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In an earlier note [arXiv:2301.00295] it was shown that there is an upper bound to the number of disjoint Hopf links (and certain related links) that can be embedded in the unit cube where there is a fixed separation required between the components within each copy of the Hopf link. The arguments relied on multi-linear properties of linking number and certain other link invariants. Here we produce a very similar upper bound for all non-trivial links by a more-general, entirely geometric, argument (but one which, unlike the original, has no analog in higher dimensions). Shortly after the initial paper, [arXiv:2308.08064] proved lower bounds which broadly cover this wider class.

Kolmogorov and Arnold, in answering Hilbert's 13th problem (in the context of continuous functions), laid the foundations for the modern theory of Neural Networks (NNs). Their proof divides the representation of a multivariate function into two steps: The first (non-linear) inter-layer map gives a universal embedding of the data manifold into a single hidden layer whose image is patterned in such a way that a subsequent dynamic can then be defined to solve for the second inter-layer map. I interpret this pattern as "minor concentration" of the almost everywhere defined Jacobians of the interlayer map. Minor concentration amounts to sparsity for higher exterior powers of the Jacobians. We present a conceptual argument for how such sparsity may set the stage for the emergence of successively higher order concepts in today's deep NNs and suggest two classes of experiments to test this hypothesis.

A closed 3-manifold $M$ may be described up to some indeterminacy by a Heegaard diagram $\mathcal{D}$. The question "Does $M$ smoothly embed in $\mathbb{R}^4$?'' is equivalent to a property of $\mathcal{D}$ which we call $\textit{doubly unlinked}$ (DU). This perspective leads to an enhancement of Hantzsche's embedding obstruction.

The digital landscape provides a dynamic platform for political discourse crucial for understanding shifts in public opinion and engagement especially under authoritarian governments This study examines YouTube user behavior during the Russian-Ukrainian war analyzing 2168 videos with over 36000 comments from January 2022 to February 2024 We observe distinct patterns of participation and gender dynamics that correlate with major political and military events Notably females were more active in antigovernment channels especially during peak conflict periods Contrary to assumptions about online engagement in authoritarian contexts our findings suggest a complex interplay where women emerge as pivotal digital communicators This highlights online platforms role in facilitating political expression under authoritarian regimes demonstrating its potential as a barometer for public sentiment.

Quantum dynamics that explore an unexpectedly small fraction of Hilbert space is inherently interesting. Integrable systems, quantum scars, MBL, hidden tensor structures, and systems with gauge symmetries are examples. Beyond dimension and volume, spectral features such as an $O(1)$-density of periodic eigenvalues, or other spectral features, can also imply observable recurrence. Low volume dynamics will recur near its initial state $| \psi_0\rangle$ more rapidly, i.e. $\lVert\mathrm{U}^k | \psi_0\rangle - | \psi_0\rangle \rVert < \epsilon$, is more likely to occur for modest values of $k$, when the (forward) orbit $\operatorname{closure}(\{\mathrm{U}^k\}_{k=1,2,\dots})$ is of relatively low dimension $d$ and relatively small $d$-volume. We describe simple quantum algorithms to detect such approximate recurrence. Applications include detection of certain cases of hidden tensor factorizations $\mathrm{U} \cong V^\dagger(\mathrm{U}_1\otimes \cdots \otimes \mathrm{U}_n)V$. "Hidden" refers to an unknown conjugation, e.g. $\mathrm{U}_1 \otimes \cdots \otimes \mathrm{U}_v \rightarrow V^\dagger(\mathrm{U}_1 \otimes \cdots \otimes \mathrm{U}_n)V$, which will obscure the low-volume nature of the dynamics. Hidden tensor structures have been observed to emerge both in a high energy context of operator-level spontaneous symmetry breaking [FSZ21a, FSZ21b, FSZ21c, SZBF23], and at the opposite end of the intellectual world in linguistics [Smo09, MLDS19]. We collect some observations on the computational difficulty of locating these structures and detecting related spectral information. A technical result, Appendix A, is that the language describing unitary circuits with no spectral gap (NUSG) around 1 is QMA-complete. Appendix B connects the Kolmogorov-Arnold representation theorem to hidden tensor structures.

