About Me

I am an Assistant Professor in the Center for Applied Mathematics at Tianjin University, specializing in algebraic topology and homotopy theory. My name in Chinese is 张宇.

My research focuses on computational aspects of stable homotopy theory, including Adams spectral sequences, stable homotopy groups of spheres, and higher algebraic structures. I develop and apply computational methods to solve problems in homotopy theory.

Quick Links: CV | 中文简历 | Adams \(E_2\) Charts

This homepage was last updated on Nov 1, 2025.

Academic Positions

Assistant Professor (2023–present)
Tianjin University, Tianjin, China.

Postdoctoral Researcher (2020–2023)
Nankai University, Tianjin, China.

Education

Ph.D. in Mathematics (2014–2020)
The Ohio State University, Columbus, Ohio, USA
Dissertation Advisor: John E. Harper

B.S. in Mathematics (2010–2014)
Peking University, Beijing, China
Dissertation Advisor: Huijun Fan

Research Interests

My research program in algebraic topology focuses on:

Computational stable homotopy theory: Machine-assisted computations of Adams spectral sequences and stable homotopy groups of spheres at odd primes
Spectral sequences: Development, explicit computation, and applications of various spectral sequences in homotopy theory
Higher structures: Model categories and higher structures in stable homotopy theory
Structured ring spectra: Homotopy theory of structured ring spectra and their applications

Current Research Focus: I am currently developing computational frameworks for Adams spectral sequence calculations at odd primes and creating interactive tools to visualize and explore the structure of Adams spectral sequences.

Adams \(E_2\) Charts

This interactive visualization displays the Adams Spectral Sequence \(E_2\) page for the sphere \(S^0\) at odd primes. This is joint work in progress with Weinan Lin, and the visualization may be updated as our work progresses. Launch Adams \(E_2\) Charts Features:

Interactive zoom and pan navigation
Support for primes 3, 5, 7, and 11
Visualization of product relationships with \(a_0\) and \(h_0\)

Visualization Guide:

Bullets: Represent additive basis elements in the \(E_2\) page
Lines: Show multiplication by \(a_0\) and \(h_0\) (we display only these specific products)
Pan: Click and drag, or use arrow keys
Zoom: Mouse wheel, pinch gesture, or +/- keys
Select element: Click on any bullet to see its products highlighted in green

This visualization is based on recent machine-generated computational results from our ongoing research program.

Publications

The secondary periodic element \(\beta_{p^2/p^2-1}\) and its applications.
Joint with J. Hong and X. Wang. Sci. China Math. 68, 207-222 (2025).
SharedIt link | journal

Detecting nontrivial products in the stable homotopy ring of spheres via the third Morava stabilizer algebra.
Joint with X. Wang, J. Wu, and L. Zhong. Proc. Amer. Math. Soc. 152 (2024), 4521-4536.
pdf | journal

A correspondence between higher Adams differentials and higher algebraic Novikov differentials at odd primes.
Joint with X. Wang. Proc. Amer. Math. Soc. 151 (2023), 5087-5096.
pdf | journal

The \(p\)-primary subgroup of the cohomology of \(BPU_n\) in dimension \(2p+6\).
Joint with Z. Zhang, L. Zhong. Topology Appl. 338 (2023): 108642.
pdf | journal

Some nontrivial secondary Adams differentials on the fourth line.
Joint with X. Wang and Y. Wang. New York J. Math. 29 (2023) 687-707.
pdf | journal

Homotopy pro-nilpotent structured ring spectra and topological Quillen localization.
J. Homotopy Relat. Struct. 17, 511-523 (2022).
pdf | SharedIt link | journal

The \(p\)-primary subgroups of the cohomology of \(BPU_n\) in dimensions less than \(2p+5\).
Joint with X. Gu, Z. Zhang and L. Zhong. Proc. Amer. Math. Soc. 150 (2022), 4099-4111.
pdf | journal

Topological Quillen localization of structured ring spectra.
Joint with J. Harper. Tbilisi Mathematical Journal 12(3):69-91, 2019.
pdf | journal

Conference

2023 Topology and its Applications Conference

We are pleased to announce the upcoming 2023 Topology and its Applications Conference, scheduled from May 12 to May 16 at Nankai University in Tianjin.

Algebraic topology, an important branch of mathematics, has witnessed significant progress in recent years. Topology has also found useful applications in fields such as chemical compounds, biological proteins, and social networks for data analysis. This conference aims to bring together distinguished experts in the field of topology and applied topology, promoting the development of topological applications and providing a platform for sharing insights and research findings on the latest advancements in this area.

For further information about the conference, please visit the conference website.

Lecture

An intuitive introduction to motivic homotopy theory

In May and June of 2021, I gave an online lecture series on motivic homotopy theory. This is intended to be a leisure introduction to the most basic setup and relevant concepts in motivic homotopy theory, from a homotopy theorist's point of view.

Lecture Materials:

Lecture 1: Overview, Category theory, and Simplicial sets. Slides
Lecture 2: Model category. Slides
Lecture 3: Scheme. Slides
Lecture 4: Unstable motivic homotopy theory. Slides
Lecture 5: Stable motivic homotopy theory. Slides

Videos of the lectures are available for download here. Password: hny3

The lectures were given in Chinese.

Teaching

As instructor at TJU

Calculus I, Autumn 2024.

As instructor at NKU

Probability and Statistics for Liberal Arts, Spring 2022.
Advanced Mathematics for Liberal Arts, Autumn 2021.

As graduate teaching associate at OSU

Math 2177: Mathematical Topics for Engineers, Autumn 2018.
Math 1172: Engineering Mathematics A, Autumn 2017.
Math 1172: Engineering Mathematics A, Autumn 2016.
Math 1152: Calculus II, Spring 2016.