We call two manifolds cut and paste equivalent if I can cut one in pieces and glue it back together to obtain the other. Cut and paste groups of manifolds are known as SK groups, with refinement the SKK groups (controlled cut and paste groups). In my joint work with Mona Merling, Laura Murray, Carmen Rovi, and Julia Semikina, we upgrade these classical invariants to  a spectrum, using a new formulation of K-theory called K-theory of squares, which recovers SK groups (now with boundary) on connected components. We recover the Euler characteristic invariant as a map of spectra to the K-theory of the integers. In further joint work with Carmen Rovi, and Julia Semikina, we moreover categorify cobordism cut and paste invariants using cube shaped diagrams of manifolds. 

SKK invariants are important for the classification of topological quantum field theories (TQFTs). In my joint work with Luuk Stehouwer and Simona Veselá, we approach questions about the computation of SKK groups with different tangential structures by means of a short exact sequence comparing SKK and bordism groups, with kernel given by the integers modded out by the Euler characteristic of manifolds of this tangential structure of one dimension higher. This leads to two questions, namely, when does there exist a manifold with odd Euler characteristic given a tangential structure and dimension, and when is the short exact sequence split. We resolve these questions in a wide range of cases and compute SKK groups for tangential structures relevant to physics, thus classifying not necessarily unitary TQFT's in dimensions 1-5.

The first question in the above project is closely related to my work on k-orientable manifolds - manifolds for which the Stiefel-Whitney classes vanish in ranks 0<i<2^k. A k-orientable manifold has an even Euler characteristic unless its dimension is a multiple of 2^{k+1}. These manifolds also exist in all these dimensions for 0,1,2 and 3-orientable but whether a 4-orientable manifold with odd Euler characteristic exists is still an open problem extending the Hopf invariant one problem, see also this note.

By a nested manifold we mean a manifold with a submanifold of strictly lower dimension, which could itself have a submanifold, et cetera. Nested cobordisms are cobordisms of nested manifolds that respect this nesting structure. In my mathematics PhD thesis I computed the homotopy type of the space of nested manifolds in R^n. In my joint work with Maxine Calle, Laura Murray, Natalia Pacheco-Tallaj, Carmen Rovi, and Shruthi Sridhar-Shapiro, we give a general formula for obtaining the generators of a nested cobordism category and we give a generator-relation presentation for the cobordism category of cylinders with one-dimensional cobordisms living on them, or "striped cylinders".

My PhD student Alba Sendón Blanco is working on the Pontryagin-Thom construction for nested manifolds. My bachelor student Rolf Vlierhuis considered SK groups of nested manifolds.

Apart from manifold topology, I also study applications of topology in biology and palaeobiology. For example, my joint work with Lewis Marsh, Otto Sumray, Thomas M. Carroll, Xin Lu, Helen M. Byrne, Heather A. Harrington studied the problem of selecting relevant genes in a single cell transcriptomics dataset. We worked around the assumption of discrete cell types, instead taking into account continuous variation between cells as well, by using methods from spectral graph theory. Eigenscores in particular use the lowest eigenvectors of the graph Laplacian to embed the gene signals in a low-dimensional space gene space in which genes that are more "relevant" for the dataset have greater norm (see the picture on the left).

In ongoing work with Gillian Grindstaff we study spaces of "nested" phylogenetic trees that model co-evolution, for example of hosts and parasites. 

During my palaeontology PhD I studied the morphology and growth of early complex life of the Ediacara biota. My work with Martin Brasier, Frankie Dunn and Alex Liu concluded on the basis of growth measurements and models that the iconic organism Dickinsonia was likely related to metazoa (animals).