We construct a compact PL 5-manifold $M$ (with boundary) which is homotopy equivalent to the wedge of eleven 2-spheres, $\vee^{}_{1 1}S^2$, which is "spineless", meaning $M$ is not the regular neighborhood of any 2-complex PL embedded in $M$. We formulate a related question about the existence of exotic smooth structures on 4-manifolds which is of interest in relation to the deformation conjecture for 2-complexes, also known as the generalized Andrews-Curtis conjecture., Comment: 7 pages. Version 2 is a substantial revision: the main theorem is strengthened, and an open question is formulated about the existence of exotic smooth structures on certain 4-manifolds. Version 3: improved exposition, updated references

The Bishop-Gromov theorem is a comparison theorem of differential geometry that upperbounds the growth of volume of a geodesic ball in a curved space. For many spaces, this bound is far from tight. We identify a major reason the bound fails to be tight: it neglects the effect of shear. By using higher curvature invariants to lowerbound the average shear, we are able to place tighter-than-Bishop-Gromov upperbounds on the late-time growth rates of geodesic balls in homogeneous spaces with non-positive sectional curvature. We also provide concrete guidance on how our theorem can be generalized to inhomogeneous spaces, to spaces with positive sectional curvatures, and to intermediate and short times. In arXiv:2209.09288 we discovered an enhancement to the BG theorem that was strongest at early times, and that relied upon additive properties of families of Jacobi equations; in this paper we find a different enhancement at late times that connects to multiplicative properties of families of Jacobi equations. A novel feature shared by both papers is the consideration of families of equations that are not coupled but whose coefficients are correlated., Comment: 33 pages

This note initiates an investigation of packing links into a region of Euclidean space to achieve a maximal density subject to geometric constraints. The upper bounds obtained apply only to the class of homotopically essential links and even there seem extravagantly large, leaving much working room for the interested reader.

The Bishop-Gromov theorem upperbounds the rate of growth of volume of geodesic balls in a space, in terms of the most negative component of the Ricci curvature. In this paper we prove a strengthening of the Bishop-Gromov bound for homogeneous spaces. Unlike the original Bishop-Gromov bound, our enhanced bound depends not only on the most negative component of the Ricci curvature, but on the full spectrum. As a further result, for finite-volume inhomogeneous spaces, we prove an upperbound on the average rate of growth of geodesics, averaged over all starting points; this bound is stronger than the one that follows from the Bishop-Gromov theorem. Our proof makes use of the Raychaudhuri equation, of the fact that geodesic flow conserves phase-space volume, and also of a tool we introduce for studying families of correlated Jacobi equations that we call "coefficient shuffling"., Comment: 41 pages, 5 figures

We illustrate some formal symmetries between Quadratic Funding (Buterin et al., 2019), a mechanism for the (approximately optimal) determination of public good funding levels, and the Born (1926) rule in Quantum Mechanics, which converts the wave representation into a probability distribution, through a bridging formulation we call "Quantum Quartic Finance". We suggest further directions for investigating the practical utility of these symmetries. We discuss potential interpretations in greater depth in a companion blog post.

In 1952 Bing astonished the mathematical world with his wild involution on $S^3$. It has been among the most seminal examples in topology. The example depends on finding shrinking homeomorphisms of Bing's decomposition of $S^3$ into points and arcs. If Bing's original homeomorphisms are varied, Bing's original wild involution changes by conjugation, which preserves some analytic properties \cite{fs22} while altering others. In 1988, Bing published a second paper "Shrinking Without Lengthening," answering a question that one of the present authors posed to him in an effort to understand the geometry of the entire conjugacy class. In this paper we produce a counterintuitive construction, namely, a method to shrink the Bing decomposition doing almost nothing at all--neither lengthening much nor rotating much.

