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Solving and visualizing the potential roots of complex functions is essential in both theoretical and applied domains, yet often computationally intensive. We present a hardware-accelerated algorithm for complex function roots density graph…
We adapt Bar-Natan's scanning algorithm for fast computations in (even) Khovanov homology to odd Khovanov homology. We use a mapping cone construction instead of a tensor product, which allows us to deal efficiently with the more…
Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the corresponding Seiberg-Witten Floer homology groups…
Fast convolution algorithms, including Winograd and FFT, can efficiently accelerate convolution operations in deep models. However, these algorithms depend on high-precision arithmetic to maintain inference accuracy, which conflicts with…
A quantum algorithm for approximating efficiently 3--manifold topological invariants in the framework of SU(2) Chern-Simons-Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided. The model of…
We give a quantum algorithm for evaluating formulas over an extended gate set, including all two- and three-bit binary gates (e.g., NAND, 3-majority). The algorithm is optimal on read-once formulas for which each gate's inputs are balanced…
Atmospheric tomography, the problem of reconstructing atmospheric turbulence profiles from wavefront sensor measurements, is an integral part of many adaptive optics systems. It is used to enhance the image quality of ground-based…
For a knot K and its knot Floer complex CFK^-(K), we introduce an algorithm to compute the bordered Floer bimodule of the complement of the knot and its meridian. The grading of the module computes spin^c-summands of a meridional knot in…
We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to…
In this paper, we write down a special Heegaard diagram for a given product three manifold $\Sigma_g\times S^1$. We use the diagram to compute its perturbed Heegaard Floer homology.
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It carries information about the Floer homology…
We define a notion of Heegaard Floer homology for three dimensional orbifolds with arbitrary cyclic singularities, generalizing the recent work of Biji Wong where the singular locus is assumed to be connected.
Bordered Floer homology is an invariant for 3-manifolds with boundary, defined by the authors in 2008. It extends the Heegaard Floer homology of closed 3-manifolds, defined in earlier work of Zolt\'an Szab\'o and the second author. In…
The Heegaard Floer correction term ($d$-invariant) is an invariant of rational homology 3-spheres equipped with a Spin$^c$ structure. In particular, the correction term of 1-surgeries along knots in $S^3$ is a ($2\mathbb{Z}$-valued) knot…
Let L be a link in an integral homology three-sphere. We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated to L, which we call a complete system of hyperboxes for L. Roughly, a…
We propose a rank-$k$ variant of the classical Frank-Wolfe algorithm to solve convex optimization over a trace-norm ball. Our algorithm replaces the top singular-vector computation ($1$-SVD) in Frank-Wolfe with a top-$k$ singular-vector…
This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a…
We analyze the running time of the Hartigan-Wong method, an old algorithm for the $k$-means clustering problem. First, we construct an instance on the line on which the method can take $2^{\Omega(n)}$ steps to converge, demonstrating that…
We give a precise description of splicing formulas from a previous paper in terms of knot Floer complex associated with a knot in homology sphere.
Let $\Omega$ be a bounded domain of $\mathbb{R}^3$ whose closure $\overline{\Omega}$ is polyhedral, and let $\mathcal{T}$ be a triangulation of $\overline{\Omega}$. Assuming that the boundary of $\Omega$ is sufficiently regular, we provide…