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 CAT* embedded surfaces in 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.

*Here CAT ∈ {TOP, DIFF}.

Research Interests — For Non-Mathematicians

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

Take for example the 3-dimensional doughnut-shaped universe here, and the two circles inside it. In some sense, both are very simple curves — they don't twist around themselves into any kind of knot. But they are different, since you can only fill in one of them to get a disc. (Hint: it's the blue one). In our universe, every unknotted curve can be filled in with a disc, so the red curve is a kind of shape can't exist in our universe! (Probably, I'm not a cosmologist.)

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 space. 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, and also useful in that they fine enough to distinguish very similar surfaces.

Conference & Workshop Attendances

2025/12 - Workshop on Surfaces in 4-Manifolds, U. Regensburg
2025/10 - 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.

Conference / Invited 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.
"The metaplectic representation is faithful" - Algebra seminar, U. of Oxford

Seminars / Reading Groups I've Organised

2025 - Gauge Theoretic Invariants [link]. 10 week reading group, building up some theory and applications of Seiberg-Witten invariants.

Email: simeon.[lastname]@glasgow.ac.uk

Find my CV here.

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