In 1952 Bing published a wild (not topologically conjugate to smooth) involution $I$ of the 3-sphere $S^3$. But exactly how wild is it, analytically? We prove that any involution $I^h$, topologically conjugate to $I$, must have a nearly exponential modulus of continuity. Specifically, given any $\alpha>0$, there exists a sequence of $\delta$'s converging to zero, $\delta > 0$, and points $x,y \in S^3$ with dist$(x,y) < \delta$, yet dist$(I^h(x), I^h(y)) > \epsilon$, where $\delta^{-1} = e^{\left(\frac{\epsilon^{-1}}{\log^{(1+\alpha)}(\epsilon^{-1})}\right)}$, and dist is the usual Riemannian distance on $S^3$. In particular, $I^h$ stretches distance much more than a Lipschitz function ($\delta^{-1} = c\epsilon^{-1}$) or a H\"{o}lder function ($\delta^{-1} = c^\prime(\epsilon^{-1})^{p}$, $1 < p < \infty$). Bing's original construction and known alternatives (see text) for $I$ have a modulus of continuity $\delta^{-1} > c \sqrt{2}^{\epsilon^{-1}}$, so the theorem is reasonably tight -- we prove the modulus must be at least exponential up to a polylog, whereas the truth may be fully exponential. Actually, the functional for $\delta^{-1}$ coming out of the proof can be chosen slightly closer to exponential than stated here (see Theorem 1). Using the same technique we analyze a large class of ``ramified'' Bing involutions and show, as a scholium, that given any function $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$, no matter how rapid its growth, we can find a corresponding involution $J$ of the 3-sphere such that any topological conjugate $J^h$ of $J$ must have a modulus of continuity $\delta^{-1}(\epsilon^{-1})$ growing faster than $f$ (near infinity). There is a literature on inherent differentiability (references in text) but as far as the authors know the subject of inherent modulus of continuity is new.

Motivated by applications to pulling back quantum cellular automata from one manifold to another, we study the existence of immersions between certain smooth 4-manifolds. We show that they lead to a very interesting partial order on closed 4-manifolds., Comment: 43 pages, with an appendix by Alan W. Reid

The co-emergence of locality between the Hamiltonian and initial state of the universe is studied in a simple toy model. We hypothesize a fundamental loss functional for the combined Hamiltonian and quantum state and minimize it by gradient descent. This minimization yields a tensor product structure simultaneously respected by both the Hamiltonian and the state, suggesting that locality can emerge by a process analogous to spontaneous symmetry breaking. We discuss the relevance of this program to the arrow of time problem. In our toy model, we interpret the emergence of a tensor factorization as the appearance of individual degrees of freedom within a previously undifferentiated (raw) Hilbert space. Earlier work [5, 6] looked at the emergence of locality in Hamiltonians only, and found strong numerical confirmation of that raw Hilbert spaces of $\dim = n$ are unstable and prefer to settle on tensor factorization when $n=pq$ is not prime, and in [6] even primes were seen to "factor" after first shedding a small summand, e.g. $7=1+2\cdot 3$. This was found in the context of a rather general potential functional $F$ on the space of metrics $\{g_{ij}\}$ on $\mathfrak{su}(n)$, the Lie algebra of symmetries. This emergence of qunits through operator-level spontaneous symmetry breaking (SSB) may help us understand why the world seems to consist of myriad interacting degrees of freedom. But understanding why the universe has an initial Hamiltonian $H_0$ with a many-body structure is of limited conceptual value unless the initial state, $|\psi_0\rangle$, is also structured by this tensor decomposition. Here we adapt $F$ to become a functional on $\{g,|\psi_0\rangle\}=(\text{metrics})\times (\text{initial states})$, and find SSB now produces a conspiracy between $g$ and $|\psi_0\rangle$, where they simultaneously attain low entropy by settling on the same qubit decomposition.

The paper describes a funding mechanism called Quadratic Finance (QF) and deploys a bit of calculus to show that within a very clean and simple linear model QF maximizes social utility. They differentiate the social utility function. The mathematical content of this note is that by taking one further derivative, one may also deduce that QF is the unique solution.

In parts I [6] and II [7] we studied how metrics $g_{ij}$ on $\mathfrak{su}(n)$ may spontaneously break symmetry and $\textit{crystallize}$ into a form which is kaq, $\textit{knows about qubits}$. We did this for $n=2^N$ and then away from powers of 2. Here we address the Fermionic version and find kam metrics, these $\textit{know about Majoranas}$. That is, there is a basis of principal axes $\{H_k\}$ of which is of homogeneous Majorana degree. In part I, we searched unsuccessfully for functional minima representing crystallized metrics exhibiting the Brown-Susskind penalty schedule, motivated by their study of black hole scrambling time. Here, by segueing to the Fermionic setting we find, to good approximation, kam metrics adhering to this schedule on both $\mathfrak{su}(4)$ and $\mathfrak{su}(8)$. Thus, with this preliminary finding, our toy model exhibits two of the three features required for the spontaneous emergence of spatial structure: 1. localized degrees of freedom and, 2. a preference for low body-number (or low Majorana number) interactions. The final feature, 3. constraints on who may interact with whom, i.e. a neighborhood structure, must await an effective analytic technique, being entirely beyond what we can approach with classical numerics.

