arXiv subjects

Jonathan Zung

Publications and source records attributed to Jonathan Zung.

16 recordsLinked to original sources

arXiv2026

Ziggurats, taut foliations, and contact structures

We study the geography of taut foliations on compact $3$-manifolds with toroidal boundary components. The central object of our work is the set of boundary multislopes realized by foliations transverse to a fixed flow on such a manifold. We prove that these sets exhibit remarkable structural properties (rationality, rigidity, and convexity) which motivate the name ziggurats. Our main tool, of independent interest, is a two-way correspondence between foliations and contact structures on $3$-manifolds with boundary: we generalize the Eliashberg-Thurston theorem, which produces pairs of positive and negative contact structures from foliations, and a construction of the first author, which builds foliations from such contact pairs. Using both directions of this correspondence, we bring contact-geometric methods to bear on the architecture of ziggurats.

math.GT↗

arXiv2026

Finiteness of veering triangulations

We show that a finite volume cusped hyperbolic 3-manifold admits at most finitely many veering triangulations, and in fact this number is bounded above by the number of Giroux torsion free tight contact structures. This resolves Kirby problem K3 3.21e. More generally, for any 3-manifold $M$ and any link $L$, we show finiteness for the family of pseudo-Anosov flows which admits a \emph{strictly positive Birkhoff section relative to $L$}. Combined with work of Li, this implies that a fixed closed $3$-manifold admits at most finitely many pseudo-Anosov flows without perfect fits. This resolves Kirby problem K3 3.21d.

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arXiv2026

Quasimorphisms and Pseudo-Anosov flows

We describe two connections between the theory of quasimorphisms and pseudo-Anosov flows without perfect fits on closed hyperbolic 3-manifolds. First we show that for every such flow $X$, there are quasimorphisms whose coarse restriction to each flowline of $\tilde{X}$ (the lifted flow in the universal cover) are uniform quasi-isometries to $\mathbb{R}$ -- such quasimorphisms are said to be *adapted* to $X$; and that the space of quasimorphisms $Q_X$ adapted to $X$ is an open convex cone in the space of all quasimorphisms on $\pi_1(M)$. Second, we obtain upper bounds on the exponential growth rate of closed orbits in such flows, both in the hyperbolic metric and in a word metric; quasimorphisms play a key role in obtaining the estimates in the second case.

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arXiv2025

Pseudo-Anosov flows on hyperbolic L-spaces

We prove that for each $n\in\mathbb{N}$ there is a hyperbolic L-space with $n$ pseudo-Anosov flows, no two of which are orbit equivalent. These flows have no perfect fits and are thus quasigeodesic. In addition, our flows admit positive Birkhoff sections, which we argue implies that they give rise to $n$ universally tight contact structures whose lifts to any finite cover are non-contactomorphic. This argument involves cylindrical contact homology together with the work of Barthelm\'e, Frankel, and Mann on the reconstruction of pseudo-Anosov flows from their closed orbits. These results answer more general versions of questions posed by Calegari and by Min and Nonino.

math.GT↗

arXiv2024

Veering triangulations and transverse foliations

We present a combinatorial approach to the existence of foliations and contact structures transverse to a given pseudo-Anosov flow. Let $\varphi$ be a transitive pseudo-Anosov flow on a closed oriented 3-manifold. Our main technical result is that every codimension 1 foliation transverse to $\varphi$ is carried by a single branched surface coming from a veering triangulation. Combined with recent breakthrough work of Massoni, this reduces the existence problem for transverse foliations to something like the feasibility of a system of inequalities (rather than equations!) over $Homeo_+([0,1])$. As a proof of concept, we show that for the hyperbolic, fibered, non-L-space knot $10_{145}$, the natural pseudo-Anosov flow on the slope $s$ Dehn surgery admits a transverse foliation for $s\in (-\infty, 3)$, but does not admit such a foliation for $s\in [5,\infty)$. The negative result is part of a more general Milnor--Wood type phenomenon which puts limitations on some well known methods for constructing taut foliations on Dehn surgeries.

