Period 1
9-11 July
Organizers: Philipp Grohs, Gitta Kutyniok, Roman Vershynin
The workshop focuses on algorithms that allow computers to learn based on data, in particular to automatically learn to recognize complex patterns and make intelligent decisions based on data. Thus, the topic of the workshop is closely related to fields such as statistics, probability theory, data mining, pattern recognition, artificial intelligence, adaptive control, and theoretical computer science.
Organizers: David Krieg, Klaus Ritter, Jan Vibyral
Information-based complexity (IBC) studies how many pieces of information are required to solve a (numerical) problem up to a prescribed error tolerance. The problems considered include function approximation and learning, numerical integration, optimization, or the solution of PDEs and SDEs. It is of particular interest how the complexity increases with the dimensionality of the problem (cf. curse of dimensionality versus tractability) and with the desired accuracy (cf. rate of convergence). The IBC workshop interacts naturally with several other workshops, including “Foundations of Data science and Machine Learning”, “Approximation Theory”, or “Stochastic Computation” as we study similar topics but from other perspectives. The two semi-plenary talks that we plan to include shall introduce the most important aspects of the field also to researchers from other communities.
Organizers: Paola Antonietti, Ralf Hiptmair, Markus Melenk
The Foundations of Numerical PDEs Workshop brings together researchers to discuss the latest advances in the numerical analysis of partial differential equations (PDEs). Focus topics of the workshop include
– Structure-preserving discretization of Hilbert complexes, including and generalizing FEEC (Finite Element Exterior Calculus) and offering a framework that unifies and extends traditional finite element methods, and forms the foundation of stable and robust numerical methods.
– Novel discretization techniques on polytopal meshes to improve flexibility, accuracy and efficiency of numerical methods for PDE-based multiphysics models in complex domains.
– Recent breakthroughs in the numerical modeling of wave propagation problems, including wave-number explicit analysis and wave-number robust numerical schemes.
The presentations at the workshop will address both theoretical developments, algorithmic challenges, and novel applications of numerical methods.
This workshop will have natural connections the the other FoCM workshops “Geometric Integration and Computational Mechanics” and “Geometric Integration and Computational Mechanics”.
Organizers: Ulrich Bauer, Michael Kerber, Xavier Pennec
Organizers: Saugata Basu, Greg Blekherman, Antonio Lerario
This workshop will explore complexity-theoretic and probabilistic aspects of real number computations, with a focus on numerical equation solving, optimization, and the geometry of solution sets. Topics include condition numbers, random real algebraic systems, and the distribution of points on manifolds. We will also address new directions involving effective o-minimality and positive, hyperbolic and Lorentzian polynomials.
Our workshop naturally connects with Computational Algebraic Geometry, particularly around shared interests in solution structure and symbolic-numeric techniques. There is also strong overlap with Numerical Linear Algebra, especially regarding condition numbers, optimization, and the stability of algorithms. Joint sessions or panels could explore the interplay between algebraic structure, numerical behavior, and computational complexity, enriching discussions across disciplines.
Organizers: Daan Huybrechs, Erik Koelink, Teresa Perez
The workshop aims to offer a multi-faceted overview of contemporary research in orthogonal polynomials and special functions. A number of talks focus on the analysis of orthogonal polynomials themselves, such as asymptotic analysis of large degree behaviour. A second class of topics cover generalizations of orthogonal polynomials, such as polynomials of several variables, matrix-valued orthogonal polynomials or polynomials over curves and surfaces. Finally, polynomials as a tool in applications feature in talks on numerical simulations and other computational settings.
The study of orthogonal polynomials links to several branches of mathematics. In applications, they play a pivotal role in approximation methods and numerical simulations. As part of the
latter the workshop links to the other workshops “Approximation Theory and Computational Harmonic Analysis” and “Foundations of Numerical PDEs”, and to a lesser extent also with “Foundations of Data Science and Machine Learning” and “Information-based Complexity”.
Organizers: Lenaic Chizat, Young-Heon Kim, Jan Maas
Optimal transport has a broad scope of applications from biology, finance, to machine learning/AI, This workshop will bring researchers at the frontier of computational optimal transport. Major directions to be considered include stability of the optimal solutions, numerical methods for stochastic optimal transport, and geometry of the space of stochastic processes. Participants in our workshop will interact with those in “Foundations of Data Science and Machine Learning”, “Information-Based Complexity”, “Stochastic Computation”, “Continuous Optimization”, “Computational Dynamics”, “Inverse Problems”, among others.
