Related papers: Predicting Euler Characteristics and Constructing …
Accurate delineation of fine-scale structures is a very important yet challenging problem. Existing methods use topological information as an additional training loss, but are ultimately making pixel-wise predictions. In this paper, we…
Magnetic skyrmions are mesmerizing spin textures with peculiar topological and dynamical properties, typically the product of competing interactions in ferromagnets, and with great technological potential [1-5]. Researchers have long…
Seeking effective neural networks is a critical and practical field in deep learning. Besides designing the depth, type of convolution, normalization, and nonlinearities, the topological connectivity of neural networks is also important.…
We use some basic properties of binomial and Stirling numbers to prove that the Euler characteristic is, essentially, the unique numerical topological invariant for compact polyhedra which can be expressed as a linear combination of the…
Complex spin textures in itinerant electron magnets hold promises for next-generation memory and information technology. The long-ranged and often frustrated electron-mediated spin interactions in these materials give rise to intriguing…
We construct deformation invariants of $2|1$-dimensional Euclidean field theories valued in a cohomology theory approximating topological modular forms. This implies several results anticipated by Stolz and Teichner and gives the first…
Most noninvasive imaging techniques utilize electromagnetic or acoustic waves originating from multiple locations and directions to identify hidden geometrical structures. Surprisingly, it is also possible to image hidden voids and…
Tensegrity structures possess intrinsic geometric symmetries that govern their dynamic behavior. However, most existing physics-informed neural network (PINN) approaches for tensegrity dynamics do not explicitly exploit these symmetries,…
Topological integral transforms have found many applications in shape analysis, from prediction of clinical outcomes in brain cancer to analysis of barley seeds. Using Euler characteristic as a measure, these objects record rich geometric…
Constitutive modeling lies at the core of mechanics, allowing us to map strains onto stresses for a material in a given mechanical setting. Historically, researchers relied on phenomenological modeling where simple mathematical…
Images of line drawings are generally composed of primitive elements. One of the most fundamental elements to characterize images is the topology; line segments belong to a category different from closed circles, and closed circles with…
Extracting informative features from images has been of capital importance in computer vision. In this paper, we propose a way to extract such features from images by a method based on algebraic topology. To that end, we construct a…
Skyrmions are topological quasi-particles characterised by local spin textures, which are considered to be robust against structural deformation. N\'eel and Bloch states are famous examples of skyrmions, which exhibit radical and chiral…
In this paper we study an emerging class of neural networks based on the morphological operators of dilation and erosion. We explore these networks mathematically from a tropical geometry perspective as well as mathematical morphology. Our…
We compute the Euler characteristics of the recently discovered series of Gothic Teichm\"{u}ller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichm\"{u}ller curves. Contrary…
Quantum simulation with controllable many-body platforms offers a powerful route to exploring complex phases and dynamics that are difficult to access in natural materials. Among these, topological spin textures such as skyrmions are…
Various topological techniques and tools have been applied to neural networks in terms of network complexity, explainability, and performance. One fundamental assumption of this line of research is the existence of a global (Euclidean)…
We study the numerical topology of the clique complex $K_n=\mathrm{Cl}(G_n)$, where $G_n$ is the partition graph on the set of integer partitions of $n$. Building on the previously established homotopy equivalence $K_n \simeq \vee^{\,b_n}…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
Macroscopic spin ensembles possess brain-like features such as non-linearity, plasticity, stochasticity, selfoscillations, and memory effects, and therefore offer opportunities for neuromorphic computing by spintronics devices. Here we…