Thursday,
9 July
09:15 – 09:30 Welcome Address (by Radu Ioan Boț and Evelyne Hubert)
09:30 – 10:30 [HS 01]
![[HS 01]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS01.png){kind=link}
Levels in arrangements (and the dual notion of k-sets) play a fundamental role in discrete and computational geometry and are a natural generalization of convex polytopes (which correspond to the 0-level). We will survey some classical results from the combinatorial theory of convex polytopes, including the Dehn–Sommerville Relations, McMullen’s Upper Bound Theorem, and the g-Theorem, due to Stanley and Billera–Lee, which completely characterizes the face numbers of simple polytopes. It is natural to wonder whether and to what extent this theory can be generalized to face numbers of levels in arrangements. We will discuss some new results in this direction, including applications to crossing numbers of graphs, as well as various conjectures and open problems.
Ulrich Wagner (chaired by Michael Kerber)
10:30 – 11:00 Coffee Break
11:00 – 12:00 [HS 01]
I will report on recent progress in the study of random lozenge tilings of a hexagon with periodic weightings.
Numerical simulations of such tilings show an asymptotic separation of three phases, that are known as frozen, rough and smooth (or solid, liquid and gaseous).
The study of these phenomena requires tools from a number of areas of mathematics, as will be explained in the talk, and that are outlined below.
In the first step, one uses the combinatorics of the model to derive explicit formulas for an equivalent random particle system. This system is a determinantal point process, which means that many probabilities can be expressed as determinants of a correlation kernel. The correlation kernel has a double contour integral formula that involves matrix-valued orthogonal polynomials (MVOP).
Secondly, one interprets the matrix valued orthogonality as orthogonality on an algebraic curve. The spectral curve is a Harnack curve as pointed out by Kenyon, Okounkov and Sheffield. The contour integrals are transformed to the spectral curve.
The third step is the study of an equilibrium problem on the Harnack curve. The result is an equilibrium measure that on the one hand gives the asymptotic distribution of the MVOP and on the other hand provides a way to construct a double cover of the Harnack curve. The three phases are distinguished by the location of zeros of certain meromorphic differentials on the double cover.
The final step is a steepest descent/saddle point analysis of the contour integrals on the Harnack curve.
Further detailed simulations reveal interesting transitions when the number of smooth regions increases.
Arno Kuijlaars (chaired by Erik Koelink)
12:00 – 14:00 Lunch Break
14:00 – 16:00
16:00 – 16:30 Coffee Break & Poster Sessions
16:30 – 18:30
Friday,
10 July
09:30 – 10:30 [HS 01]
This talk considers the numerical solution of high-frequency scattering problems modelled by the Helmholtz equation with a bounded obstacle. Although the analysis of this problem dates back at least 50 years, over the past decade or so, tools and techniques from semiclassical analysis have provided a new perspective and been used to settle several long-standing open problems in this area. Semiclassical analysis works in phase space (i.e., position and frequency) and describes rigorously the extent to which solutions of high-frequency PDEs are dictated by the properties of the corresponding geometric-optic rays.
The goal of this talk is to showcase (to a general audience) some of the numerical-analysis results about finite-element methods, boundary-element methods, and domain-decomposition methods recently obtained using semiclassical techniques.
Euan Spence (chaired by Ralf Hiptmair)
10:30 – 11:00 Coffee Break
11:00 – 11:10 FoCM News (by Evelyne Hubert and Gregorio Malajovich)
11:10 – 12:00 [HS 01]
The Adapted Wasserstein distance has been proven to be a suitable metric for comparing stochastic processes while accounting for the evolution of information in time, especially for the purpose of stochastic dynamic optimization in a context of model uncertainty. In this talk I will first recall the motivation and intuition behind such distance for general processes, and then restrict the attention to the space of Gaussian processes. I will introduce the adapted Bures-Wasserstein space, analogue of the classical Bures-Wasserstein space in optimal transport for the setting of stochastic processes, and provide a geometric analysis of such space. This includes full description of tangent spaces and geodesic curves, as well as explicit characterization of solutions to the adapted transport problem. I will conclude by presenting a sensitivity analysis w.r.t. the adapted Bures-Wasserstein distance.
