Stochastic Analysis and Probability • Uncertainty Quantification • Mathematical Biology – Collective dynamics Number Theory • Non-commutative geometry • Quantum Computation • Topological Modular Forms • Efficient congruency in resilience Highlights of the emerging interdisciplinary links for each research area are documented below with the full documented outputs included in Appendix Three. Algebra • Algebraic methods in data analysis (Persistent Cohomology) • Computer Science 9
• Constraint satisfaction problems. • Algebraic characterisation • Statistical Mechanics – diagram algebras, correlation functions, Lie theory, representation theory. • Theoretical Physics and Representation Theory Combinatorics • Computer science: Constraint satisfaction (St. Andrews) and also connections with model theory (e.g. MacPherson, Leeds). • Algorithm Design – Structural graph theory (width parameters) (ERC goals outside UK) • Algorithms: Computer Science – Microsoft + many top places, ERC grants. • Confirmation theory – Error correcting codes • Computational Complexity Geometry and Topology • Computer Vision • Molecular Biology • High energy Physics/Quantum Physics • String Theory • Topological Data Analysis • Robotics – Robotic Motion and Robotic Vision • Networks • Cryptology (Heilbronn) • Molecular Biology • Machine learning and data analysis Logic • Formal verification of software/hardware • Theory of programming languages • Quantum information 10
• Databases and big data Mathematical Analysis • Imaging • Physics – Information Theory • Theoretical Physics • Statistical Mechanics • Materials Science • Financial • Engineering Number Theory • Physics • Computer Science – Algorithmic aspects • Additive Combinatorics • Complexity Theory • Cryptology – Heilbronn Funding • Optics • Quantum Chaos • String Theory • Statistical Mechanics This session provided context for the links we know exist between the research areas of the Mathematical Sciences taxonomy and beyond. These findings are critical to highlight the importance and impact that mathematical research has on adjacent disciplines. Previous Successes in Pure Mathematics As Pure Mathematical research is renowned for being unpredictable and its true impact may not be elucidated for decades from its inception, a discussion was held to obtain a cross cutting perspective from the community on examples of successful research which have highlighted Pure Mathematics in recent decades. A non-exhaustive list of the examples highlighted is listed: • Fermat’s Last theorem 11
• Poincare Conjecture • Mordell Conjecture • Classification of finite simple groups • Mori Theory • Influence of Theoretical Physics (Two ways) • Work by Ben Green and Terrence Tao • Cryptography • Birch Swinnerton-Dyer Conjectures.