I am interested in algebraic geometry, representation theory, low-dimensional topology, number theory and mathematical physics.

Currently, the main focus of my research is in harmonic analysis on p-adic groups and how it is influenced by so called braid homology theories. I also care about the related arithmetic geometry. This comes close to some "physical mathematics" like 3d SUSY QFTs and string theory, sometimes featuring in my research. Some other (math) things I'm thinking about are double affine Hecke algebras, elliptic cohomology, and three-manifold invariants.

My research is supported by the Finnish Academy of Science and Letters.

articles:

• (with Minh-Tam Trinh) The Hilb-vs-Quot conjecture.
  Submitted, available at arXiv:2310:19633.
• (with Cheng-Chiang Tsai) Shalika germs for tamely ramified elements in GL(n).
  Submitted, available at arXiv:2209.02509. (Video 1, Video 2, slides.)
• (with Nicolas Hemelsoet and Anna Lachowska) On the affine Springer fibers inside the invariant center of the small quantum group.
  Submitted, available at arXiv:2205.09700. (Video.)
• (with Eugene Gorsky and Alexei Oblomkov) The affine Springer fiber-sheaf correspondence.
  Submitted to Advances in Math., revision in preparation in response to referee report. Available at arXiv:2204.00303. (Video, slides.)
• (with Pablo Boixeda Alvarez and Ivan Losev) Appendix to: Affine Springer fibers, Procesi bundles, and Cherednik algebras.
  arXiv:2104.09543. Duke Math. J. 173.5 (2024): 807-872.
• (with Eugene Gorsky and José Simental Rodríguez) IHÉS notes on Algebra and geometry of link homology.
  arXiv:2108:10356. Bulletin of the LMS 55.2 (2023): 537-591..
• (with Niklas Garner) Generalized affine Springer theory and Hilbert schemes on planar curves.
  arXiv:2004.15024, IMRN, 8, 6402-6460 (2023).
• Unramified affine Springer fibers and isospectral Hilbert schemes.
  arXiv:1808.02278, Selecta Mathematica (New Series) 26, 61 (2020).
• (with Yuzhe Bai and Eugene Gorsky) Quadratic ideals and Rogers-Ramanujan recursions.
  arXiv:1805.01593, The Ramanujan Journal 52 (2020): 67-89.
• Hecke correspondences for Hilbert schemes of reducible locally planar curves.
  arXiv:1711.06444, Algebraic Geometry (Foundation Compositio Mathematica) 6.5 (2019).

other:

• Sage code to compute Shalika germs, orbital integrals, etc. for GL(n).
• Some hand-written notes on Maulik's proof of the OS/DHS conjecture.
• (with Gurbir Dhillon) Proof of the hard Lefschetz theorem. Chapter 20 of Introduction to Soergel Bimodules
  RSME Springer Series (2020).
• Koszul algebras and resolutions. M.Sc. thesis, Aalto University, May 2014. Available on request. See also arXiv:1412:3542.
• Steady states in chemical reaction networks. B.Sc. thesis, Aalto University, January 2014. Available on request.
• Lecture notes from the course "Combinatorics" at Aalto University, 2014. Taught by Alexander Engström. (Warning: typos)

• Ikshu Neithalath (Postdoc based at the University of Helsinki, jointly with Kari Vilonen, 2024-).
• Lizao Ye (Postdoc, 2024-).
• Emilia Takanen (PhD student, 2024-): Emilia is working on sheaf-theoretic Floer homology.
• Miika Rankaviita (PhD student, 2024-).
• Lassi Virtanen (MSc student, 2024): Lassi is writing his MSc thesis on Schmid's nilpotent orbit theorem.
• Lauri Karanko (BSc student, 2023): Cyclic sieving in rational q,t-Catalan combinatorics.
• Susanna Terron (MSc student, 2023): Dynamical Quantum Groups.
• Hana Ephremidze (BSc student, 2023, now in Bonn): Algebraic closure of local fields in characteristic p>0. Hana's thesis won the Prix Junior Maryam Mirzakhani 2023.

fall 2024 (Aalto):

• Differential and Integral Calculus 2 (Period I)

spring 2024 (Aalto):

• Algebraic Geometry 1 (Period IV)