Related papers: Order-detection, representation-detection, and app…
In joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves in a punctured torus. In particular, knot Floer homology is captured by this invariant.…
We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Sch{\"o}nflies theorem.…
This paper presents methods to compare high order networks, defined as weighted complete hypergraphs collecting relationship functions between elements of tuples. They can be considered as generalizations of conventional networks where only…
A large class of topological orders can be understood and classified using the string-net condensation picture. These topological orders can be characterized by a set of data (N, d_i, F^{ijk}_{lmn}, \delta_{ijk}). We describe a way to…
While robotic manipulation of rigid objects is quite straightforward, coping with deformable objects is an open issue. More specifically, tasks like tying a knot, wiring a connector or even surgical suturing deal with the domain of…
The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of…
We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the…
Let $M_1$ and $M_2$ be closed connected orientable $3$-manifolds. We classify the sets of smooth and piecewise linear isotopy classes of embeddings $M_1\sqcup M_2\rightarrow S^6$.
We prove homological stability for sequences of "oriented configuration spaces" as the number of points in the configuration goes to infinity. These are spaces of configurations of n points in a connected manifold M of dimension at least 2…
This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can…
We study when geometric simplicity of decision boundaries, used here as a notion of interpretability, can conflict with accurate approximation of axis-aligned decision trees by shallow neural networks. Decision trees induce rule-based,…
We suggest a new algorithm to estimate representations of compact Lie groups from finite samples of their orbits. Different from other reported techniques, our method allows the retrieval of the precise representation type as a direct sum…
A method for detecting and approximating fault lines or surfaces, respectively, or decision curves in two and three dimensions with guaranteed accuracy is presented. Reformulated as a classification problem, our method starts from a set of…
Superfluid phases of $^3$He discovered in 1972 opened the new area of the application of topological methods to condensed matter systems. Due to the multi-component order parameter which characterizes the broken $SO(3)\times SO(3)\times…
We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology, and can…
The deep connection among braids, knots and topological physics has provided valuable insights into studying topological states in various physical systems. However, identifying distinct braid groups and knot topology embedded in…
We introduce an approach based on moving frames for polygon recognition and symmetry detection. We present detailed algorithms for recognition of polygons modulo the special Euclidean, Euclidean, equi-affine, skewed-affine and similarity…
The vast majority of symmetry-protected topological (SPT) states are difficult to detect, which often leads to their misidentification as ordinary or topologically trivial phases. In this work, we propose a general framework for detecting…
Capsule networks aim to parse images into a hierarchy of objects, parts and relations. While promising, they remain limited by an inability to learn effective low level part descriptions. To address this issue we propose a way to learn…
In the study of high-dimensional data, it is often assumed that the data set possesses an underlying lower-dimensional structure. A practical model for this structure is an embedded compact manifold with boundary. Since the underlying…