Period 2
13-15 July
Organizers: Ben Adcock, Albert Cohen, Karlheinz Gröchenig
This workshop explores the interplay between the approximation of functions and computational harmonic analysis. It focuses on the mathematical foundations of cutting-edge algorithms in areas such as image and signal processing, numerical analysis, and data science.
This workshop intersects most directly with the Foundations of Data Science and Machine Learning, Information-Based Complexity, and Foundations of Numerical PDEs workshops.
Organizers: Mireille Bossy, Nawaf Bou-Rabee, David Cohen
The Stochastic Computation workshop highlights recent advances at the intersection of stochastic processes and numerical analysis. Core topics include:
– the design and analysis of algorithms for stochastic differential equations (SDEs and SPDEs);
– theoretical insights and practical advances in accelerated Markov chain Monte Carlo (MCMC) methods; and,
– emerging directions such as rough path theory, uncertainty quantification for random PDEs, backward SDEs (BSDEs), and stochastic optimization techniques used in machine learning (e.g., stochastic gradient descent).
We anticipate strong synergies with the Foundations of Data Science and Machine Learning and Geometric Integration and Computational Mechanics workshops. In particular, the rapidly evolving field of diffusion-based generative modeling presents an exciting new frontier, blending ideas from stochastic numerics, high-dimensional sampling, and machine learning. These connections promise rich opportunities for cross-pollination, especially in the development of new structure-preserving algorithms and scalable inference techniques with rigorous mathematical foundations. The extent of interaction will naturally depend on the participants and confirmed speakers.
Organizers: Yussef Marzouk, Daniel Sanz-Alonso, Carola-Bibiane Schönlieb
This workshop will focus on inverse problems – surveying recent theoretical and algorithmic advances in optimization – based, statistical, and machine learning approaches, as well as the interplay among these approaches.
Organizers: Melvin Leok, Sigrid Leyendecker, Ari Stern
Dynamical systems arising in mechanics exhibit rich geometric structure (symmetries, conservation laws, etc.), posing unique challenges for the design and analysis of computational methods. This workshop focuses on recent advances in structure-preserving numerical algorithms for such systems.
Interactions with other workshops:
Foundations of Data Science and Machine Learning
Potential interactions include dynamical aspects of machine learning (integration of neural ODEs, symplectic methods for adjoint systems, etc.) and structure-preserving machine learning (learning dynamics of mechanical systems with structure, scientific machine learning, etc.).
Foundations of Numerical PDEs
Potential interactions include structure-preserving methods for PDEs (e.g., finite element exterior calculus) and methods for PDEs arising in computational mechanics (Hamiltonian PDEs, Hamilton–Jacobi equations, continuum mechanics of solids and fluids, etc.).
Stochastic Computation
Potential interactions include structure-preserving methods for stochastic differential equations, topics arising in computational statistical mechanics, Hamiltonian Monte Carlo, etc.
Computational Dynamics
Potential interactions related to geometric structure preservation in computational dynamics and dynamical aspects of numerical integrators.
Organizers: Türkü Özlüm Celik, Frank Sottile, Seth Sullivant
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: Folkmar Bornemann, Ioana Dumitriu, Joel A. Tropp
A workshop on random matrix theory, algorithms and applications that focuses on the theory and applications of random matrices. It is unique in the sense the “users” and “producers” of random matrix theory from the entire breadth of science and engineering are being brought together.
Our speakers will speak about the role of random matrices in a variety of areas that include statistical signal processing, machine learning, numerical algorithms, analysis, wireless communications, array processing, computational electromagnetics, uncertainty quantification, optics, and multivariate statistics.
Participants will interact with the workshops on Stochastic Computation, Special Functions and Orthogonal Polynomials, Numerical Linear Algebra, Foundations of Data Science and Machine Learning, inter alia.
Organizers: Sabrina Kunzweiler, Elisa Lorenzo Garcia, Stefano Marseglia
Elliptic curves play a very important role in modern mathematics with applications ranging from the very theoretical to the very computational. For example, they are the backbone of Wiles proof of Fermat’s Last Theorem and of one of the most widely used cryptosystem.
More recently, isogenies of elliptic curves and abelian varieties gained attention as well.
They are used in different cryptographic schemes that are proposed in the context of post-quantum cryptography.
To make these schemes efficient, and to evaluate their security, it is important to understand the underlying mathematics.
Our workshop will gather experts working on the computational tools involving elliptic curves, abelian varieties and isogenies, as well as experts working on the cryptographic applications.