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arXiv2024

Pseudo-Anosov representatives of stable Hamiltonian structures

A pseudo-Anosov homeomorphism of a surface is a canonical representative of its mapping class. In this paper, we explain that a transitive pseudo-Anosov flow is similarly a canonical representative of its stable Hamiltonian class. It follows that there are finitely many pseudo-Anosov flows admitting positive Birkhoff sections on any given rational homology 3-sphere. This result has a purely topological consequence: any 3-manifold can be obtained in at most finitely many ways as $p/q$ surgery on a fibered hyperbolic knot in $S^3$ for a slope $p/q$ satisfying $q\geq 6$, $p\neq 0, \pm 1, \pm 2 \mod q$. The proof of the main theorem generalizes an argument of Barthelm\'e--Bowden--Mann.

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arXiv2024

Zeta functions and the Fried conjecture for smooth pseudo-Anosov flows

To a transitive pseudo-Anosov flow $\varphi$ on a $3$-manifold $M$ and a representation $\rho$ of $\pi_1(M)$, we associate a zeta function $\zeta_{\varphi,\rho}(s)$ defined for $\Re s \gg 1$, generalizing the Anosov case. For a class of ``smooth pseudo-Anosov flows'', we prove that $\zeta_{\varphi,\rho}(s)$ has a meromorphic continuation to $\mathbb{C}$. We also prove a version of the Fried conjecture for smooth pseudo-Anosov flows which, under some conditions on $\rho$, relates $\zeta_{\varphi,\rho}(0)$ to the Reidemeister torsion of $M$. Finally we prove a topological analogue of the Dirichlet class number formula. In order to deal with singularities, we use $C^\infty$ versions of the approaches of Rugh and Sanchez--Morgado, based on Markov partitions.

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arXiv2024

Expansion and torsion homology of 3-manifolds

A Riemannian manifold is a called a good rational expander in dimension $i$ if every $i$-cycle bounds a rational $i+1$-chain of comparatively small volume. We construct 3-manifolds which are good expanders in all dimensions. On the other hand, we show that expanders must be topologically complicated: they must have lots of torsion homology. We also give some applications to topological overlap problems, constructing examples of 3-manifolds with large width over $\mathbb R^2$.

math.GT↗

arXiv2024

Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms

We study two quantifications of being a homology sphere for hyperbolic 3-manifolds, one geometric and one topological: the spectral gap for the Laplacian on coclosed 1-forms and the size of the first torsion homology group. We first construct a sequence of closed hyperbolic integer homology spheres with volume tending to infinity and a uniform coclosed 1-form spectral gap. This answers a question asked by Lin--Lipnowski. We also find sequences of hyperbolic rational homology spheres with the same properties that geometrically converge to a tame limit manifold. Moreover, we show that any such sequence must have unbounded torsion homology growth. Finally we show that a sequence of closed hyperbolic rational homology 3-spheres with uniformly bounded rank and a uniform coclosed 1-form spectral gap must have torsion homology that grows exponentially in volume.

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arXiv2021

Reeb flows transverse to foliations

Let $\mathcal F$ be a co-oriented $C^2$ foliation on a closed, oriented 3-manifold. We show that $T\mathcal F$ can be perturbed to a contact structure with Reeb flow transverse to $\mathcal F$ if and only if $\mathcal F$ does not support an invariant transverse measure. The resulting Reeb flow has no contractible orbits. This answers a question of Colin and Honda. The main technical tool in our proof is leafwise Brownian motion which we use to construct good transverse measures for $\mathcal F$; this gives a new perspective on the Eliashberg--Thurston theorem.

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arXiv2020

Taut foliations, left-orders, and pseudo-Anosov mapping tori

For a large class of 3-manifolds with taut foliations, we construct an action of $\pi_1(M)$ on $\mathbb{R}$ by orientation preserving homeomorphisms which captures the transverse geometry of the leaves. This action is complementary to Thurston's universal circle. Applications include the left-orderability of the fundamental groups of every non-trivial surgery on the figure eight knot. Our techniques also apply to at least 2598 manifolds representing 44.7% of the non-L-space rational homology spheres in the Hodgson-Weeks census of small closed hyperbolic 3-manifolds.