Based on joint works with D. Bartl, A. Grass, S. Hou and G. Pammer.
Beatrice Acciaio (chaired by Young-Heon Kim)
12:00 – 14:00 Lunch Break
14:00 – 16:00
16:00 – 16:30 Coffee Break & Poster Session
16:30 – 18:30
Evening at a Traditional Viennese Wine Tavern
19:00 – 22:00, ➜ Neustift am Walde 66
Saturday,
11 July
09:30 – 10:30 [HS 01]
The primary task of many applications is approximating/estimating a function through samples drawn from a probability distribution on the input space. The deep approximation is to approximate a function by compositions of many layers of simple functions, that can be viewed as a series of nested feature extractors. The key idea of deep learning network is to convert layers of compositions to layers of tuneable parameters that can be adjusted through a learning process, so that it achieves a good approximation with respect to the input data. In this talk, we shall discuss mathematical theory behind this new approach and approximation rate of deep network; we will also show how this new approach differs from the classic approximation theory, and how this new theory can be used to understand and design deep learning networks.
Zuowei Shen (chaired by Weizhu Bao)
10:30 – 11:00 Coffee Break
11:00 – 11:05 Award Ceremony (presented by Erich Novak)
Joseph F. Traub Prize for Achievement in Information-Based Complexity and Joseph F. Traub Information-Based Complexity Young Researcher Award
11:05 – 12:00 [HS 01]
In the “Complexity of Bézout’s Theorem”, Shub and Smale proposed a geometric point of view on polynomial system solving, in which the complexity of continuation methods is governed by the length of paths in a suitable condition metric. This idea suggests a broader question: what is the right geometry on a space of zero sets of a family of analytic functions, or more generally on a space of analytic cycles?
In this talk, I will describe an approach to this question based on optimal transport. The basic idea is to associate to each analytic cycle its normalized volume measure in the ambient space. In this way, a family of cycles gives a family of probability measures, and the Wasserstein distance measures the optimal cost of moving one such measure into another, taking the ambient geometry into account. Deforming one cycle into another then becomes a constrained transport problem: the mass is allowed to move, but only through measures that still come from cycles in the same family.
This leads to a natural inner Wasserstein distance on spaces of algebraic and analytic cycles. On the regular part of the cycle space, this distance recovers familiar differential-geometric structures of Weil–Petersson type. At the same time, it extends beyond the regular locus, incorporating singular cycles as finite-distance points of the same metric geometry.
I will also discuss how this Wasserstein geometry connects back to the original motivation from polynomial system solving. In simple families, the discriminant appears as a conical singularity of the transport metric, and minimizing geodesics with regular endpoints avoid it.
The talk is based on joint works with Paolo Antonini, Fabio Cavalletti, and Lorenzo Cecchi.
Antonio Lerario (chaired by Gregorio Malajovich)
12:00 – 14:00 Lunch Break
14:00 – 16:00
16:00 – 16:30 Coffee Break & Poster Sessions
16:30 – 18:30
Sunday,
12 July
Conference Dinner – Festive Evening at Palais Niederösterreich
17:00 – 22:00, ➜ Herrengasse 13
Monday,
13 July
09:30 – 09:35 Award Ceremony (by Karlheinz Gröchenig)
Vasil A. Popov Prize
09:35 – 10:30 [HS 01]
Generative flows, in particular flow matching provides state-of-the-art performance for sampling from (conditional) continuous distributions.
The talk gives a brief introduction into the topic. In particular, we address recent research on learning the latent distribution and
extend the approach to spherical flows for sampling from discrete distributions. Large language models (LLMs) based on autoregressive decoding are the prevailing approach to text generation. An alternative to sequential generation is to apply diffusion or flow-based generative modeling to produce the entire sequence at once. We study the problem of learning generative models for discrete distributions in a continuous embedding space. Whereas prior approaches typically operate on the Euclidean space or the probability simplex, we instead work on the n-sphere. There the von Mises-Fisher (vMF) distribution induces a natural noise process and admits a closed-form conditional score.
