Hailun Zheng

activity

20152026

most citedMinimal flag triangulations of lower-dimensional manifolds

3 citations · 15 across the 29 of their papers we have counts for

collaborators

29 papers

math.CO2026

Unimodality of Independence Polynomials for Sufficiently Large Forests

Ethan X. Fang, Junwei Lu, Eran Nevo +2

We prove that the independence sequence of every sufficiently large forest is unimodal. The result follows from establishing log-concavity on a central interval of the sequence, al…

math.CO2026

On Unavoidable Faces of High-Dimensional Polytopes

Jesús A. De Loera, Ethan X. Fang, Shengtao Guo +2

Kalai's cube--simplex conjecture asserts that for all positive integers $ℓ, k$ , there is an integer $f (ℓ, k)$ such that every polytope of dimension at least $f (ℓ, k)$ has eit…

math.CO2026

Sharp Bounds on the Independence Number of Simplicial Spheres

Jesús A. De Loera, Ethan X. Fang, Junwei Lu +2

We study the maximum size of an independent set in the graph of a simplicial sphere. Let $β (d, n)$ denote this maximum over all simplicial $(d - 1)$ -spheres on $n$ vertices, and let $…

math.CO2026

Lower bounds on the $g$ -numbers of spheres without large missing faces

Isabella Novik, Hailun Zheng

We establish several new lower bounds on the $g$ -numbers of simplicial spheres without large missing faces. For this class of spheres, we derive bounds on the $g$ -numbers in terms…

math.CO2026

Simplicial spheres with $g_{k} = 1$

Isabella Novik, Hailun Zheng

For $d \geq 4$ , Kalai (1987) characterized all simplicial $(d - 1)$ -spheres with $g_{2} = 0$ , and for $k \geq 2$ and $d \geq 2 k$ , Murai and Nevo (2013) characterized all simplicial $(d - 1)$ -…

math.AG2025

The geometric Merkurjev-Panin Conjecture for the Cox category

Daniel Erman, Andrew Hanlon, Gaku Liu +1

We show that a strong version of the geometric Merkurjev-Panin conjecture holds for the Cox category of a projective toric variety. That is, we prove that the full strong exception…

math.CO2025

Missing faces of neighborly and nearly neighborly polytopes and spheres

Isabella Novik, Hailun Zheng

For a $(d - 1)$ -dimensional simplicial complex $Δ$ and $1 \leq i \leq d$ , let $f_{i - 1}$ be the number of $(i - 1)$ -faces of $Δ$ and $m_{i}$ be the number of missing $i$ -faces of $Δ$ . In th…

math.CO2024

Transversal numbers of simplicial polytopes, spheres, and pure complexes

Isabella Novik, Hailun Zheng

We prove new upper and lower bounds on transversal numbers of several classes of simplicial complexes. Specifically, we establish an upper bound on the transversal numbers of pure…

math.CO2023

Affine stresses, inverse systems, and reconstruction problems

Satoshi Murai, Isabella Novik, Hailun Zheng

A conjecture of Kalai asserts that for $d \geq 4$ , the affine type of a prime simplicial $d$ -polytope $P$ can be reconstructed from the space of affine $2$ -stresses of $P$ . We prove…

math.CO2023

The merging operation and $(d - i)$ -simplicial $i$ -simple $d$ -polytopes

Isabella Novik, Hailun Zheng

We define a certain merging operation that given two $d$ -polytopes $P$ and $Q$ such that $P$ has a simplex facet $F$ and $Q$ has a simple vertex $v$ produces a new $d$ -polytope $P\…

math.CO2022

Neighborly spheres and transversal numbers

Isabella Novik, Hailun Zheng

We survey several old and new problems related to the number of simplicial spheres, the number of neighborly simplicial spheres, the number of centrally symmetric simplicial sphere…

math.CO2022★ 2 cited

Affine stresses: the partition of unity and Kalai's reconstruction conjectures

Isabella Novik, Hailun Zheng

Kalai conjectured that if $P$ is a simplicial $d$ -polytope that has no missing faces of dimension $d - 1$ , then the graph of $P$ and the space of affine $2$ -stresses of $P$ determine…

math.CO2021

Reconstructing simplicial polytopes from their graphs and affine $2$ -stresses

Isabella Novik, Hailun Zheng

A conjecture of Kalai from 1994 posits that for an arbitrary $2 \leq k \leq ⌊ d /2 ⌋$ , the combinatorial type of a simplicial $d$ -polytope $P$ is uniquely determined by th…

math.CO2021

Many neighborly spheres

Isabella Novik, Hailun Zheng

The result of Padrol asserts that for every $d \geq 4$ , there exist $2^{Ω (n l o g n)}$ distinct combinatorial types of $⌊ d /2 ⌋$ -neighborly simplicial $(d - 1)$ -spheres with…

