Structures and proofs—algebra, number theory, topology, and geometry.
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Ivy Delaney has been working with neural networks and machine learning since the mid-nineties, back when a couple of hidden layers and a long afternoon of training counted as ambitious. She has watched the field go from academic curiosity to…
By Supat Charoensappuech, in collaboration with Gemini 3.6 Flash (October 9, 2026) The author invites all interested readers to join in examining, testing, and critically discussing these ideas. This article is intended as the beginning of a conversation, not the end. 1. Introduction: From…
Two independent Nature experiments guided by Xue-Jia Yu's theory show that topological states can survive at critical points where the energy gap closes.
The source data that support the findings of this study are available from the corresponding author on request. Source data are provided with this paper. The code used to evaluate the conclusions of this study is available from the corresponding…
Topology: A Robust Framework Protected by Topological Invariants To understand the significance of this work, we first need to discuss the concept of topology. However, this framework appears almost entirely incompatible with topological phases of matter. Critical points are gapless,…
Nature Physics (2026) Cite this article Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016). Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017). Watanabe, H., Po, H. C. & Vishwanath, A. Structure and topology of band structures in…
You can’t say much about modal logic in general. You have to be more specific to get anywhere. You have to choose some axioms. Ideally the axioms you need for your application correspond to a named set of axioms that…
The previous post discussed a superficial connection between modal logic and topology, that both give use the terms regular and normal to indicate added sets of axioms. McKinsey and Tarski developed a deeper connection between modal logic and topology that…
Quick Answer & Key Insight: Why Does 1/998001 Generate All 3-Digit Numbers? Yes, 1/998001 generates every 3-digit sequence. Because 998001 equals $999^2$, the decimal expansion cascades through consecutive integers, but carrying digits at the 998th block creates an unexpected loop trap that skips 998. See the mathematical…
Chia-Yi Ju1,2,3,4 and Szu-Ming Chen5 Featured image: The evolution of a general quantum state depends heavily on how its parameter path winds around an exceptional point. [1] Bo Peng, Şahin Kaya Özdemir, Weijian Chen, Franco Nori, and Lan Yang. ``What is and what…