In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class. Here we apply this viewpoint to geometry and initiate a program of classifying homogeneous metrics on group manifolds by their long-distance properties. We show that many metrics on low-dimensional Lie groups have markedly different short-distance properties but nearly identical distance functions at long distances, and provide evidence that this phenomenon is even more robust in high dimensions. An application of these ideas of particular interest to physics and computer science is complexity geometry--the study of quantum computational complexity using Riemannian geometry. We argue for the existence of a large universality class of definitions of quantum complexity, each linearly related to the other, a much finer-grained equivalence than typically considered. We conjecture that a new effective metric emerges at larger complexities that describes a broad class of complexity geometries, insensitive to various choices of microscopic penalty factors. We discuss the implications for recent conjectures in quantum gravity., Comment: v2: journal version. new title (previously "Effective Geometry, Complexity, and Universality")

We consider a model of quantum computation using qubits where it is possible to measure whether a given pair are in a singlet (total spin $0$) or triplet (total spin $1$) state. The physical motivation is that we can do these measurements in a way that is protected against revealing other information so long as all terms in the Hamiltonian are $SU(2)$-invariant. We conjecture that this model is equivalent to BQP. Towards this goal, we show: (1) this model is capable of universal quantum computation with polylogarithmic overhead if it is supplemented by single qubit $X$ and $Z$ gates. (2) Without any additional gates, it is at least as powerful as the weak model of "permutational quantum computation" of Jordan [14, 18]. (3) With postselection, the model is equivalent to PostBQP., Comment: To be published in Quantum Journal

This article is not a proof of the Poincar\'{e} conjecture but a discussion of the proof, its context, and some of the people who played a prominent role. It is a personal, anecdotal account. There may be omission or transpositions as these recollections are 40 years old and not supported by contemporaneous notes, but memories feel surprisingly fresh. I have not looked up old papers to check details of statements; this article is merely a download from my current mental state.

We give a procedure for "reverse engineering" a closed, simply connected, Riemannian manifold with bounded local geometry from a sparse chain complex over $\mathbb{Z}$. Applying this procedure to chain complexes obtained by "lifting" recently developed quantum codes, which correspond to chain complexes over $\mathbb{Z}_2$, we construct the first examples of power law $\mathbb{Z}_2$ systolic freedom. As a result that may be of independent interest in graph theory, we give an efficient randomized algorithm to construct a weakly fundamental cycle basis for a graph, such that each edge appears only polylogarithmically times in the basis. We use this result to trivialize the fundamental group of the manifold we construct., Comment: 32 pages, 7 figures; v2 minor clarifications, improved result using improved codes as input; v3 minor clarifications and corrections

We explore the emergence of many-body physics from quantum mechanics via spontaneous symmetry breaking. To this end, we study potentials which are functionals on the space of Hamiltonians enjoying an unstable critical point corresponding to a random quantum mechanical system (the Gaussian unitary ensemble), but also less symmetrical local minima corresponding to interacting systems at the level of operators., Comment: Published in JHEP

We introduce and study the notion of filling links in 3-manifolds: a link L is filling in M if for any 1-spine G of M which is disjoint from L, $\pi_1(G)$ injects into $\pi_1(M\smallsetminus L)$. A weaker "k-filling" version concerns injectivity modulo k-th term of the lower central series. For each k>1 we construct a k-filling link in the 3-torus. The proof relies on an extension of the Stallings theorem which may be of independent interest. We discuss notions related to "filling" links in 3-manifolds, and formulate several open problems. The appendix by C. Leininger and A. Reid establishes the existence of a filling hyperbolic link in any closed orientable 3-manifold with $\pi_1(M)$ of rank 2., Comment: 22 pages, 5 figures. v4: typos corrected

