I am an assistant professor in the Department of Mathematics at Vrije Universiteit Amsterdam, part of the Center for Topology and Applications Amsterdam (CTA^2). I am a regular visitor of the Center for Systems Biology Dresden. My main field of research is algebraic topology. In particular I study manifold invariants, cobordism categories and topics in mathematical physics such as topological quantum field theories, as well as applications of topology in the life sciences and in palaeobiology.
Together with Gijs Heuts, Magdalena Kędziorek, Inbar Klang, Lennart Meier and Steffen Sagave, I am part of the NWO XL consortium Symmetry on the interface of topology and higher algebra, and we are hiring several positions and organising a variety of scientific events between 2025 and 2031.
I am also part of the PRELIFE network researching the origin of life. This is a large-scale consortium with 62 researchers working on all aspects of origin of life research, including astronomy, biology, chemistry, earth and (exo)planetary sciences, educational science, information sciences, mathematics, and physics, funded by the Nationale Wetenschapsagenda (NWA). We are hiring 15 PhD positions and 6 non-scientific personnel. My project, co-supervised with Mark van Zuilen and Otto Sumray, applies topological data analysis to the problem of distinguishing biogenic fossils from abiogenic features in the rock record.
In 2022-2025 I held a VENI grant titled Shapes inside shapes in mathematics, physics, and biology.
Before this, I was a postdoctoral research associate at the Centre for Topological Data Analysis at the University of Oxford working with Heather Harrington and Helen Byrne. Before this I was a postdoc at the Max Planck Institute for Mathematics in Bonn and a postdoc at the Department of Mathematical Sciences at Copenhagen. I was a DPhil student at the Mathematical Institute of the University of Oxford, working with Ulrike Tillmann on topology of manifolds and cobordism categories. I was also a DPhil student at the Earth Sciences Department of the University of Oxford working with the late Martin Brasier on morphology of Ediacaran organisms. Before that I wrote my master's thesis on quantum gravity with Renate Loll at the Institute for Theoretical Physics of Utrecht University and Tom Ilmanen at the Department of Mathematics of ETH Zürich.
I am a co-organiser of the VU General Mathematics Colloquium
Within the Center for Topology and Applications Amsterdam (CTA^2), Thomas Rot organises CTA^3G
Topology Intercity Seminar (TopICS) is the seminar for our consortium Symmetry on the interface of topology and higher algebra
European Autumn School of Topology (EAST), happening every September, is a yearly PhD school in algebraic topology and homotopy theory of which the organisation is now part of Symmetry on the interface of topology and higher algebra
Together with artist duo Firma Kluit, we are working on the exciting collaborative artwork Dwaalweg, an artistic installation that tells the story of what it is like to do research in mathematics, inspired by my research on nested manifolds.
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).
Current and past PhD students, postdocs and students:
PhD students:
Alba Sendón Blanco (jointly supervised with Thomas Rot)
Bernat Jordà Carbonell (jointly supervised with Mark van Zuilen and Otto Sumray)
Hester Breman (unofficial, main supervisor Rainer Goebel)
Postdocs:
Guy Boyde (on a VENI)
Hannah Santa Cruz Baur (working also with Magnus Botnan and Senja Barthel)
Master/bachelor students:
Rolf Vlierhuis
Positions:
As part of the Symmetry on the interface of topology and higher algebra I will be hiring a PhD student in algebraic topology to start in 2026, as well as a postdoc joint with Inbar Klang to start in 2027.
See also my Google scholar page.
with Luuk Stehouwer and Simona Veselá, preprint arXiv:2504.07917 (2025). 68 pages.
with Maxine Calle, Laura Murray, Natalia Pacheco-Tallaj, Carmen Rovi, and Shruthi Sridhar-Shapiro, Topology and its Applications, in press, 109448 (2025). 38 pages.
On the importance of visuo-spatial thinking for research mathematicians
with Hester Breman, Mikkel Willum Johansen, Henrik Kragh Sørensen, and Rainer Goebel, Canadian Journal of Experimental Psychology/Revue canadienne de psychologie expérimentale 79(2), 137–145 (2025).
with Carmen Rovi, and Julia Semikina, preprint arXiv:2210.00682, to appear in Algebraic & Geometric Topology (2024). 31 pages.
with Lewis Marsh, Otto Sumray, Thomas M. Carroll, Xin Lu, Helen M. Byrne, Heather A. Harrington, Entropy 24(8), 1116 (2022). 32 pages.
with Mona Merling, Laura Murray, Carmen Rovi, and Julia Semikina, Topology and its Applications, Vol. 318, 108105 (2022). 22 pages.
with COvid-19 Multi-omics Blood ATlas (COMBAT) Consortium (203 authors), Cell, 185(5), 916-938 (2022). 82 pages.
with Henrik Kragh Sørensen, Mikkel Willum Johansen, and Hester Breman, 1st Mathematical Reasoning in General Artificial Intelligence Workshop, ICLR (2021). 8 pages.
International Mathematical Research Notices, 2020(14), 4496-4511 (2020). 12 pages.
with Martin D. Brasier, Fraces S. Dunn and Alexander G. Liu, Proceedings of the Royal Society B: Biological Sciences, 284(1862), 20171348 (2017). 9 pages.
MATHEMATICS
DPhil thesis Mathematics, Trinity College, University of Oxford. Supervised by Ulrike Tillmann (2018). 111 pages.
PALAEONTOLOGY
DPhil thesis Earth Sciences, St. Cross College, University of Oxford. Supervised by Martin Brasier (2015). 312 pages.
PHYSICS
MSc thesis Mathematics & Theoretical Physics, Utrecht University. Supervised by Renate Loll, Tom Ilmanen, and Gunther Cornelissen (2012). 66 pages.
Primordial Gravitational Waves
BSc thesis Mathematics & Physics, Utrecht University. Supervised by Tomislav Prokopec, Jan Smit (2009). 51 pages.
with Thomas Rot. Lecture notes for the Mastermath course Algebraic Topology II 2024-2025. 138 pages.
Research note. arXiv:2007.05451 (2020) 17 pages.
Research essay (2011). 16 pages.
Evolution of Vertebrates through the eyes of parasitic flatworms
Research essay (2011).
I developed the course Elements of Geometry for 2nd year mathematics students, that is taught as an intensive course at VU in June (thus far in 2024 and 2025).
In spring 2025 I taught Algebraic Topology II on Fiber Bundles and Characteristic Classes together with Thomas Rot. Lecture notes available here.
In autumn 2026 I teach Algebraic Topology I together with Inbar Klang. We use lecture notes from Steffen Sagave, available here.
I enjoy drawing and painting, below are some examples of my work.