cmengca23

Cheng Meng's Homepage

Cheng Meng

Yau Mathematical Sciences Center, Tsinghua University
Office: Shuangqing Building A, 743
Yau Mathematical Sciences Center, Tsinghua University, Beijing, China 100084
Email: cheng319 at purdue dot edu(expired), cmengca95 at outlook.com, cheng319000 at mail.tsinghua.edu.cn

About Me

I am a postdoctoral fellow at Yau Mathematical Sciences Center from Fall 2024. I received my PhD from Purdue in December 2023. My advisors are Giulio Caviglia and Linquan Ma. My research interests are in commutative algebra and its interactions with algebraic geometry, homological algebra and combinatorics.

I worked on various problems in commutative algebra involving graded rings and modules, Grobner basis, homological invariants, multiplicity and characteristic p methods. One of my recent work gives a way to compute the h-function of principal ideals, which gives the Hilbert-Kunz multiplicity and F-signature of the quotient hypersurface ring. The statement of the main theorem uses Riemann-Stieltjes integral and its proof is rooted in the representation theory of k[T] developed by Han-Monsky. Geometry and Fractal analysis also appear in the study. As an application of the main result, I prove an inequality on the Hilbert-Kunz multiplicity of Fermat quadrics. In another recent collaborative work, I also find many explicit Stillman bounds whose existence is proved by Ananyan and Hochster.

Recently I am looking into the following problems:

  • Representation of commutative Artinian algebras;
  • Computations on Grobner basis.
    • I am also interested in linkages, Rees algebras in characteristic p and other problems on F-singularities.

    Commutative Algebra Student Seminar

    In Spring 2022 I organized a commutative algebra student seminar. For more information about the seminar, please see here.

    ICTP Summer School

    In Summer 2022 I was a tutor in the ICTP summer school "Graduate Course on Tight Closure of Ideals and its Applications"

    Teaching

    Past

  • Teaching assistant for Commutative Algebra, Spring 2025.
  • Teaching assistant for Algebra I, Fall 2024.
  • Recitation instructor for MA 261, Fall 2023.
  • Recitation instructor for MA 165, Fall 2019. There is a link on this course.
    • I have been the grader for:
  • MA 546 and MA 558, Fall 2022.
  • MA 553 and MA 558, Spring 2021.
  • MA 553, MA 557 and MA 304, Fall 2020.
  • MA 553 and MA 558, Spring 2020.
  • MA 453 and MA 546, Spring 2019.
  • MA 450 and MA 553, Fall 2018.
  • MA 453, Spring 2018.
  • MA 527, Fall 2017.
  • MA 303, Spring 2017.
  • MA 304, Fall 2016.

    • Publications and preprints

    9.Finiteness and infiniteness of gradings of Noetherian rings. Arxiv

    8.Explicit Stillman bounds for all degrees (with Giulio Caviglia and Yihui Liang). Arxiv

    7.Analysis in Hilbert-Kunz theory. Arxiv

    6.Asymptotic colengths for families of ideals: an analytic approach (with Sudipta Das). Arxiv

    5.h-function, Hilbert-Kunz density function and Frobenius-Poincaré function (with Alapan Mukhopadhyay). PDF Arxiv

    4.Strongly Lech-independent ideals and Lech's conjecture PDF Arxiv

    3.Local cohomology tables of sequentially almost Cohen-Macaulay modules J. Algebra 665 (2025), 596–627. PDF Arxiv

    2.Restrictions on Hilbert coefficients give the depth of a prime ideal inside the polynomial ring PDFArxiv

    1.G-graded irreducibility and the index of reducibility, Comm. Algebra 48 (2020), no. 2, 826–832. PDF Arxiv

    Links

    My CV
    My photo

    This page was last updated on Aug 6, 2025.