Homepage of Mingchen Xia
PhD student at Chalmers tekniska högskola, Göteborg, Sweden. My advisor is Robert Berman.
Domain of research: Complex geometry, Kähler geometry, pluripotential theory, non-Archimedean geometry
Email address: email@example.com. I do not reply to any email that begins with Dear (Cher, Hejsan) Xiam
My name in Chinese characters (in case you want to write to me in Chinese): 夏铭辰(Simplified)/夏銘辰(Traditional)
I'm learning Shimura varieties. Updated on April 5, 2022
This is a collection of problems arising from my own research that may be of interest to people outside my domain.
If you know the solutions to any of the following problems, please let me know.
Flat pull-back of currents on complex analytic spaces
Compute mixed volumes of big line bundles from Okounkov bodies
By a theorem of Jow (Adv. 223.4 2010), information of all Okounkov bodies determines all numerical information of line bundles. This problem asks for explicit formulae.
Errare humanum est.
All of my preprints can be found on arXiv. See also my Google scholar page.
On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows Analysis & PDE 14-6 (2021), 1951--1976. Journal link, arXiv:1901.07889
Pluripotential-theoretic stability thresholds arXiv:2012.12039
If you do not want to get into the technical details of this paper, you may read the third item under My notes.
In the published version (=arXiv version 2), I removed results concerning non-Archimedean L-functionals, as they seem to be slightly off-topic. The readers could find these results in the arXiv version 1.
Integration by parts formula for non-pluripolar product arXiv:1907.06359
This paper was the first proof of the integration by parts formula. However, a better approach was found later on by Lu, so this paper is no longer important.
[*]Mabuchi geometry of big cohomology classes with prescribed singularities arXiv:1907.07234
[*]The closures of test configurations and algebraic singularity types (with Tamás Darvas) Advances in Mathematics, 397 (2022), 108198. Journal link,
[*]The volume of pseudoeffective line bundles and partial equilibrium (with Tamás Darvas) arXiv:2112.03827
Partial Okounkov bodies and Duistermaat--Heckman measures of non-Archimedean metrics arXiv:2112.04290
This is a slightly better version than the arXiv version. Several typos are fixed, and one additional reference is added.
There is a slight mistake in the arXiv version: In Section 4, the standing hypothesis should be that the classes are big instead of pseudo-effective.
Non-Archimedean geometry and algebraic geometry
Papers marked with [*] are those which I consider as containing new ideas. The other papers contain non-trivial results based on existing techniques.
[*]On Liu morphisms in non-Archimedean geometry arXiv:2106.08032
Analytic Bertini theorem arXiv:2110.14971
In the beamer with the same name, I sketched the proof of the analytic Bertini theorem for fibrations.
Works in progress
If I were to choose two things that I hate the most in the world, those must be teaching and following a math course. --- Wozijibiande (Famous philosopher)
I do NOT know MatLab AT ALL and refuse to learn it. I hereby apologize to all of my students and furture students at Chalmers.
2020 MVE255 Matematisk analys i flera variabler --- Teaching assistant for the MatLab part.
2020 D1/T1/DV-GU Introduktionkurs --- No students.
2020 TMV211 Inledande diskret matematik --- Teaching assistant.
2019 MVE041 Flervariablelmatematik --- Teaching assistant for the MatLab part.
2019 TMV210 Inledande diskret matematik --- Teaching assistant.
2018 TMV210 Inledande diskret matematik --- Teaching assistant.
Legal links for working mathematicians
Illegal links for working mathematicians
- Libgen If you hate Elvasier or like free knowledge, please download books from these links.
- Sci-hub Sci-hub is getting blocked in many countries recently. If the link fails to work, please try to change the domain name.