physics.app-ph2020

Hybrid Polymer-Garnet Materials for All-Solid-State Energy Storage Devices

Juan C. Verduzco, John N. Vergados, Alejandro Strachan +1

Hybrid electrolyte materials comprising polymer-ionic salt matrixes embedded with garnet particles constitute a promising class of materials for the realization of all-solid-state…

math.CO2020

New families of highly neighborly centrally symmetric spheres

Isabella Novik, Hailun Zheng

In 1995, Josckusch constructed an infinite family of centrally symmetric (cs, for short) triangulations of $3$ -spheres that are cs- $2$ -neighborly. Recently, Novik and Zheng extende…

math.CO2019

A new family of triangulations of $R P^{d}$

Lorenzo Venturello, Hailun Zheng

We construct a family of PL triangulations of the $d$ -dimensional real projective space $R P^{d}$ on $Θ ((\frac{1 + 5}{2})^{d + 1})$ vertices for every $d \geq 1$ . This impr…

math.CO2019★ 3 cited

Minimal flag triangulations of lower-dimensional manifolds

Christin Bibby, Andrew Odesky, Mengmeng Wang +3

We prove the following results on flag triangulations of 2- and 3-manifolds. In dimension 2, we prove that the vertex-minimal flag triangulations of $R P^{2}$ and $\mathbb{S…

math.CO2019★ 3 cited

Highly neighborly centrally symmetric spheres

Isabella Novik, Hailun Zheng

In 1995, Jockusch constructed an infinite family of centrally symmetric $3$ -dimensional simplicial spheres that are cs- $2$ -neighborly. Here we generalize his construction and show…

math.MG2019★ 1 cited

A new valuation on polyhedral cones

Hailun Zheng, Michał Zydor

We define a new family of valuations on polyhedral cones valued in the space of bounded polyhedra.

math.CO2018

Centrally symmetric and balanced triangulations of $S^{2} \times S^{d - 3}$ with few vertices

Alexander Wang, Hailun Zheng

A small triangulation of the sphere product can be found in lower dimensions by computer search and is known in few other cases: Klee and Novik constructed a centrally symmetric tr…

math.CO2018★ 1 cited

Face enumeration on flag complexes and flag spheres

Hailun Zheng

We give a survey on the recent results and problems on the face enumeration of flag complexes and flag simplicial spheres, with an emphasis on the characterization of face vectors…

math.CO2018★ 1 cited

The upper bound theorem for flag homology 5-manifolds

Hailun Zheng

We prove that among all flag homology $5$ -manifolds with $n$ vertices, the join of $3$ circles of as equal length as possible is the unique maximizer of all the face numbers. The s…

math.CO2017

A lower bound theorem for centrally symmetric simplicial polytopes

Steven Klee, Eran Nevo, Isabella Novik +1

Stanley proved that for any centrally symmetric simplicial $d$ -polytope $P$ with $d \geq 3$ , $g_{2} (P) \geq (2 d) - d$ . We provide a characterization of centrally symmetric $d$ -…

Unimodality of Independence Polynomials for Sufficiently Large Forests

On Unavoidable Faces of High-Dimensional Polytopes

Sharp Bounds on the Independence Number of Simplicial Spheres

Lower bounds on the g-numbers of spheres without large missing faces

Simplicial spheres with gk​=1

The geometric Merkurjev-Panin Conjecture for the Cox category

Missing faces of neighborly and nearly neighborly polytopes and spheres

Transversal numbers of simplicial polytopes, spheres, and pure complexes

Affine stresses, inverse systems, and reconstruction problems

The merging operation and (d−i)-simplicial i-simple d-polytopes

Neighborly spheres and transversal numbers

Affine stresses: the partition of unity and Kalai's reconstruction conjectures

Reconstructing simplicial polytopes from their graphs and affine 2-stresses

Many neighborly spheres

Hybrid Polymer-Garnet Materials for All-Solid-State Energy Storage Devices

New families of highly neighborly centrally symmetric spheres

A new family of triangulations of RPd

Minimal flag triangulations of lower-dimensional manifolds

Highly neighborly centrally symmetric spheres

A new valuation on polyhedral cones

Centrally symmetric and balanced triangulations of S2×Sd−3 with few vertices

Face enumeration on flag complexes and flag spheres

The upper bound theorem for flag homology 5-manifolds

A lower bound theorem for centrally symmetric simplicial polytopes

Lower bounds on the $g$ -numbers of spheres without large missing faces

Simplicial spheres with $g_{k} = 1$

The merging operation and $(d - i)$ -simplicial $i$ -simple $d$ -polytopes

Reconstructing simplicial polytopes from their graphs and affine $2$ -stresses

A new family of triangulations of $R P^{d}$

Centrally symmetric and balanced triangulations of $S^{2} \times S^{d - 3}$ with few vertices