State-of-the-art deep learning systems such as TensorFlow and PyTorch tightly couple the model with the underlying hardware. This coupling requires the user to modify application logic in order to run the same job across a different set of resources, thereby limiting the choice of hardware for a given workload and potentially forcing the user to forgo more efficient hardware configurations. We propose VirtualFlow, a system leveraging a novel abstraction called virtual node processing to decouple the model from the hardware. In each step of training or inference, the batch of input data is split across virtual nodes instead of hardware accelerators (e.g. GPUs and TPUs). Mapping multiple virtual nodes to each accelerator and processing them sequentially effectively time slices the batch, thereby allowing users to reduce the memory requirement of their workloads and mimic large batch sizes on small clusters. Using this technique, VirtualFlow enables many new use cases, such as reproducing training results across different hardware, resource elasticity, and heterogeneous training. In our evaluation, our implementation of VirtualFlow for TensorFlow achieved strong convergence guarantees across different hardware with out-of-the-box hyperparameters, up to 48% lower job completion times with resource elasticity, and up to 42% higher throughput with heterogeneous training., Comment: 12 pages, 29 figures

In recent years, emerging storage hardware technologies have focused on divergent goals: better performance or lower cost-per-bit. Correspondingly, data systems that employ these technologies are typically optimized either to be fast (but expensive) or cheap (but slow). We take a different approach: by architecting a storage engine to natively utilize two tiers of fast and low-cost storage technologies, we can achieve a Pareto-efficient balance between performance and cost-per-bit. This paper presents the design and implementation of PrismDB, a novel key-value store that exploits two extreme ends of the spectrum of modern NVMe storage technologies (3D XPoint and QLC NAND) simultaneously. Our key contribution is how to efficiently migrate and compact data between two different storage tiers. Inspired by the classic cost-benefit analysis of log cleaning, we develop a new algorithm for multi-tiered storage compaction that balances the benefit of reclaiming space for hot objects in fast storage with the cost of compaction I/O in slow storage. Compared to the standard use of RocksDB on flash in datacenters today, PrismDB's average throughput on tiered storage is 3.3$\times$ faster and its read tail latency is 2$\times$ better, using equivalently priced hardware.

Classical results of Milnor, Wood, Mather, and Thurston produce flat connections in surprising places. The Milnor-Wood inequality is for circle bundles over surfaces, whereas the Mather-Thurston Theorem is about cobording general manifold bundles to ones admitting a flat connection. The surprise comes from the close encounter with obstructions from Chern-Weyl theory and other smooth obstructions such as the Bott classes and the Godbillion-Vey invariant. Contradiction is avoided because the structure groups for the positive results are larger than required for the obstructions, e.g. $\operatorname{PSL}(2,\mathbb{R})$ versus $\operatorname{U}(1)$ in the former case and $C^1$ versus $C^2$ in the latter. This paper adds two types of control strengthening the positive results: In many cases we are able to (1) refine the Mather-Thurston cobordism to a semi-$s$-cobordism (ssc) and (2) provide detail about how, and to what extent, transition functions must wander from an initial, small, structure group into a larger one. The motivation is to lay mathematical foundations for a physical program. The philosophy is that living in the IR we cannot expect to know, for a given bundle, if it has curvature or is flat, because we can't resolve the fine scale topology which may be present in the base, introduced by a ssc, nor minute symmetry violating distortions of the fiber. Small scale, UV, "distortions" of the base topology and structure group allow flat connections to simulate curvature at larger scales. The goal is to find a duality under which curvature terms, such as Maxwell's $F \wedge F^\ast$ and Hilbert's $\int R\ dvol$ are replaced by an action which measures such "distortions." In this view, curvature results from renormalizing a discrete, group theoretic, structure.

