Boyde hopes it will provide a new set of connections through which the many parts of mathematics where diagram algebras play a role can interact.
Applying new methods
Over the last four years, a variety of new topological techniques, concerning shape and space, have been developed to study algebraic properties of objects called diagram algebras. These algebras originate in physics and appear in a wide variety of contexts in algebra, geometry, and knot theory. This project will apply the new methods, first to study stability and instability properties in algebra, and second to investigate other applications in a variety of contexts, including knot theory, and the geometry of surfaces.
Topological tools for algebraic problems
Topology began around 1900 with Poincaré and Noether, who used algebra to study shapes and spaces. Since then, this relationship has reversed in some cases: topological tools now help study algebra. Often, to study algebraic objects topologically, researchers use algebraic homology theories. ''You can think of these as a sort of scanner, like an x-ray, and they give you a topological picture of your algebraic input. “Taking a scan” here really means doing a calculation, and these calculations are often exceedingly difficult. Many mathematicians around the world, in many different areas, work on problems that involve this sort of calculation'', Boyde explains.
Unexpected discovery
Recently, researchers made key advances in computing homology for diagram algebras, algebraic structures where elements are geometric diagrams. In joint work with Boyd, Randal-Williams and Sroka, they discovered unexpected structure: a new way of breaking it into small pieces, and an, even more surprising, way of multiplying elements. This makes both computation and expression of results much more efficient.
New topological insights into knots and algebra
Mathematicians now have access to a lot of new topological information about diagram algebras. Therefor Boyde want to figure out what that can tell us about the other parts of mathematics where these algebras play a role (principally knot theory and representation theory). Boyde: ''The project has two main goals: first, to investigate what additional information about knots can be uncovered by studying the homology of diagram algebras beyond what is captured by the Jones polynomial. And second, to develop and apply a newly discovered method (a spectral sequence, in collaboration with Sroka) for computing Hochschild homology. A more powerful and detailed homological tool that reveals deeper geometric structures and plays a key role in areas such as support varieties in representation theory.''
Boyde will join the Center for Topology and Applications Amsterdam for the upcoming three years.
Veni
The NWO Veni grant, of up to 320.000 euros, is awarded to excellent researchers who have recently obtained their PhD, to conduct independent research and develop their ideas for a period of three years. Laureates are at the start of their scientific career and display a striking talent for scientific research.