linear model depending on the set of observed coordinates.
Aviva Szpirglas:
Title: Kernel of the Hermite symmetric interpolation
Abstract: Let P be a polynomial of degree p. We define an interpolation basis associate with P for the vector space of symmetric polynomials of multi degree at most (k,…,k). Then is defined a linear mapping from the vector space of symmetric polynomials in (p-k) variables in the vector space of symmetric polynomials of multi degree at most (k,…,k): if g is a symmetric polynomial in (p-k) variables, its image is the symmetric polynomial in (p-k) variables with multi degree at most (k,…,k) which is the Hermite interpolation of g. Note that this mapping is a projection. We study the kernel ideal of this mapping. We determine generators for this ideal which are symmetrical polynomials.
Nelly Villamizar:
Title: Koszul Homology and Splines on Three-Dimensional Fans
Abstract: Splines are piecewise polynomial functions defined over partitions of real domains satisfying prescribed smoothness conditions. Beyond their role in approximation theory, spline spaces are closely connected to questions in commutative algebra and algebraic geometry. In this talk, we study spline spaces associated to fans arising from central hyperplane arrangements in three-dimensional space, with prescribed smoothness along each hyperplane.
Our main result shows that, for generic hyperplane arrangements (meaning that no three hyperplanes meet along a common line), the dimension of the spline space in each degree can be expressed in terms of the zeroth and first homology modules of the Koszul complex of powers of the linear forms defining the arrangement. We use this connection to compute spline dimensions in high degrees, as well as in all degrees for generic arrangements with at most five hyperplanes. We also determine the Hilbert function for generic arrangements with sufficiently many hyperplanes and constant smoothness. As an application, we compute the dimensions of the C0 and C1 spline spaces in all degrees for generic arrangements.
This is joint work with Carles Checa, Michael DiPasquale, Pablo Mazón, Thái Thành Nguyen, Liana Sega, Prajwal Udanshive, and Adam Van Tuyl.
Title: AI interactions with real algebraic geometry and topology
Abstract: Artificial intelligence (AI) is increasingly affecting our day-to-day mathematical research. The “Leiden Declaration on Artificial Intelligence and Mathematics”—published on June 2, 2026 and endorsed by the International Mathematical Union — addresses many key questions currently being raised. A few weeks later a spectacular success of AI was the disproval of the famous Jacobian Conjecture. What is the current impact of AI in real algebraic geometry and topology ? Discussion led by Nidhi Kaihnsa and Marie-Françoise Roy.