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arXiv2017

An Error Detection and Correction Framework for Connectomics

We define and study error detection and correction tasks that are useful for 3D reconstruction of neurons from electron microscopic imagery, and for image segmentation more generally. Both tasks take as input the raw image and a binary mask representing a candidate object. For the error detection task, the desired output is a map of split and merge errors in the object. For the error correction task, the desired output is the true object. We call this object mask pruning, because the candidate object mask is assumed to be a superset of the true object. We train multiscale 3D convolutional networks to perform both tasks. We find that the error-detecting net can achieve high accuracy. The accuracy of the error-correcting net is enhanced if its input object mask is "advice" (union of erroneous objects) from the error-detecting net.

cs.CV↗

arXiv2017

Superhuman Accuracy on the SNEMI3D Connectomics Challenge

For the past decade, convolutional networks have been used for 3D reconstruction of neurons from electron microscopic (EM) brain images. Recent years have seen great improvements in accuracy, as evidenced by submissions to the SNEMI3D benchmark challenge. Here we report the first submission to surpass the estimate of human accuracy provided by the SNEMI3D leaderboard. A variant of 3D U-Net is trained on a primary task of predicting affinities between nearest neighbor voxels, and an auxiliary task of predicting long-range affinities. The training data is augmented by simulated image defects. The nearest neighbor affinities are used to create an oversegmentation, and then supervoxels are greedily agglomerated based on mean affinity. The resulting SNEMI3D score exceeds the estimate of human accuracy by a large margin. While one should be cautious about extrapolating from the SNEMI3D benchmark to real-world accuracy of large-scale neural circuit reconstruction, our result inspires optimism that the goal of full automation may be realizable in the future.

cs.CV↗

arXiv2017

Deep Learning Improves Template Matching by Normalized Cross Correlation

Template matching by normalized cross correlation (NCC) is widely used for finding image correspondences. We improve the robustness of this algorithm by preprocessing images with "siamese" convolutional networks trained to maximize the contrast between NCC values of true and false matches. The improvement is quantified using patches of brain images from serial section electron microscopy. Relative to a parameter-tuned bandpass filter, siamese convolutional networks significantly reduce false matches. Furthermore, all false matches can be eliminated by removing a tiny fraction of all matches based on NCC values. The improved accuracy of our method could be essential for connectomics, because emerging petascale datasets may require billions of template matches to assemble 2D images of serial sections into a 3D image stack. Our method is also expected to generalize to many other computer vision applications that use NCC template matching to find image correspondences.

cs.CV↗

arXiv2017

A correlation game for unsupervised learning yields computational interpretations of Hebbian excitation, anti-Hebbian inhibition, and synapse elimination

Much has been learned about plasticity of biological synapses from empirical studies. Hebbian plasticity is driven by correlated activity of presynaptic and postsynaptic neurons. Synapses that converge onto the same neuron often behave as if they compete for a fixed resource; some survive the competition while others are eliminated. To provide computational interpretations of these aspects of synaptic plasticity, we formulate unsupervised learning as a zero-sum game between Hebbian excitation and anti-Hebbian inhibition in a neural network model. The game formalizes the intuition that Hebbian excitation tries to maximize correlations of neurons with their inputs, while anti-Hebbian inhibition tries to decorrelate neurons from each other. We further include a model of synaptic competition, which enables a neuron to eliminate all connections except those from its most strongly correlated inputs. Through empirical studies, we show that this facilitates the learning of sensory features that resemble parts of objects.

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arXiv2013

On the Longest Paths and the Diameter in Random Apollonian Networks

We consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n-3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) every path in a RAN has length o(n), refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a path of length (2n-5)^{log 2/log 3}, and that the expected length of its longest path is Omega(n^0.88). Finally, we prove that a.a.s. the diameter of a RAN is asymptotic to c log n, where c \approx 1.668 is the solution of an explicit equation.

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