Following Haken and Casson-Gordon, it was shown in [Sc] that given a reducing sphere or boundary-reducing disk E in a Heegaard split manifold M, the Heegaard surface T can be isotoped so that it intersects E in a single circle. Here we show that when this is achieved by two different positionings of T, one can be moved to the other by a sequence of 1) isotopies of T rel E 2) pushing a stabilizing pair of T through E and 3) eyegelass twists of T. The last move is inspired by one of Powell's proposed generators for the Goeritz group., Comment: This version improves exposition and corrects typos, following referee's comments

A school of thought contends that human decision making exhibits quantum-like logic. While it is not known whether the brain may indeed be driven by actual quantum mechanisms, some researchers suggest that the decision logic is phenomenologically non-classical. This paper develops and implements an empirical framework to explore this view. We emulate binary decision-making using low width, low depth, parameterized quantum circuits. Here, entanglement serves as a resource for pattern analysis in the context of a simple bit-prediction game. We evaluate a hybrid quantum-assisted machine learning strategy where quantum processing is used to detect correlations in the bitstreams while parameter updates and class inference are performed by classical post-processing of measurement results. Simulation results indicate that a family of two-qubit variational circuits is sufficient to achieve the same bit-prediction accuracy as the best traditional classical solution such as neural nets or logistic autoregression. Thus, short of establishing a provable "quantum advantage" in this simple scenario, we give evidence that the classical predictability analysis of a human-generated bitstream can be achieved by small quantum models., Comment: 10 pages, 2 figures, 12 tables

We show that all PL manifolds of dimension $\geq 3$ have spines similar to Bing's house with two rooms. Beyond this we explore approximation rigidity and an $h$-principle., Comment: 13 pages, 8 figures

We construct Bing houses in all dimensions $n \geq 3$, obtaining non-separating PL immersions of $\mathbb{S}^n \rightarrow\mathbb{R}^{n+1}$., Comment: 16 pages, 17 figures. Appendix added

We consider the group structure of quantum cellular automata (QCA) modulo circuits and show that it is abelian even without assuming the presence of ancillas, at least for most reasonable choices of control space; this is a corollary of a general method of ancilla removal. Further, we show how to define a group of QCA that is well-defined without needing to use families, by showing how to construct a coherent family containing an arbitrary finite QCA; the coherent family consists of QCA on progressively finer systems of qudits where any two members are related by a shallow quantum circuit. This construction applied to translation invariant QCA shows that all translation invariant QCA in three dimensions and all translation invariant Clifford QCA in any dimension are coherent., Comment: 25 pages, 10 figures

Trusted executions environments (TEEs) such as Intel(R) SGX provide hardware-isolated execution areas in memory, called enclaves. By running only the most trusted application components in the enclave, TEEs enable developers to minimize the TCB of their applications thereby helping to protect sensitive application data. However, porting existing applications to TEEs often requires considerable refactoring efforts, as TEEs provide a restricted interface to standard OS features. To ease development efforts, TEE application developers often choose to run their unmodified application in a library OS container that provides a full in-enclave OS interface. Yet, this large-TCB development approach now leaves sensitive in-enclave data exposed to potential bugs or vulnerabilities in third-party code imported into the application. Importantly, because the TEE libOS and the application run in the same enclave address space, even the libOS management data structures (e.g. file descriptor table) may be vulnerable to attack, where in traditional OSes these data structures may be protected via privilege isolation. We present EnclaveDom, a privilege separation system for large-TCB TEE applications that partitions an enclave into tagged memory regions, and enforces per-region access rules at the granularity of individual in-enclave functions. EnclaveDom is implemented on Intel SGX using Memory Protection Keys (MPK) for memory tagging. To evaluate the security and performance impact of EnclaveDom, we integrated EnclaveDom with the Graphene-SGX library OS. While no product or component can be absolutely secure, our prototype helps protect internal libOS management data structures against tampering by application-level code. At every libOS system call, EnclaveDom then only grants access to those internal data structures which the syscall needs to perform its task.

We discuss a possible definition for "$k$-width" of both a closed $d$-manifold $M^d$, and on embedding $M^d \overset{e}{\hookrightarrow} \mathbb{R}^n$, $n > d \ge k$, generalizing the classical notion of width of a knot. We show that for every 3-manifold 2-width$(M^3) \le 2$ but that there are embeddings $e_i: T^3 \hookrightarrow \mathbb{R}^4$ with 2-width$(e_i) \to \infty$. We explain how the divergence of 2-width of embeddings offer a tool to which might prove the Goeritz groups $G_g$ infinitely generated for $g \geq 4$. Finally we construct a homeomorphism $\theta_g: G_g \to \mathrm{MCG}(\underset{g}{\#} S^2 \times S^2)$, suggesting a potential application of 2-width to 4D mapping class groups.

