Qianyi's Thesis
Qianyi's Thesis
We were asked to write an undergraduate thesis in ECNU 25Spring. I was in a D-module seminar led by Dima in 24Fall. I wanted to learn more about this topic, so I planned to read the first part of the reference [1] thoroughly, and wrote a detailed exposition of proof of the Riemann-Hilbert correspondence.
We mainly followed [1][2][3], here is some record of questions I asked Dima when reading(I stopped typing questions after March).
In 24 Fall semester, I attended a seminar on the same topic, directed by Dima.
12/30/24 Thesis Proposal Defense, here is the Slide.
5/17/25 Thesis Defense, here is the Slide, honored with Outstanding Undergraduate Thesis.
A 70-page exposition of the proof of the Riemann–Hilbert correspondence, defended in June 2025. Here's a small content.
History of Hilbert 21th problem
I D-modules
Notion of D-modules
Defining four Derived Functors
Kashiwara's Equivalence
Characteristic Varieties and Bernstein's inequality
Duality functor and De Rham functor
II Connections
Meromorphic connections, regularity
Degline's RH correspondence
Translating regularity to algebraic varieties
III Kashiwara's Riemann-Hilbert correspondence
Preservation of holonomicity (Sketch)
Minimal extensions
Proof of Kashiwara's RH correspondence (Sketch)
One semester is not enough for such long content, so I am still learning and understanding this.
[1] Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki. D-Modules, Perverse Sheaves, and Representation Theory. Vol. 236. Progress in Mathematics. Trans-
lated by Toshiyuki Tanisaki. Boston, MA: Springer-Verlag, 2008.
[2] Schnell, C. (2014). An introduction to D-modules. Retrieved from https://www.math.stonybrook.edu/~cschnell/pdf/notes/d-modules.pdf
[3] Oda, T. (n.d.). Holonomic D-modules. Retrieved from https://www.math.ucla.edu/~tomokioda0723/Holonomic_D.pdf