Mathematical Endeavours

Research Interests — For Mathematicians

At its most broad, my interests lie in geometric topology with a low-dimensional skew. More accurately, I like to think about manifolds in dimensions 4±1 and embeddings with codimension ≈2. 

My current focus is on equivalence relations on embedded surfaces in (smooth or topological) 4-manifolds that are weaker than isotopy, like cobordism and concordance. I'm mostly thinking about non-orientable (or at least unoriented) surfaces at the moment, but am generally interested in finding out to what extend embedded surfaces can be reasonably classified. 

Research Interests — For Non-Mathematicians

The type of shapes that can exist in a given "universe" depend on the shape of that "universe". 

Once you move up a dimension from curves and look at surfaces in 4-dimensional spaces,  things get even more complicated. Generally, we can't hope to find all of the possible surfaces in a given 4-dimensional universe. But we might be able to if we weaken what we mean for two surfaces to be "equivalent". This is sort of like saying that we can't write down every integer, but we can if we call two integers "equivalent" if their last 10 digits are the same. I'm interested in finding out which types of equivalence are useful, in that they are coarse enough to be computable, but fine enough to distinguish very similar surfaces.

Conference & Workshop Attendances

• [*25/12 - Workshop on Surfaces in 4-Manifolds, U. Regensburg]

• [*25/10 - Trisections and Related Topics, Research School, CIRM

• 2025/09 - Mapping Class Groups of Non-Simply Connected 4-Manifolds, U. of Glasgow

• 2025/05 - Links in Dimensions 3 & 4, ICERM.

• 2025/04 - UK Topology In Low-Dimensions Event (TILDE), Newcastle U.

Conferences Talks I've Given

• 2025/09/04 "Concordance and cobordism of surfaces" - Lightning talk at Mapping Class Groups of Non-Simply Connected 4-Manifolds, U. of Glasgow

• 2025/05/15 "Concordance and cobordism of (non-orientable) surfaces" - Lightning talk at Links in Dimensions 3 & 4, ICERM.

Seminar Talks I've Given

• "Even dimensional L-groups" - 9th seminar in series about surgery theory, U. of Glasgow

• "The s-cobordism theorem" - 2nd seminar " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ."

• "Kirby diagrams" - 2nd seminar in series about Kirby calculus, U. of Glasgow

• "Introduction to surgery" - AGQ Example Showcase, The Maxwell Institute

• "The metaplectic representation is faithful" - Algebra seminar, U. of Oxford