Hailun Zheng

I am an assistant professor at the University of Hawai'i at Manoa. My research lies in geometric combinatorics & polytope theory. My non-mathematical hobbies include history, photography, and occasionally writing martial art novels. [CV]

Email: hailunz@hawaii.edu

Publications

1. ​Minimal balanced triangulations of sphere bundles over the circle. SIAM J. Discrete Math. 30(2016), 1259-1268.[pdf]

2. The flag upper bound theorem for 3- and 5-manifolds. Israel J. Math., 222(2017), 401-419. [pdf] (The extended abstract appears in DMCTS Proceedings of FPSAC 16.

3. A characterization of homology manifolds with $g_2\leq 2$. J. Combin. Theory Ser. A, 153(2018), 31-45. [pdf]

4. A lower bound theorem for centrally symmetric simplicial polytopes (with Steven Klee, Eran Nevo and Isabella Novik), Discrete Comput. Geom., 61(2019), 541-561. [pdf]

5. Face enumeration on flag complexes and flag spheres. Algebraic and geometric combinatorics on lattice polytopes (T. Hibi and A. Tsuchiya, eds.), World Scientific Publishing Co. Pte. Ltd, Singapore, 2019, 435-449. [pdf]

6. A new valuation on polyhedral cones (with Michal Zydor), Algebra i Analiz, 32(2020), 1–11. [pdf]

7. Ear decomposition and balanced neighborly simplicial manifolds. Electron. J. Combin., 27(2020), P1.10, 2020. [pdf]

8. Centrally symmetric and balanced triangulations of $S^2\times S^{d-3}$ with few vertices (with Alexander Wang). European J. Combin., 85(2020), 103043. The extended abstract with title ``Triangulations of the product of spheres with few vertices" appears in DMCTS Proceedings of FPSAC 19. [pdf]

9. The upper bound theorem for flag homology 5-manifolds. J. Combin. Theory Ser. A, 175(2020), 105259. [pdf]

10. Highly neighborly centrally symmetric spheres (with Isabella Novik). Adv. Math., 370(2020), 107238. [pdf]

11. The rigidity of the graphs of homology spheres minus one edge. Discrete Mathematics, 343(2020), 112135. [pdf]

12. A new family of triangulations of RP^d (with Lorenzo Venturello), Combinatorica, 41(2020), 127-146. [pdf]

13. Minimal flag triangulations of lower-dimensional manifolds (with Christin Bibby, Andrew Odesky, Mengmeng Wang, Shuyang Wang and Ziyi Zhang), Involve, 13(2020), 683-703. [pdf]

14. The stresses on centrally symmetric complexes and the lower bound theorems (with Isabella Novik),  Algebraic Combinatorics, 4(2021), 541-549. [pdf]

15.  Many neighborly spheres (with Isabella Novik). Mathematische Annalen, 388(2024), 969-984. [pdf]

16.  New families of highly neighborly centrally symmetric spheres (with Isabella Novik). Trans. Amer. Math. Soc., 375(2022), 4445-4475. [pdf]

17.  Reconstructing simplicial polytopes from their graphs and affine 2-stresses (with Isabella Novik). Israel Journal of Mathematics, 255(2023), 891-910. [pdf]

18.  Neighborly spheres and transversal numbers (with Isabella Novik). For the AMS volume on Open Problems in Algebraic Combinatorics, to accompany the 2022 conference held at UMN. [pdf]

19.   Affine stresses: the partition of unity and Kalai’s reconstruction conjectures (with Isabella Novik). To appear in Discrete Comput. Geom. [pdf]

 20.  The merging operation and (d-i)-simplicial i-simple d-polytopes (with Isabella Novik). To appear in Combinatorial Theory. [pdf]

 21.  Affine stresses, inverse systems and reconstruction problems (with Satoshi Murai and Isabella Novik). Int. Math. Res. Not., 2024, no. 10, 8540-8556. [pdf]

22. Missing faces of neighborly and nearly neighborly polytopes and spheres (with Isabella Novik). To appear in Israel Journal of Mathematics. [pdf]

23. Transversal numbers of simplicial polytopes, spheres, and pure complexes (with Isabella Novik). To appear in SIAM Journal on Discrete Mathematics. [pdf]

Preprints

1. The geometric Merkurjev-Panin Conjecture for the Cox category (with Daniel Erman, Andrew Hanlon, and Gaku Liu). Submitted.

2. Simplicial spheres with g_k=1 (with Isabella Novik). Submittied. 

3. Lower bounds on the g-numbers of spheres without large missing faces (with Isabella Novik). Submitted.

Grants

NSF Grant DMS-2535689, Geometric combinatorics on polytopes and spheres,  2025-2026. 

