This is a collaborative workshop aimed at researchers in commutative algebra who have not necessarily worked on the applications, but also at researchers in the applications with a background in commutative algebra.

The topics of this workshop are: algebraic statistics, algebraic vision, approximation theory, coding theory, connections with physics, numerical algebraic geometry, and tensors. For each topic, there will be an introductory lecture by an expert in the field. In addition, for each topic, a senior researcher working in the area will propose a project for participants to work on as a group. Ample time will be allocated for collaborative work on the group projects during the workshop. 

The names of the senior researchers who will serve as group leaders and brief descriptions of the proposed projects will be published on this webpage in due time. Each accepted participant will be assigned to a working group, according to their inclinations and interests.

Focus groups and mentors are:

Approximation Theory - Mike DiPasquale and Nelly Villamizar

The focus group on approximation theory will address problems stemming primarily from multivariate splines (that is, piecewise polynomial functions continuously differentiable to some order defined on a partition of a real domain) which can be studied via commutative algebra and algebraic geometry.

Adjacent relevant topics include interpolation and geometric modelling.  A useful way to study multivariate splines is to view them as a graded ring, module, or vector space (depending on how much structure the underlying partition allows); then study the properties of this graded object such as the Hilbert function, (Castelnuovo-Mumford) regularity, freeness, generators, and so on.  Remarkably, many of these properties are still not completely understood even for splines on relatively simple planar triangulations — for instance there is an unsolved conjecture (open for almost 30 years) regarding the dimension of the space of continuously differentiable cubic splines on planar triangulations.  Possible projects, depending on the interests of the participants, include studying the Hilbert function of spline spaces on simple triangulations or tetrahedral vertex stars (this involves some homological methods as well as studying syzygies of ideals generated by powers of linear forms and colon ideals formed from these), applying numerical algebraic geometry software to look for counterexamples to the above conjecture on continuously differentiable cubic splines, and exploring algebraic consequences of imposing supersmoothness on spline spaces.

Spectral Theory - Stephen Shipman, Frank Sottile (first two days), and Matt Faust

Natural questions concerning the spectra of operators on periodic graphs, including tight-binding models in quantum mechanics, are expressed in terms of module endomorphisms and polynomials.  Hence these problems are amenable to study using ideas from commutative algebra and algebraic geometry.  This is a new application of algebra with interesting and consequential open problems.  We will present the background, introduce the basic ideas and questions in this area, and present problems that are ripe for further study by the participants.

Tensors -  JM Landsberg and Claudiu Raicu

Recently it was realized that many important questions regarding tensors from algebraic complexity and quantum information theory are naturally addressed using advanced methods from commutative algebra. Similarly, old questions in algebraic geometry related to tensors have seen significant advances exploiting commutative algebra and representation theory. For this workshop we will focus on two problems. The first is from complexity theory and concerns the geometry of tensors with defective "geometric rank" (a recently defined rank notion for tensors) - the problem generalizes the classical study of spaces of matrices of bounded rank and amounts to studying degeneracy loci of certain maps between vector bundles on projective space. The second is from classical algebraic geometry and it is also relevant for complexity - it aims to find defining equations and syzygies for the varieties of binary forms that factor as $x^j y^{d-j}$ after a change of basis.

Coding Theory - Alberto Ravagnani and Diego Ruano 

This project focuses on the applications of commutative algebra and (algebraic) combinatorics to coding theory. Error-correcting codes are mathematical objects that cleverly add redundancy to information to protect it from noise and disturbances. There exist several classes of codes, which can be constructed and analyzed using tools from algebra, geometry, graph theory, and combinatorics more generally. Constructing and analyzing error-correcting codes often boils down to solving interesting, sometimes "exotic" problems in these areas of mathematics. The focus of this project is precisely on those problems.

Algebraic Statistics - Bernd Sturmfels (first two days), Seth Sullivant, and Max Wiesmann

Statistical models based on directed graphs appear frequently in applied settings.  When network models are restricted to discrete random variables, the model becomes a semialgebraic set, whose algebraic structure can be useful for analyzing statistical properties of the model.  This topic will consist of algebraic problems based on the study of neural networks and phylogenetic networks.  

Numerical Algebraic Geometry - Jonathan Hauenstein and Paul Breiding

Due to the ubiquity of solving systems of polynomial equations in mathematics, science, and engineering, many computational approaches have been proposed.  This project will focus on the computational paradigm of numerical algebraic geometry which uses methods from numerical analysis to study and manipulate the solutions of systems of polynomial equations.  Following an introduction to homotopy continuation and numerical algebraic geometry, participants will gain practical experience with software tools.  Two subgroups will each consider a current research topic in numerical algebraic geometry in detail.  

Algebraic Vision - Timothy Duff and Anton Leykin

Several fundamental problems in computer vision amount to reconstructing a scene and/or cameras from multiple images. This group will focus on the role of algebra in addressing such problems. Several potential research topics will be discussed. An overarching theme will be the study of algebraic varieties which constrain the possibilities for reconstruction: their dimensions, equations, and amenability to symbolic or numerical computation.

Application Status

- Deadline for applications: March 1, 2025

- Please note that only successful applicants will be contacted via email after decisions are made.

*NOTE: If you are interested in this workshop, please complete the funding application form regardless if you need funding or not. The form is to help the organizers match selected participants to the focus group/mentor(s).

  Information last updated on: January 27, 2025