Toni Annala

Email: tannala at

I'm a mathematician, and a part-time celebrity in South Korea. Currently, I'm a member at the Institute for Advanced Study.

I specialize in the investigation of geometric structures through their invariants. My latest project focuses on expanding the scope of motivic homotopy theory by removing the assumption that the affine line is contractible (video). This framework is aimed to enable applying motivic homotopy theory to cohomology theories that were previously deemed inaccessible, such as prismatic cohomology. During my doctoral studies, I studied intersection theory and its generalizations, such as algebraic cobordism, using derived algebraic geometry (video). Beyond pure mathematics, I have investigated topological structures arising in physics. In particular, in collaboration with physicists, we have investigated the topological stability and classification of vortex-link structures formed from non-Abelian vortices that could arise in, for example, liquid crystals and Bose–Einstein condesates (article).


Here are my peer-reviewed publications (appeared or to appear):

  1. Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory (joint with Marc Hoyois and Ryomei Iwasa), Journal of the American Mathematical Society.
  2. Topologically protected vortex knots and links (joint with Roberto Zamora-Zamora and Mikko Möttönen), Communications Physics.
  3. Bivariant algebraic cobordism with bundles (joint with Shoji Yokura), Algebraic Geometry (Foundation Compositio Mathematica).
  4. Cohomology of the moduli stack of algebraic vector bundles (joint with Ryomei Iwasa), Advances in Mathematics.
  5. Charge ambiguity and splitting of monopoles, (joint with Mikko Möttönen), Physical Review Research.
  6. Base independent algebraic cobordism, Journal of Pure and Applied Algebra.
  7. Algebraic Spivak's theorem and applications, Geometry & Topology.
  8. Chern classes in precobordism theories, Journal of the European Mathematical Society.
  9. Precobordism and cobordism, Algebra & Number Theory.
  10. Oriented Borel-Moore homologies of toric varieties, Annales de l'Institut Fourier.
  11. On line bundles in derived algebraic geometry, Annals of K-theory.
  12. Bivariant derived algebraic cobordism, Journal of Algebraic Geometry.
and preprints:
  1. Atiyah duality for motivic spectra (joint with Marc Hoyois and Ryomei Iwasa).
  2. Language Models as Science Tutors (joint with Alexis Chevalier et al.).
  3. Optically Induced Monopoles, Knots, and Skyrmions in Quantum Gases (joint with Tommi Mikkonen and Mikko Möttönen).
  4. Topologically Protected Vortex Knots in an Experimentally Realizable System (joint with Hermanni Rajamäki and Mikko Möttönen).
  5. Bordism invariants of colored links and topologically protected tricolorings (joint with Hermanni Rajamäki and Mikko Möttönen).
  6. Motivic spectra and universality of K-theory (joint with Ryomei Iwasa).
Here is my doctoral thesis: Here are slide(s) of talks I have given: Here are some of my older writings. Several people seem to have found my notes on resolutions of singularities; this makes me uncomfortable. However, I will keep everything online, at least for now.