A smooth embedding of a closed $3$-manifold $M$ in $\mathbb{R}^4$ may generically be composed with projection to the fourth coordinate to determine a Morse function on $M$ and hence a Heegaard splitting $M=X\cup_\Sigma Y$. However, starting with a Heegaard splitting, we find an obstruction coming from the geometry of the curve complex $C(\Sigma)$ to realizing a corresponding embedding $M\hookrightarrow \mathbb{R}^4$., Comment: 6 pages

In this note we discuss estimates for the curvature of 4-dimensional gradient Ricci soliton singularity models by applying Perelman's point selection, a fundamental result of Cheeger and Naber, and topological lemmas., Comment: Revised version, as appeared in Advances in Mathematics 372 (2020), article number 107303. We would like to thank the referee for their suggestions

Third-party code plays a critical role in IoT applications, which generate and analyze highly privacy-sensitive data. Unlike traditional desktop and server settings, IoT devices mostly run a dedicated, single application. As a result, vulnerabilities in third-party libraries within a process pose a much bigger threat than on traditional platforms. We present Pyronia, a fine-grained access control system for IoT applications written in high-level languages. Pyronia exploits developers' coarse-grained expectations about how imported third-party code operates to restrict access to files, devices, and specific network destinations, at the granularity of individual functions. To efficiently protect such sensitive OS resources, Pyronia combines three techniques: system call interposition, stack inspection, and memory domains. This design avoids the need for application refactoring, or unintuitive data flow analysis, while enforcing the developer's access policy at run time. Our Pyronia prototype for Python runs on a custom Linux kernel, and incurs moderate performance overhead on unmodified Python applications., Comment: 19 pages, 3 figures

We extend the classical definition of {\it width} to higher dimensional, smooth codimension 2 knots and show in each dimension there are knots of arbitrarily large width.

We study the effect of Nielsen moves and their geometric counterparts, handle slides, on good boundary links. A collection of links, universal for 4-dimensional surgery, is shown to admit Seifert surfaces with trivial Lagrangian. They are good boundary links, with Seifert matrices of a more general form than in known constructions of slice links. We show that a certain more restrictive condition on Seifert matrices is sufficient for proving the links are slice. We also give a correction of a Kirby calculus identity in \cite{FK2}, useful for constructing surgery kernels associated to link-slice problems.

HTTP-based video streaming is a key application on the Internet today, comprising the majority of Internet traffic today. Yet customers remain dissatisfied with video quality, resulting in lost revenue for content providers. Recent studies have blamed this on the adaptive bitrate selection (ABR) algorithm used by client players, claiming it interacts poorly with TCP when the video buffer is full, which causes it to underestimate available network bandwidth. We show that the root cause of the problem lies in the data plane, and that even a perfect control plane (ABR) algorithm is not enough to guarantee video flows their fair share of network bandwidth. Namely, it is the sequential download of (small) video segments that is at fault, as they disrupt the normal interaction between TCP congestion control and router queue occupancy. We analytically derive the throughput of a video flow as a function of download size and network conditions, and use this to develop an adaptive algorithm for selecting the download size. Combined with pipelining, our approach achieves near-optimal throughput and fast bitrate adaptation, regardless of the control plane algorithm. We implement our approach as a DASH video player called Sprint, and evaluate it against state-of-the-art proposals from the literature as well as deployed players from Netflix, Youtube, Hulu, and Amazon. Sprint consistently achieves above 90% of its fair-share throughput, while the previous state-of-the-art exhibits high variability (e.g., from 31% to close to fair share depending on the network conditions). Industry players often achieve below 50% of their fair share.

$\pi/8$ phase gates (magic gates or T-gates) are crucial to augment topological systems based on Majorana zero modes to full quantum universality. We present a scheme based on a combination of projective measurements and non-adiabatic evolution that effectively cancels smooth control errors when implementing phase gates in Majorana-based systems. Previous schemes based on adiabatic evolution are susceptible to problems arising from small but finite dynamical phases that are generically present in topologically unprotected gates. A measurement-only approach eliminates dynamical phases. For non-protected gates, however, forced-measurement schemes are no longer effective which leads to low success probabilities of obtaining the right succession of measurement outcomes in a measurement-only implementation. We show how to obtain a viable measurement-based scheme which dramatically increases the success probabilities by evolving the system non-adiabatically with respect to the degenerate subspace in between measurements. We outline practical applications of our scheme in recently proposed quantum computing designs based on Majorana tetrons and hexons., Comment: 13 pages, 8 figures. v2: minor updates, added figure of Y-junction