Exploiting the radial symmetry of the vMF density, we reduce the continuity equation on the n-sphere to a scalar ODE in the cosine similarity,
whose unique bounded solution determines the velocity of the spherical flow.
This is joint work with J. Chemseddine, R. Duong and G. Kornhardt (TU Berlin).
Gabriele Steidl (chaired by Youssef Marzouk)
10:30 – 11:00 Coffee Break
11:00 – 11:10 Award Ceremony (by Albert Cohen & Teresa Krick)
Szanto Medal
11:10 – 12:00 [HS 01]
Two-dimensional (2-D) fluids produce fascinating patterns of swirling motion. How and why the patterns emerge are long-standing questions, first addressed in the 19th century. Countless researchers have since contributed to innovative techniques, but the basic question of swirling 2-D motion and its long-term behavior remains largely open. The numerical approach to this problem naturally leads to matrix hydrodynamics, where quantization theory is used to obtain structure-preserving spatial discretizations—an idea developed by Vladimir Zeitlin in the early 1990s. Recently, this idea was revived in several directions, some of which I will present in my talk.
Klas Modin (chaired by Melvin Leok)
12:00 – 14:00 Lunch Break
14:00 – 16:00
16:00 – 16:30 Coffee Break & Poster Session
16:30 – 18:30
Tuesday,
14 July
09:30 – 10:30 [HS 01]
Looking around us, many surfaces including the Earth are no plain Euclidean domains but special cases of Riemannian manifolds. Uncertain physical phenomena on these surfaces can for example be described by random fields and stochastic partial differential equations. In this talk, I will introduce these stochastic quantities on hypersurfaces. Furthermore, I will discuss their efficient simulation and show how the obtained samples can be used to generate random shapes and time-evolving stochastic manifolds.
Annika Lang (chaired by Gabriel Lord)
10:30 – 11:00 Coffee Break
11:00 – 11:10 Award Ceremony (by Evelyne Hubert and Elena Celledoni)
The Stephen Smale Prize
11:10 – 12:00 [HS 01]
Finite element exterior calculus (FEEC) has demonstrated the importance of topological and geometric structure in the design and analysis of numerical methods for partial differential equations. By combining differential complexes, cohomology, and compatible discretizations, FEEC has provided new perspectives on problems ranging from electromagnetism to fluid and plasma dynamics. Recent developments in topological hydrodynamics further highlight the role of topology and geometry in continuum physics. Phenomena involving knots and braids in fluids and plasmas, Lie advection, magnetic relaxation, and dynamo action naturally lead to questions that are fundamentally global and asymptotic rather than local. These developments motivate a broader notion of global numerical analysis, concerned not only with finite-time approximation, but also with long-time dynamics, invariant structures, and topological constraints in fluid and plasma dynamics. The foundations of FEEC are largely built around scalar fields and differential forms, emphasizing discrete topology through the de Rham complex and antisymmetric tensor structures. At the same time, many important continuum theories are fundamentally tensorial. This suggests the need for a broader framework extending the principles of FEEC beyond differential forms toward a finite element tensor calculus (FETC), incorporating tensor symmetries, representation-theoretic structures, and discrete geometry. In this talk I will discuss several directions toward such a framework, including tensor complexes arising from Bernstein–Gelfand–Gelfand (BGG) constructions, tensor field decompositions, and discrete geometric structures. The broader goal is to develop structure-preserving computational frameworks that faithfully reflect the geometry, topology, and dynamics underlying complex physical systems.
