Canonical stratifications along bisheaves

A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of the sheaf to the stalks of the cosheaf. In this note we describe how, given a bisheaf constructible (i.e., locally constant) with respect to a triangulation of its underlying space, one can explicitly determine the coarsest stratification of that space for which the bisheaf remains constructible.Comment: 10 pages; this is the Final Version which appeared in the Proceedings of the 2018 Abel Symposium on Topological Data Analysi

Ergodicity of the zigzag process

The zigzag process is a Piecewise Deterministic Markov Process which can be used in a MCMC framework to sample from a given target distribution. We prove the convergence of this process to its target under very weak assumptions, and establish a central limit theorem for empirical averages under stronger assumptions on the decay of the target measure. We use the classical "Meyn-Tweedie" approach. The main difficulty turns out to be the proof that the process can indeed reach all the points in the space, even if we consider the minimal switching rates

Probing the stability of the spin liquid phases in the Kitaev-Heisenberg model using tensor network algorithms

We study the extent of the spin liquid phases in the Kitaev-Heisenberg model using infinite Projected Entangled-Pair States tensor network ansatz wave functions directly in the thermodynamic limit. To assess the accuracy of the ansatz wave functions we perform benchmarks against exact results for the Kitaev model and find very good agreement for various observables. In the case of the Kitaev-Heisenberg model we confirm the existence of 6 different phases: N\'eel, stripy, ferromagnetic, zigzag and two spin liquid phases. We find finite extents for both spin liquid phases and discontinuous phase transitions connecting them to symmetry-broken phases.Comment: 9 pages, 7 figures. Adjusted notation in equations 4-8. Added bond labeling to lower panel in figure 1. Included missing acknowledgement

An asynchronous leapfrog method II

A second order explicit one-step numerical method for the initial value problem of the general ordinary differential equation is proposed. It is obtained by natural modifications of the well-known leapfrog method, which is a second order, two-step, explicit method. According to the latter method, the input data for an integration step are two system states, which refer to different times. The usage of two states instead of a single one can be seen as the reason for the robustness of the method. Since the time step size thus is part of the step input data, it is complicated to change this size during the computation of a discrete trajectory. This is a serious drawback when one needs to implement automatic time step control. The proposed modification transforms one of the two input states into a velocity and thus gets rid of the time step dependency in the step input data. For these new step input data, the leapfrog method gives a unique prescription how to evolve them stepwise. The stability properties of this modified method are the same as for the original one: the set of absolute stability is the interval [-i,+i] on the imaginary axis. This implies exponential growth of trajectories in situations where the exact trajectory has an asymptote. By considering new evolution steps that are composed of two consecutive old evolution steps we can average over the velocities of the sub-steps and get an integrator with a much larger set of absolute stability, which is immune to the asymptote problem. The method is exemplified with the equation of motion of a one-dimensional non-linear oscillator describing the radial motion in the Kepler problem.Comment: 41 pages, 25 figure

Computing Topological Persistence for Simplicial Maps

Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under $\mathbb{Z}_2$ coefficients for a sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis. First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provides an algorithm for computing the persistence induced by simplicial maps. Our main result is that the above simple minded approach can be improved for a sequence of simplicial maps given in a monotone direction. A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses--two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally. Finally, we exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.Comment: This is the revised and full version of the paper that is going to appear in the Proceedings of 30th Annual Symposium on Computational Geometr

On 2-representation infinite algebras arising from dimer models

The Jacobian algebra arising from a consistent dimer model is a bimodule $3$-Calabi-Yau algebra, and its center is a $3$-dimensional Gorenstein toric singularity. A perfect matching of a dimer model gives the degree making the Jacobian algebra $\mathbb{Z}$-graded. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a $2$-representation infinite algebra which is a generalization of a representation infinite hereditary algebra. In this paper, we show that internal perfect matchings, which correspond to toric exceptional divisors on a crepant resolution of a $3$-dimensional Gorenstein toric singularity, characterize the property that the degree zero part of the Jacobian algebra is finite dimensional. Moreover, combining this result with the theorems due to Amiot-Iyama-Reiten, we show that the stable category of graded maximal Cohen-Macaulay modules admits a tilting object for any $3$-dimensional Gorenstein toric isolated singularity. We then show that all internal perfect matchings corresponding to the same toric exceptional divisor are transformed into each other using the mutations of perfect matchings, and this induces derived equivalences of $2$-representation infinite algebras.Comment: 28 pages, v2: improved some proof