Expository notes
Most of the results proved in these notes are about something that seems to be well-known to experts but I could not find reference. Since the proof of the results are straightforward or simple applications of an established result (if not, I would write a separate paper), feel free to use in your paper if you need.
A very confusing sign in shifted symplectic geometry (uploaded:2024-11-16, last updated:2024-11-21)
For an n-shifted symplectic vector space V with the duality isomorphism φ: V -> V^{¥vee}[n], I explain that an equality
φ = (-1)^{n(n + 1) / 2 + 1} (φ^{¥vee} [-n])^{-1}
holds if we use the standard sign convention.
Survey and announcement
3d cohomological Hall algebra for local surfaces (With Adeel Khan) (uploaded:2023-11-14)
This is an announcement on our forthcoming paper ``Derived microlocal geometry and cohomological Donaldson–Thomas theory'', though many details are already included. We use ideas from microlocal geometry to show the Joyce conjecture in the conormal case. As a consequence, we construct 3d CoHA product for local surfaces in a way that it is compatible with Kapranov--Vasserot 2d CoHA under dimensional reduction.
An introduction to cohomological Donaldson-Thomas theory (uploaded:2023-9-30, last updated:2024-4-12)
Written for the KIAS school. The main topic is the Joyce conjecture on the functorial behaviour of the DT perverse sheaves. It also discusses recent application of the CoDT theory to the cohomological study of the moduli space of Higgs bundles on Riemann surfaces.
Edit: Some materials in the survey has already become outdated. For example, the cohomological Hall algebras for 3-CY categories were constructed in Kinjo-Park-Safronov and Conjecture 6.2.2 (cohomological integrality) was proved by Bu-Davison-Ibanez Nunez-Kinjo-Padurariu. Also, there are several important topics (e.g. the CoDT theory for 3-manifolds, relation with the Langlands program,...) not covered in this survey.
For fun
Euler classes of dual bundles (uploaded:2023-11-12)
We show that the euler class of the complex dual is given by e(E^{vee}) = (-1)^{rank E} e(E) without using the existence of bundle metric. The proof is motivated by the Fourier-Sato transform.
Growth sequence of crystals and non-commutative projective spaces (in Japanese) (uploaded:2023-11-12), Published in UTokyo Repository.
Written as a part of WINGS-FMSP program. For an (n × n) matrix A with non-negative entries, we introduce ''non-commutative projective space P^A'' as a triangulated category and showed that the growth sequence of a crystal (in the real world) should be recovered as a dimension of Hom space in this category. I proposed a conjecture that we can define ''higher growth sequence'' of a crystal using this category, such that the Euler characteristic becomes a quasi-polynomial. Hope to come back to this project in the future!
Edit: Conjecture 8 (1) was solved affirmatively by Junnosuke Koizumi and Takahiro Ueoro, and a counterexample to Conjecture 8 (2) was found by Junnosuke Koizumi.