There are two schools of "measurement-only quantum computation". The first ([11]) using prepared entanglement (cluster states) and the second ([4]) using collections of anyons, which according to how they were produced, also have an entanglement pattern. We abstract the common principle behind both approaches and find the notion of a graph or even continuous family of equiangular projections. This notion is the leading character in the paper. The largest continuous family, in a sense made precise in Corollary 4.2, is associated with the octonions and this example leads to a universal computational scheme. Adiabatic quantum computation also fits into this rubric as a limiting case: nearby projections are nearly equiangular, so as a gapped ground state space is slowly varied the corrections to unitarity are small., Comment: Added some new results in section 4

We prove that compact topological 4-manifolds can be effectively presented by a finite amount of data., Comment: 6 pages

Due to its high performance and decreasing cost per bit, flash is becoming the main storage medium in datacenters for hot data. However, flash endurance is a perpetual problem, and due to technology trends, subsequent generations of flash devices exhibit progressively shorter lifetimes before they experience uncorrectable bit errors. In this paper we propose extending flash lifetime by allowing devices to expose higher bit error rates. To do so, we present DIRECT, a novel set of policies that leverages latent redundancy in distributed storage systems to recover from bit corruption errors with minimal performance and recovery overhead. In doing so, DIRECT can significantly extend the lifetime of flash devices by effectively utilizing these devices even after they begin exposing bit errors. We implemented DIRECT on two real-world storage systems: ZippyDB, a distributed key-value store backed by RocksDB, and HDFS, a distributed file system. When tested on production traces at Facebook, DIRECT reduces application-visible error rates in ZippyDB by more than 10^2 and recovery time by more than 10^4. DIRECT also allows HDFS to tolerate a 10^4--10^5 higher bit error rate without experiencing application-visible errors.

In 1980 J. Powell proposed that five specific elements sufficed to generate the Goeritz group of any Heegaard splitting of $S^3$, extending work of Goeritz on genus $2$ splittings. Here we prove that Powell's conjecture was correct for splittings of genus $3$ as well, and discuss a framework for deciding the truth of the conjecture for higher genus splittings., Comment: 39 pages; 17 figures

Training machine learning (ML) models with large datasets can incur significant resource contention on shared clusters. This training typically involves many iterations that continually improve the quality of the model. Yet in exploratory settings, better models can be obtained faster by directing resources to jobs with the most potential for improvement. We describe SLAQ, a cluster scheduling system for approximate ML training jobs that aims to maximize the overall job quality. When allocating cluster resources, SLAQ explores the quality-runtime trade-offs across multiple jobs to maximize system-wide quality improvement. To do so, SLAQ leverages the iterative nature of ML training algorithms, by collecting quality and resource usage information from concurrent jobs, and then generating highly-tailored quality-improvement predictions for future iterations. Experiments show that SLAQ achieves an average quality improvement of up to 73% and an average delay reduction of up to 44% on a large set of ML training jobs, compared to resource fairness schedulers., Comment: Appeared in the 1st SysML Conference. Full paper published in ACM SoCC 2017

We introduce a collection of 1/2-$\pi_1$-null 4-dimensional surgery problems. This is an intermediate notion between the classically studied universal surgery models and the $\pi_1$-null kernels which are known to admit a solution in the topological category. Using geometric applications of the group-theoretic 2-Engel relation, we show that the 1/2-$\pi_1$-null surgery problems are universal, in the sense that solving them is equivalent to establishing 4-dimensional topological surgery for all fundamental groups. As another application of these methods, we formulate a weaker version of the $\pi_1$-null disk lemma and show that it is sufficient for proofs of topological surgery and s-cobordism theorems for good groups.

Suppose $\delta$ is a generic immersed closed curve in the boundary of a 3-manifold M and $\delta$ is null-homotopic in M. Then $\delta$ can be displaced by a height function in a collar of the boundary so that the resulting simple closed curve in the collar bounds a disk in M., Comment: Some references and additional comments added