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Bojun Zhao

Publications and source records attributed to Bojun Zhao.

9 recordsLinked to original sources
arXiv2026

Left-orderability in Dehn fillings of pseudo-Anosov mapping tori

For pseudo-Anosov mapping tori with co-orientable invariant foliations and monodromies reversing their co-orientations, a family of taut foliations was constructed in previous work on Dehn fillings with all rational slopes outside a neighborhood of the degeneracy slope. In this paper, we prove that all such Dehn fillings have left-orderable fundamental groups. We present two approaches, both establishing left-orderability through the branching behavior of taut foliations. The first approach produces an $\mathbb{R}$-covered foliation arising from this family for each filling slope, and the second approach shows that, depending on the choice of a suitable system of arcs on $\Sigma$, the resulting foliation either has one-sided branching or is $\mathbb{R}$-covered. Consequently, the second approach associates to each Dehn filling a family of representations of its fundamental group into $\mathcal{G}_\infty$, the group of germs at infinity, whereas the first approach yields an explicit left-invariant order. As an application, combining our results with earlier work in the literature, we verify the L-space conjecture for all surgeries on the $(-2,3,2q+1)$-pretzel knot ($q \geqslant 3$) in $S^3$. From another perspective, $\mathbb{R}$-covered foliations can be produced systematically across a large family of Dehn fillings on cusped hyperbolic manifolds, and in some cusped manifolds they cover all fillings that admit co-orientable taut foliations. This expands the class of known $\mathbb{R}$-covered foliations.

math.GT↗
arXiv2026

Coarse-Grained Boltzmann Generators

Sampling equilibrium molecular configurations from the Boltzmann distribution is a longstanding challenge. Boltzmann Generators (BGs) address this by combining exact-likelihood generative models with importance sampling, but practical scalability is limited. Meanwhile, coarse-grained surrogates enable the modeling of larger systems by reducing effective dimensionality, yet often lack a reweighting procedure required to ensure asymptotically correct statistics. In this work, we propose Coarse-Grained Boltzmann Generators (CG-BGs), a framework for reduced-order generative modeling with importance sampling in coarse-grained coordinate space. CG-BGs generate samples using a flow-based model and reweight them using a learned potential of mean force (PMF). We show that the PMF can be learned from rapidly converged trajectories via enhanced sampling force matching. Experiments demonstrate that CG-BGs capture solvent-mediated interactions in highly reduced representations while substantially reducing computational cost relative to atomistic BGs, providing a practical route toward equilibrium sampling of larger molecular systems.

cs.LG↗
arXiv2024

Reconstruction of Anosov flows from infinity

Every pseudo-Anosov flow $\phi$ in a closed $3$-manifold $M$ gives rise to an action of $\pi_1(M)$ on a circle $S^{1}_{\infty}(\phi)$ from infinity \cite{Fen12}, with a pair of invariant \emph{almost} laminations. From certain actions on $S^{1}$ with invariant almost laminations, we reconstruct flows and manifolds realizing these actions, including all orientable transitive pseudo-Anosov flows in closed $3$-manifolds. Our construction provides a geometry model for such flows and manifolds induced from $\mathcal{D} \times \mathcal{D}$, where $\mathcal{D}$ is the Poincar\'e disk with $\partial \mathcal{D}$ identified with $S^{1}_{\infty}(\phi)$. In addition, our result applies to Cannon conjecture under the assumption that certain group-equivariant sphere-filling Peano curve exists, which offers a description of orientable quasigeodesic pseudo-Anosov flows in hyperbolic $3$-manifolds in terms of group actions on $\partial \mathbb{H}^{3} \times \partial \mathbb{H}^{3} \times \partial \mathbb{H}^{3}$.

