I am a fourth year Ph.D. student at the Department of Mathematics at the University of California, Davis (UC Davis) under the advisement of Eugene Gorsky. I received the M.Sc. degree in engineering physics from Aalto University in June 2014, majoring in mathematics with a minor in computer science. I am supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters and the NSF grant DMS-1700814. I am also a Fulbright grantee.
I am interested in representation theory and algebraic geometry, together with their connections to physics, low-dimensional topology and number theory. More specifically, I am interested in the representation theory of various kinds of Hecke algebras, with some emphasis on Cherednik's DAHAs. On the algebraic geometry side, I am interested in moduli spaces of sheaves on curves and surfaces.
Most recently, my research has concerned topics around affine Springer theory. More specifically, I have thought about the relationship of affine Springer theory in type A with HOMFLY homology of torus links, Hilbert schemes of points on the plane, as well as how all of these topics fit into a larger conjectural framework due to Gorsky, Oblomkov, Negut, Rasmussen, Rozansky, Shende and others. Part of my motivation also comes from physical predictions made using the recently popular 3d N=4 field theories. With my friend, Minh-Tam Trinh, we have been looking at generalizations of some of these connections to other reductive groups.
In another direction, I am interested in the geometric representation theory of degenerate/trigonometric DAHAs as developed by Lusztig-Yun, Oblomkov-Yun and Varagnolo-Vasserot. More specifically, I am trying to understand the purely outer graded case and its relation to character sheaves for symmetric spaces due to Vilonen, Xue and others.
Historically, I have had a recurring interest in commutative algebra and its relationship to other areas of math, and am happy to talk about the subject. Other projects/topics I would like to talk to people about/understand better include:
- How to prove hybrid Kazhdan-Lusztig positivity for affine Weyl groups in type A, and through this LLT and Macdonald positivity, possibly using hyperbolic localization for Soergel bimodules.
- Screening operators in rational CFT in two dimensions can be used to compute correlation functions, and together they also form quantum groups. In the minimal model case (with irrational central charge) work of Kalle Kytölä and Eveliina Peltola uses formalizations of these techniques to construct partition functions for multiple SLEs. I would like to understand the minimal models with all parameter values, in particular the connection to quantum groups at roots of unity, as well as extensions to WZW models.
If you know anything or want to know something about the above things and to talk to me about them, please do not hesitate to send me an email.
- Hecke correspondences for Hilbert schemes of reducible locally planar curves. arXiv:1711.06444. To appear in Algebraic Geometry (Foundation Compositio Mathematica).
- (with Yuzhe Bai and Eugene Gorsky) Quadratic ideals and Rogers-Ramanujan recursions. arXiv:1805.01593, May 2018. To appear in The Ramanujan Journal.
- Unramified affine Springer fibers and isospectral Hilbert schemes. arXiv:1808.02278, August 2018. Submitted. (poster)
- Blocks for the category of perverse sheaves on the cone of nilpotent symmetric matrices. In preparation.
- (with Gurbir Dhillon) Proof of the hard Lefschetz theorem. Chapter 18 of "Soergel Bimodules", To appear in the Springer RSME Series.
- 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)
- MAT 150A "Algebra". Instructor: Albert Schwarz.
- MAT 22B "ODEs". Instructor: James Bremer.
- MAT 21C "Calculus". Instructor: Francesco Rubini.
- MAT 21A "Calculus". Instructor: Brian Osserman.
- Categorification in quantum topology and beyond. January 7-18, 2019. University of Vienna
- Various workshops at the MSRI programs "Derived Algebraic Geometry" and "Birational Geometry and Moduli Spaces". Spring 2019. MSRI