 Invited talks

Combinatorics at the Confluence, an ICM satellite conference, July 20-22, 2026, Pittsburgh, USA.

Rigid structures in algebraic combinatorics and algebraic statistics, May 18-22, 2026, ICMS, Bayes Centre, Edinburgh, UK.

Matroids, rigidity, and algebraic statistics, May 17-21, 2025, ICERM, USA. 

Joint Mathematics Meetings, AMS special session on geometric and topological combinatorics, January 8-11, 2025, Seattle, USA.

AMS fall western sectional meeting, special session on topological and geometric methods in combinatorics, October 26-27, UC Riverside, USA.

Suzhou workshop on matroid theory, July 8-12, 2024, Soochow University, Suzhou, China. 

Geometric, algebraic, and topological combinatorics, December 10-15, 2023, Oberwolfach meeting, Germany.

Combinatorics and geometry of convex polyhedra, March 20-24, 2023, Simons Center, USA.

Teaching

Instructor 

   University of Hawai`i at Manoa

      Math 241    Calculus I Fall 2026      

      Math 611 Algebra (graduate level)  Fall 2026 and Fall 2025

      Math 480 Senior Seminar Spring 2026   (Click here to see more details!)

      Math 242    Calculus II Fall 2025

      Math 649T Topic course (Geometric Combinatorics)       Winter 2025

      Math 475    Combinatorial Theory Winter 2025

   University of Houston-Downtown   

      Math 1301 College Algebra   Spring 2023 and Autumn 2022

      Math 3309 Number Theory   Autumn 2022

      Math 2307 Linear Algebra   Autumn 2023 and Spring 2024

      Math 2309 Discrete Mathematics   Autumn 2023

      Math 4303 Intro to topology and functional analysis Spring 2024 

​   University of Michigan

      Math 214    Applied Linear Algebra.     Winter and Autumn 2019

      Math 115    Calculus I.                                   Autumn 2018

      Math 590    Topology.                                     Winter 2018

      Math 567    Coding theory.                         Winter  2020

​   University of Washington

      Math 324    Multi-variable Calculus.    Autumn 2016, Spring 2015

      Math 309    Linear Analysis.                      Autumn 2015

      Math 307    Differential Equations       Spring 2017, Winter 2015

      Math 124    Calculus I.                                  Summer 2014

​Teaching Assistant

   University of Copenhagen (graduate level)

      Commutative Algebra                                  Block 3, 2020-21

      Invitations to Combinatorics   Block 2, 2021-22

​   Unversity of Washington

      Math 126    Calculus III.                               Spring 2013, Autumn 2014

      Math 125    Calculus II.                                 Winter 2013

      Math 124    Calculus I.                                   Autumn 2012

Mentoring

Undergraduate summer intern at University of Hawai`i at Manoa, Summer 2026

   Project: Cubical manifolds

   Student: Sabrina Victor (Amherst College)

Senior project, University of Hawai`i at Manoa, Spring 2026

   Project: Vertex-minimal cubical surfaces [Poster]

   Student: Rowan Miller

Honors project, University of Hawai`i at Manoa, 2025-2026

   Project: DispersalNim: A Combinatorial Game on Graphs [Poster]

   Student: Adam Inamasu

Senior project, University of Houston-Downtown, 2023

    Project: Persistent topology and application in data science

    Student: Ross Mccoy

​Lab of Geometry at the University of Michigan, 2019

    Project: Lower dimensional flag triangulated manifolds [description]

    Co-mentoring with Christin Bibby [website] and Andrew Odesky [website].

Summer REU at the University of Michigan, 2018

    Project: The triangulation of the product of spheres

    Students: Alexander Wang [website], Jonathon Jacobs

Prospective students

Undergraduates: I enjoy working with undergraduate students on research projects in geometric and algebraic combinatorics. Some potential topics include the rigidity theory of polytopes and face enumeration problems for polyhedral complexes. Linear algebra is the main prerequisite; some background in enumerative combinatorics and/or topology is helpful but not required. 

Master's and PhD students: I'm happy to discuss potential master's or PhD projects. A good research direction depends on a student's background and interests so I encourage you to email me and set up a time to talk. A few things that will aid discussion: review my recent papers, review a survey paper, come with personal ideas or questions about what draws you to it, and your recent coursework. Two books I'd recommend to anyone curious about this area: Lectures on Polytopes by Günter Ziegler, and Combinatorics and Commutative Algebra by Richard Stanley (the "green book"). I also recommend attending our algebra seminar organized by Prof. Erman and Prof. Gross.