math.GT↗
arXiv2023

Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy

Let $\Sigma$ be a compact orientable surface with nonempty boundary, let $\varphi: \Sigma \to \Sigma$ be an orientation-preserving pseudo-Anosov homeomorphism, and let $M = \Sigma \times I / \stackrel{\varphi}{\sim}$ be the mapping torus of $\Sigma$ over $\varphi$. Let $\mathcal{F}^{s}$ denote the stable foliation of $\varphi$ in $\Sigma$. Let $T_1, \ldots, T_k$ denote the boundary components of $M$. With respect to a canonical choice of meridian and longitude on each $T_i$, the degeneracy locus of the suspension flow of $\varphi$ on $T_i$ can be identified with a pair of integers $(p_i; q_i)$ such that $p_i > 0$ and $-\frac{1}{2}p_i < q_i \leqslant \frac{1}{2}p_i$. Let $c_i$ denote the number of components of $T_i \cap (\Sigma \times \{0\})$. Assume that $\mathcal{F}^{s}$ is co-orientable and $\varphi$ reverses the co-orientation on $\mathcal{F}^{s}$. We show that the Dehn filling of $M$ along $\partial M$ with any multislope in $J_1 \times \ldots \times J_k$ admits a co-orientable taut foliation, where $J_i$ is one of the two open intervals in $\mathbb{R} \cup \{\infty\} \cong \mathbb{R}P^{1}$ between $\frac{p_i}{q_i + c_i}, \frac{p_i}{q_i - c_i}$ which doesn't contain $\frac{p_i}{q_i}$. For some hyperbolic fibered knot manifolds, the slopes given above contain all slopes that yield non-L-space Dehn filllings. The examples include (1) the exterior of the $(-2,3,2q+1)$-pretzel knot in $S^{3}$ for each $q \in \mathbb{Z}_{\geqslant 3}$ (see \hyperref[Kri]{[Kri]} for a previous proof), (2) the exteriors of many L-space knots in lens spaces.

math.GT↗
arXiv2022

Left orderability and taut foliations with orderable cataclysm

Let $M$ be a connected, closed, orientable, irreducible $3$-manifold. We show that: if $M$ admits a co-orientable taut foliation $\mathcal{F}$ with orderable cataclysm, then $\pi_1(M)$ is left orderable. This provides an elementary proof that $\pi_1(M)$ is left orderable if $M$ admits an Anosov flow with a co-orientable stable foliation without using Thurston's universal circle action. Furthermore, for every closed orientable 3-manifold that admits a pseudo-Anosov flow $X$ with a co-orientable stable foliation, our result applies to infinitely many of Dehn fillings along the union of singular orbits of $X$.

math.GT↗
arXiv2022

Left orderability, foliations, and transverse $(π_1,\mathbb{R})$ structures for $3$-manifolds with sphere boundary

Let $M$ be a closed orientable irreducible $3$-manifold such that $π_1(M)$ is left orderable. (a) Let $M_0 = M - Int(B^{3})$, where $B^{3}$ is a compact $3$-ball in $M$. We have a process to produce a co-orientable Reebless foliation $\mathcal{F}$ in $M_0$ such that: (1) $\mathcal{F}$ has a transverse $(π_1(M),\mathbb{R})$ structure, (2) there exists a simple closed curve in $M$ that is co-orientably transverse to $\mathcal{F}$ and intersects every leaf of $\mathcal{F}$. More specifically, given a pair $(<,Γ)$ composed of a left-invariant order "$<$" of $π_1(M)$ and a fundamental domain $Γ$ of $M$ in its universal cover with certain property (which always exists), we can produce a resulting foliation in $M - Int(B^{3})$ as above, and we can test if it can extend to a taut foliation of $M$. (b) Suppose further that $M$ is either atoroidal or a rational homology $3$-sphere. If $M$ admits an $\mathbb{R}$-covered foliation $\mathcal{F}_0$, then there is a resulting foliation $\mathcal{F}$ of our process in $M - Int(B^{3})$ such that: $\mathcal{F}$ can extend to an $\mathbb{R}$-covered foliation $\mathcal{F}_{extend}$ of $M$, and $\mathcal{F}_0$ can be recovered from doing a collapsing operation on $\mathcal{F}_{extend}$. Here, by a collapsing operation on $\mathcal{F}_{extend}$, we mean the following process: (1) choosing an embedded product space $S \times I$ in $M$ for some (possibly non-compact) surface $S$ such that $S \times \{0\}, S \times \{1\}$ are leaves of $\mathcal{F}_{extend}$ (notice that $\mathcal{F}_{extend} \mid_{S \times I}$ may not be a product bundle), (2) replacing $\mathcal{F}_{extend} \mid_{S \times I}$ by a single leaf $S$. (c) We conjecture that there always exists a resulting foliation of our process in $M - Int(B^{3})$ which can extend to a taut foliation in $M$.

math.GT↗
arXiv2019

The extending surfaces of immersions into surfaces

S. Blank solved the question of classifying immersed circles in $\mathbb{R}^{2}$ that extend to immersed disks, and how many topologically inequivalent disks can be extended. The quetions of various cases in $2$-dimension have already been solved by generalizing his method. In this paper, we give a new way, which is straightforward for the questions, and we determine all topological equivalence classes of immersed surfaces bounded by an arbitrary immersed circle in a closed oriented surface.

math.GT↗