Kaibo Hu (chaired by Elena Celledoni)
12:00 – 14:00 Lunch Break
14:00 – 16:00
16:00 – 16:30 Coffee Break & Poster Session
16:30 – 18:30
Mayor’s Reception at Vienna City Hall
19:00 – 22:00, ➜ Friedrich-Schmidt-Platz 1
Wednesday,
15 July
09:30 – 10:30 [HS 01]
This talk explores o-minimal optimization, a framework for treating a wide class of structured problems, including piecewise linear, semi-algebraic, and more generally, problems defined by simple analytic formulas. This makes it particularly well suited to modern computational languages such as Python. After giving an overview of what o-minimal structures are and why they are fundamental for optimization, we will focus on a simple but versatile tool called the projection formula established in collaboration with Daniilidis, Lewis and Shiota. This formula bridges two views on nonsmoothness: for geometers, nonsmooth functions are smooth once the domain is partitioned into manifolds; for optimizers, this is not necessary, since subdifferentials already carry the right amount of first-order information. We will show how the projection formula yields powerful results, including nonsmooth Sard-type theorems, a nonsmooth Lojasiewicz inequality, a nonsmooth chain rule, and a formal subdifferential framework known as conservative calculus. These results, in turn, have notable algorithmic implications for proximal-gradient and Lagrangian-like methods, stochastic subgradient methods, and automatic differentiation.
Jérôme Bolte (chaired by Radu Ioan Boț)
10:30 – 11:00 Coffee Break
11:00 – 12:00 [HS 01]
By a landmark result of Faltings, the set of rational points on a smooth projective curve of genus 2 or more is known to be finite. Yet determining this finite set of points remains a major open problem in number theory. We give a survey of computational methods addressing this problem, after the work of Chabauty, Coleman, and Kim and discuss several notable examples.
Jennifer Balakrishnan (chaired by Martin Sombra)
12:00 – 14:00 Lunch Break
14:00 – 16:00
16:00 – 16:30 Coffee Break & Poster Session
16:30 – 18:30
Thursday,
16 July
09:30 – 10:30 [HS 01]
In a smooth world, curves, surfaces, and more generally, manifolds intersect each other transversely, if at all; tangencies are rare events that cannot be observed in experiments. In dynamical systems theory, therefore, it was long assumed that tangencies between invariant manifolds occur at isolated points when a parameter is varied, and the transition from tame to chaotic dynamics is mediated by a single tangential event. Recent theoretical work by Bonatti, Diaz, and others has shown that the boundary between tame and chaotic dynamics is, actually, more like a thick grey world that challenges our geometric intuition: tangencies may occur robustly, which is called wild chaos. This type of dynamics requires at least three dimensions for discrete-time systems, or four for a system of ordinary differential equations. This higher dimensionality has been an impediment to our understanding of how steady states, periodic solutions, and their invariant manifolds organise wild chaotic dynamics.
In this talk, I will present numerical methods that we developed to help us understand what wild chaos and robust tangencies look like, how they arise, and why this matters for applications. Our focus will be on three-dimensional smooth maps with the counter-intuitive property that their one-dimensional invariant manifolds behave like surfaces: they cannot be avoided by other smooth curves. I will present our algorithm for computing such one-dimensional manifolds to extremely long arclengths. I will also explain how our implementation allows us to construct pseudo-orbits, which are used to detect and continue families of connecting orbits that lie at the intersection of different manifolds.
Hinke Osinga (chaired by Georg Gottwald)
10:30 – 11:00 Coffee Break
11:00 – 12:00 [HS 01]
Quantum signal processing (QSP) provides a representation of scalar polynomials of degree d as products of matrices in SU(2), parameterized by (d+1) real numbers known as phase factors. It is the mathematical foundation of quantum singular value transformation (QSVT), which has emerged as a unifying framework in quantum computing, with applications in scientific computation including Hamiltonian simulation, linear systems, and eigenvalue problems. This talk presents recent advances in the mathematical and numerical analysis of QSP, including extensions beyond polynomials, the complexity of constructing the underlying phase factors, and the stability of these constructions in finite precision. These developments reveal an unexpected bridge between quantum computation and several classical areas of mathematics, including nonlinear Fourier analysis, fast polynomial arithmetic, and Gaussian elimination for matrices with displacement structure.
Lin Lin (chaired by Joel Tropp)
12:00 – 14:00 Lunch Break
14:00 – 16:00
16:00 – 16:30 Coffee Break & Poster Session
16:30 – 18:30
Friday,
17 July
09:30 – 10:30 [HS 01]
A basic problem in Enumerative Combinatorics is to count the number of objects with fixed size in a class of objects, as for instance, to enumerate the number of plane trees of size n for all positive integer n. If the structure of the class of objects is recursive then one can obtain some recurrence relations for the counting sequence. In many cases, this recurrence relations translates into a functional equation that defines the generating series attached to the counting problem.
