Konferenzen  >  Mathematik  >  Computer-gestütztes Beweisen

Institute for Pure and Applied Mathematics (IPAM), UCLA

A number of core technologies in computer science are based on formal methods, that is, a body of methods and algorithms that are designed to act on formal languages and formal representations of knowledge. Such methods include interactive proof assistants, automated reasoning systems (including first-order theorem provers and satisfiability solvers), computer algebra systems, and knowledge representation and database systems. Methods based on machine learning have also led to the discovery of new mathematical results. The goal of this workshop is to bring mathematicians and computer scientists together to explore potential applications of these technologies to the domain of pure mathematics, building upon the exciting recent developments in this subject.

CIRM – Centre International de Rencontres Mathématiques

The debate between Hilbert and Brouwer, which took place at the beginning of the 20th century, is without doubt one of the most important events in the history of foundations of mathematics. In that debate Brouwer rejected transcendent proof methods and insisted on the exclusive use of intuitionist (or constructive) arguments for a conceptually correct development of mathematics. Hilbert, on the contrary, viewed these transcendental and non-effective proofs as the very essence of mathematical thinking. He introduced proof theory in the hope to justify these transcendent methods by purely effective means. While Gödel’s second incompleteness theorem is generally said to contravene Hilbert’s hopes, it is remarkable that Gödel himself insisted on the fact that his result does not contradict at all Hilbert’s program. In fact, the impact of Gödel’s result on Hilbert’s program was the topic of a discussion between Gödel, Herbrand and Neumann about the fundamental question of the scope of constructive mathematics. Recent results in different fields of mathematics, such as proof theory, type theory, constructive algebra and categorical logic, shed new light on this question. For instance, proof theory seems to indicate precise limits to intuitionistic type theory, while on the other hand the Univalence Axiomallows for a rather unexpected extension of constructive methods. The goal of this workshop is precisely to bring together experts in these different fields to evaluate the impact of those new results on foundations of mathematics.

American Institute of Mathematics (AIM)

This workshop, sponsored by AIM and the NSF, will be devoted to solving open problems regarding the stability of nonlinear waves using computer assisted methods of proof. Some results using computer assisted methods of proof include Hilbert's 18th problem and Smale's 14th problem. Researchers have developed efficient methods for obtaining rigorous error bounds for numerical approximations of heteroclinic and homoclinic connections between fixed points in ODE systems. These rigorous computation methods establish the existence and uniqueness of the solution in addition to providing a tight, completely rigorous error bound. Thus, these rigorous computations can be used to prove Theorems. Similarly, researchers have developed efficient and robust numerical methods for computing quantities that relay information about the spectral stability of one-dimensional traveling wave solutions to PDEs, which in turn yields information regarding the nonlinear stability. This workshop will help researchers identify collaborative opportunities to use rigorous computation to prove theorems regarding open problems in nonlinear wave theory.

Computer assisted proofs for stability analysis of nonlinear waves. Stability of multi-dimensional non-planar traveling waves. Rigorous computation of center manifolds

Angewandte Mathematik (allgem.),    Analysis

Mathematical Sciences Research Institute (MSRI)

Computational proof assistants now make it possible to develop global, digital mathematical libraries with theorems that are fully checked by computer. This summer school will introduce students to the new technology and the ideas behind it, and will encourage them to think about the goals and benefits of formalized mathematics. Students will learn to use the Lean interactive proof assistant, and by the end of the session they will be in a position to formalize mathematics on their own, join the Lean community, and contribute to its mathematical library.

Kurse und Veranstaltungen für Studenten der Mathematik,    Algebra

in conjunction with ACNS 2023

Over the years, many automated methods have been proposed to help designers and cryptanalysts prove security bounds of symmetric-key primitives against various attacks such as differential or integral cryptanalysis. These include the use of mixed integer linear programming (MILP), Boolean satisfiability problem (SAT) solvers and their extension, satisfiability modulo theories (SMT) solvers, as well as constraint programming (CP) solvers. More recently, data-driven techniques that rely on machine and deep learning have also been proposed, leading to attacks that rival that of their classical counterparts. The aim of the ADSC workshop is to provide an international forum for researchers to explore and further push the boundaries of these automated and data-driven methods. We welcome submissions on the application, improvement and efficient implementation of these methods.

Mathematical Sciences Research Institute (MSRI)

This summer school will provide an introduction to the field of probabilistic proofs and the beautiful mathematics behind it, as well as prepare students for conducting cutting-edge research in this area.

Hausdorff Center for Mathematics – hcm (Universität Bonn)

The last decade has witnessed tremendous advances in both interactive and automated theorem proving, and we are arguably on the doorstep of a new era, in which interactive theorem provers validate ground-breaking mathematical research in a reasonably short time, as shown in Peter Scholze's Liquid Tensor Experiment. This new area is driven both by new software and by a growing community of users. In addition, we have seen the advent of new software that guides mathematicians in finding proofs, helps them develop new conjectures or even generates a proof or part of a proof with minimal human input. This HSM will highlight both these developments.

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)