Despite the substantial progress in the computer-aided verification of computer systems, ensuring the correctness of programs implementing algebraic operations is still an open problem. This problem remains unsolved even when we restrict consideration to loops that are non-nested, without exit conditions, and/or only use limited (polynomial) arithmetic.Such programs naturally arise in compiler optimization, cryptography, cybersecurity, control theory, and probabilistic reasoning. This talk will present classes of computer systems for which we automatically can solve the challenge of proving programs error-free. Key to our setting is the combination of algebraic techniques with static code analysis, allowing us to even turn some unsolvable verification challenges into solvable ones.

Linear differential equations with polynomial coefficients in a complex variable appear naturally in fields such as combinatorics, algebraic geometry, and mathematical physics. Their numerical solution, and in particular that of so-called connection problems between regular singular points, finds applications that include the high-precision computation of values of Feynmann integrals, volumes of semi-algebraic sets, or more generally periods of algebraic varieties, the asymptotic expansion of linearly recurrent sequences, and the factorization of linear differential operators. In this talk, I will give a demonstration of a numerical ODE solver that focuses on this very special class of equations, supports arbitrary-precision computations, and provides rigorous error bounds along with the numerical results. I will also demonstrate software for some of the applications listed above built on top of this solver.

The time-honored tool for modeling cause-effect relationships is a Bayesian network. This model postulates noisy functional dependencies among random variables according to a directed graph. Bayesian networks enjoy widespread use despite two shortcomings: (1) different causal specifications may define the same class of probability distributions, so cause-effect relationships cannot be learned reliably from observational data alone; and (2) if the directed graph contains cycles (“feedback loops”), the model loses desirable statistical properties such as global identifiability of parameters. In this talk I want to introduce an emergent alternative paradigm in causal modeling which aims to address these issues. The interactions of random variables are modeled by stochastic diffusion processes in equilibrium. This temporal perspective easily accommodates feedback loops. I will cover the algebraic nature of these models, their independence structure and recent identifiability results.

Take an algebraic object M. We want to find algorithms that decide whether a given statement (in a suitable language) about M is true. I will discuss how dynamical systems have recently been used to produce such decidability and undecidability results, including the following. (1) The monadic second-order theory of (N; <, {2n : n ≥ 0}, {3n : n ≥ 0}) is decidable. (2) The first-order theory of (N; <, +, {2n : n ≥ 0}, {3n : n ≥ 0}) is undecidable. (3) The first-order theory of (N; <, n ↦ |τ(n)|), where τ(n) is the Ramanujan tau function, is undecidable.

Precise predictions in particle physics rely on the computation of multiloop scattering amplitudes in quantum field theory. In this talk, I will discuss how algebraic algorithms facilitate these calculations. I will review methods for determining and evaluating Feynman integrals, the fundamental building blocks of scattering amplitudes. To manage the complexity of their coefficients, techniques such as finite-field sampling and multivariate partial fraction decomposition play a central role. I will present recent progress in this area and highlight open questions where a deeper structural understanding could further advance our predictive capabilities in fundamental physics.

Take an algebraic object M. We want to find algorithms that decide whether a given statement (in a suitable language) about M is true. I will discuss how dynamical systems have recently been used to produce such decidability and undecidability results, including the following. (1) The monadic second-order theory of (N; <, {2n : n ≥ 0}, {3n : n ≥ 0}) is decidable. (2) The first-order theory of (N; <, +, {2n : n ≥ 0}, {3n : n ≥ 0}) is undecidable. (3) The first-order theory of (N; <, n ↦ |τ(n)|), where τ(n) is the Ramanujan tau function, is undecidable.

We warmly invite all registered participants to a joint workshop dinner generously sponsored by Max Planck Institute for Software Systems

Quantum field theory does a good job of predicting the behaviour of small numbers of quantum particles. These calculations involve the combinatorics of graphs, which becomes intractable as the number of particles increases. We introduce a new approach to setting up these calculations using piecewise linear functions coming from tropical geometry. Surprisingly, we find this is well adapted for the limit of _large_ numbers of particles, where the calculations resemble the statistics of random walks. The results suggest that quantum field theory has a broader domain of application than we otherwise expect.

Differential dynamical systems appear in many applications in modeling and control theory. Frequently, for such dynamical systems, the task is to compute the minimal differential equation satisfied by a chosen coordinate. This task arises in particular in applications where only part of the solution data can be experimentally observed. As such, it is an important special case of more general differential elimination problems. We will observe two approaches to tackle the computation of such a minimal equation in the evaluation-interpolation fashion.

First, we will present a new result, which provides explicit bounding hyperplanes for the Newton polytope of such a minimal equation. The obtained bounds are based on the degrees of a given dynamical system and are proven to be sharp in “more than half the cases”. We demonstrate an evaluation-interpolation algorithm for computing minimal differential equations for dynamical systems resulting from these bounds.

Second, we will discuss a new approach of computing so-called mixed fiber polytopes in the problem of polynomial elimination. We demonstrate the increase in practical performance of our algorithm compared to existing methods and discuss an application of our work to differential elimination.

This is a joint work with Gleb Pogudin and Rafael Mohr

Algebraic techniques, such as Gröbner bases, have a long and successful history in automatic geometric theorem proving. Recently, these methods have found a new and promising area of application in the context of proving operator statements. Operator statements are first-order formulas involving identities of matrices or, more generally, of linear operators. In this talk, we present how the correctness of an operator statement can be translated into an algebraic statement about noncommutative polynomials, which can be verified automatically using computer algebra. Moreover, we discuss how to certify the invalidity of false operator statements by constructing explicit counterexamples.

I will explain how expanding dynamical systems can be efficiently encoded using group theory and finite state automata; and, in fact, how these two domains correspond tightly to each other. I will focus on complex dynamics — iteration of a rational map on the Riemann sphere — and show how fundamental questions in this domain can be answered, both theoretically and in practice.

Many defining functions in dynamics are represented by multivariate real polynomials. Under this assumption, key properties of dynamical systems often have an algebraic interpretation such as the question of nonnegativity of real, multivariate polynomials. Nonnegativity is not only a classical question in real algebraic geometry. It can also be tackled via polynomial optimization; a thriving field in the past 25 years. In this talk I present two instances highlighting this connection of polynomial optimization and dynamics, which we investigate in recent and ongoing projects: First, deciding mono- vs. multistationarity in 2- and n-site phosphorylation networks in terms of their kinematic parameters (joint work with E. Feliu, N. Kaihnsa, and O. Y¨ur¨uk). Second, computing Lyapunov functions for polynomial ODE systems (joint work with J. Heuer and N. Rieke). Interestingly, both of these results moreover lead to (different) combinatorial follow-up problems, which I will also briefly explain. This talk is meant as an invitation to the audience to further explore the intersection of polynomial optimization, dynamical systems, and combinatorics.