The complexity of the enumerative sequence can thus be captured by the algebraic nature of the generating series. Indeed, determining whether the generating functions is rational, algebraic, satisfies a linear differential equation or a polynomial one, will provide precise information about the asymptotic behavior of the counting sequence or yield new recurrence relations.
For the generating functions satisfying a so-called functional equation in one or two catalytic variables, the work of many researchers has led to the development of a general strategy for addressing this classification problem. This strategy combines methods from combinatorics, computer algebra, analysis, probability, differential algebra, and Galois theories. In this talk, I will present some of these methods by illustrating them with combinatorial examples.
Charlotte Hardouin (chaired by Carlos d’Andrea)
10:30 – 11:00 Coffee Break
11:00 – 11:10 FoCM Journal News and Call for Venue for Next Conference
11:10 – 12:00 [HS 01]
What structural graph properties are useful in designing efficient algorithms? Tree decompositions have been successfully used for this purpose, provided that the decomposition is not too “complicated” (has bounded width).
Recently, new notions of complexity of graph decompositions have been introduced, designed specifically to tackle a particular algorithmic problem.
Tree decompositions have traditionally been used in the context of forbidden graph minors; studying them in connection with graph containment relations of more local flavor (such as induced subgraph or induced minors) is a relatively new research direction.
In this talk we will discuss recent progress in this area, touching on both the classical notion of bounded width, and the newer measures of complexity
Maria Chudnovsky (chaired by Kurusch Ebrahimi-Fard)
12:00 – 14:00 Lunch Break
14:00 – 16:00
16:00 – 16:30 Coffee Break & Poster Session
16:30 – 18:30
City Cruise on the Danube
19:15 – 22:45, ➜ Reichsbrücke bridge, 2nd district
Saturday,
18 July
09:30 – 10:30 [HS 01]
Many eigenvalue problems are too large, depend nonlinearly on the spectral parameter, or arise from infinite-dimensional operators, so computing the whole spectrum is not a reasonable goal. In practice, the spectral information of interest often lies in a prescribed region of the complex plane. Contour methods are eigensolvers designed for exactly this task: they compute eigenvalues in a region by turning spectral information inside a curve into linear solves on its boundary. The resulting algorithms are local, parallelizable, and accurate, and they are among the state-of-the-art methods in several applications.
In this talk, I will introduce the basic idea behind contour methods and explain how it extends from matrices to differential operators, nonlinear eigenvalue problems, and multiparameter eigenvalue problems. In the multiparameter setting, new complications appear as the residue calculus behind classical one-parameter contour methods no longer applies directly. I will derive a matrix-valued residue formula from Grothendieck residue theory and show how it leads to the first practical contour method for computing multiparameter eigenvalues in a prescribed region.
Alex Townsend (chaired by Vanni Noferini)
10:30 – 11:00 Coffee Break
11:00 – 12:00 [HS 01]
Applications in physics, biology, and chemistry often involve PDE systems with multiple interacting components, such as gas mixtures, competing populations, or chemical reactants. These are typically described by nonlinear reaction–diffusion systems. Their analysis and numerical approximation are complicated by strong nonlinear coupling, cross-diffusion effects, and structural constraints such as positivity, boundedness, and entropy dissipation, often in the absence of positive definiteness of the diffusion operator.
Exploiting the entropy structure underlying these systems, we develop structure-preserving discretizations based on nonlinear transformations to entropy variables that ensure positivity and boundedness of the solutions at the discrete level. We focus on local discontinuous Galerkin schemes, where auxiliary variables permit a reformulation of the problem in which nonlinearities are removed from differential operators and interface terms. This approach naturally accommodates high-order approximations, enables localized and parallel evaluation of nonlinear operators, and yields discrete entropy stability.
Ilaria Perugia (chaired by Rob Stevenson)
12:00 – 14:00 Lunch Break
14:00 – 16:00
16:00 – 16:30 Coffee Break & Poster Session
16:30 – 18:30