Period 3
16-18 July
Organizers: Radu Boţ, Nicolas Boumal, Katya Scheinberg
This edition of the Continuous Optimization workshop aims to bring together leading experts from diverse backgrounds and career stages to discuss recent advances in the field. The program spans algorithms, theory and applications. Given the wide applicability of optimization in engineering (including machine learning as a prominent example) and the commonalities in mathematical tools used in other foundational fields (e.g., complexity theory, variational analysis, the dynamical systems view of iterative algorithms, the prevalence of various flavors of geometry and more), we expect renewed interactions with several other workshops, including Foundations of Data Science and Machine Learning, Computational Optimal Transport, Inverse Problems, Computational Algebraic Geometry, Random Matrices, Quantum Information and Quantum Algorithms, Computational Dynamics, and Numerical Linear Algebra.
Organizers: Angela Capel Cuevas, Richard Kueng, Lin Lin
Organizers: Michael Feischl, Christian Kreuzer, Rob Stevenson
Adaptivity in numerical PDEs refers to the data-dependent adaptation of the numerical discretisation of PDEs with the aim of improving the quality-cost balance. It is often only adaptivity that allows precise numerical simulation of complex physical processes. As the target function is typically unknown, adaptivity is usually organised iteratively, with decisions being made in each iteration based on a posteriori error indicators to adaptively improve the next iteration. Because of the early recognized enormous advantages of adaptive methods in practical applications it has become an indispensable tool in scientific computing and engineering sciences. In addition, a well-founded mathematical theory for basic modelling problems has been developed over the last three decades, which has been recently successively extended to larger problem classes, such as time-dependent problems, and includes more and more aspects, such as adaptive numerical solvers and strictly equivalent a posteriori bounds.
The workshop aims to present the latest developments in practical applications and theoretical findings in the field of adaptivity.
This workshop will have connections the the FoCM workshops “Approximation Theory and Computational Harmonic Analysis”, “Foundations of Numerical PDEs”, and “Numerical Linear Algebra”.
Organizers: Georg Gottwald, Eva Miranda, Cristobal Rojas
This workshop will bring together mathematicians working on computational dynamics, spanning a wide range from computability theory to data driven modelling. The workshop will ask questions when and how dynamical systems can be used to perform computation in the sense of Turing machines, including undecidability. Recent advances have shown that fluid flows can in principle be used to perform computations, implying that their future dynamics will be unpredictable. A probabilistic approach to dynamical systems which aims at understanding its statistical properties rather than the exact evolution of a single trajectory, offers an alternative to overcome the predictability barrier of certain dynamical systems. Set-oriented methods allow for the characterization of global statistical properties and detection of coherence in the flow, when the dynamical system is accessible only through some partial observations. The convergence properties of these methods have only recently been explored, giving rise to improved algorithms.
Organizers: Georg Regensburger, Sonia Rueda, Jacques-Arthur Weil
The workshop explores algebraic and geometric foundations as well as symbolic algorithms for questions related to differential and functional equations, as well as their applications.
Favorite topics have included differential algebra and elimination, Lie symmetry, differential invariants, moving frames, integrability, differential Galois theory, local and closed form solutions, normal form algorithms, operator algebras and combinatorics.
We wish this workshop to be a forum for ideas, techniques and applications . We thus would like to encourage the speakers to contextualize their contribution for a diverse audience.
Other workshop related with differential equations mostly focus on numerical or probabilistic methods; Symbolic Analysis in the main forum for symbolic treatment of differential and functional equations.
The FOCM conference as a whole should provide several other areas of interest for participants of the Symbolic Analysis workshop.
Organizers: Vanni Noferini, Elizaveta Rebova, Françoise Tisseur
This workshop will survey recent progress in numerical linear algebra, such as, the modern approaches to low-rank approximation, least squares problems, linear systems and spectral estimation. Topics of interest include sparse and structured matrix computations, fast deterministic and randomized algorithms, complexity analysis, and the growing role of numerical linear algebra in computational mathematics, data science, and optimization.
Organizers: Kurusch Ebrahimi-Fard, Claudia Malvenuto, Christoph Spiegel
Combinatorics and graph theory are fundamental to computational mathematics, supplying the language and discrete structures on which algorithms, complexity analysis and data models are built. This workshop will highlight new results and recent developments in algebraic, enumerative and geometric combinatorics, with a particular focus on advanced algebraic structures and random graphs. Another focus will be on extremal and structural combinatorics, including graph-limit theory and logic-based techniques that expose how local constraints govern the behaviour of large networks.
This workshop will have connections to other FoCM workshops, including “Foundations of Data Science and Machine Learning”, “Special Functions and Orthogonal Polynomials”, “Random Matrices”, “Computational Number Theory”, and “Quantum Information and Quantum Algorithms”.