Leila Sloman writes about five mathematicians at ETH Zurich who, while pursuing a different problem in late 2025, stumbled upon a simple proof of the supercritical sharpness conjecture for all infinite transitive graphs—a decades-old question in percolation theory about how quickly networks flood past a critical threshold. The proof confirms that above the critical probability, fluid covers nearly the entire graph, resolving what researchers had called "the one remaining fortress" in the field.
- Percolation theory originated from Rosalind Franklin's 1940s work on coal porosity at the British Coal Utilization Research Association
- Oded Schramm, who co-initiated the study of percolation on transitive graphs with Itai Benjamini, died in a hiking fall in 2008 at age 46
- The key insight was reordering a standard "sprinkling" technique—analyzing the sprinkled edges first rather than last—which both simplified the proof and made it general enough for all transitive graphs
- The subcritical half of the sharpness conjecture had already been proved in 2007 by Antunović and Veselić
- A major open question remains: what happens exactly at the critical probability on three-dimensional lattices
- Percolation theory originated from Rosalind Franklin's 1940s work on coal porosity at the British Coal Utilization Research Association
- Oded Schramm, who co-initiated the study of percolation on transitive graphs with Itai Benjamini, died in a hiking fall in 2008 at age 46
- The key insight was reordering a standard "sprinkling" technique—analyzing the sprinkled edges first rather than last—which both simplified the proof and made it general enough for all transitive graphs
- The subcritical half of the sharpness conjecture had already been proved in 2007 by Antunović and Veselić
- A major open question remains: what happens exactly at the critical probability on three-dimensional lattices
A new study published in American Antiquity reveals that early Native Americans used two-sided dice in games of chance over 12,000 years ago, predating known Old World dice by millennia. By applying a morphological test to archaeological artifacts, researcher Robert Madden identified hundreds of "binary lots" used in structured, rule-based games. These activities suggest that Ice Age hunter-gatherers understood and relied on random outcomes long before formal probability theory emerged. Rather than commercial gambling, these games likely served social functions, fostering reciprocal relationships and gifting between different groups through fair, one-on-one competition.
This is an open, unconventional textbook covering mathematics, computing, and artificial intelligence from foundational principles. It's designed for practitioners seeking a deep understanding, moving beyond exam preparation and focusing on real-world application. The author, drawing from years of experience in AI/ML, has compiled notes that prioritize intuition, context, and clear explanations, avoiding dense notation and outdated material.
The compendium covers a broad range of topics, from vectors and matrices to machine learning, computer vision, and multimodal learning, with future chapters planned for areas like data structures and AI inference.
The compendium covers a broad range of topics, from vectors and matrices to machine learning, computer vision, and multimodal learning, with future chapters planned for areas like data structures and AI inference.
A visual introduction to probability and statistics, covering basic probability, compound probability, probability distributions, frequentist inference, Bayesian inference, and regression analysis. Created by Daniel Kunin and team with interactive visualizations using D3.js.
This notebook provides an introduction to Naive Bayes classification, covering concepts, formulas, and implementation.
A comprehensive guide covering the most critical machine learning equations, including probability, linear algebra, optimization, and advanced concepts, with Python implementations.
For decades, mathematicians have struggled to understand matrices that reflect both order and randomness, like those that model semiconductors.
A new method could change that.
Recent research has made a significant mathematical advance in understanding Anderson localization, the phenomenon where disorder in a material (like impurities in silicon) can stop electron flow. Researchers proved that, for a simplified model called band matrices, electrons do become trapped ("localized") with enough disorder. This breakthrough, achieved by Yan Yau and Jun Yin’s team, uses a new mathematical technique and brings us closer to fully understanding Anderson’s original model and designing materials with specific electronic properties. It’s a key step in understanding systems between order and randomness.
A new method could change that.
Recent research has made a significant mathematical advance in understanding Anderson localization, the phenomenon where disorder in a material (like impurities in silicon) can stop electron flow. Researchers proved that, for a simplified model called band matrices, electrons do become trapped ("localized") with enough disorder. This breakthrough, achieved by Yan Yau and Jun Yin’s team, uses a new mathematical technique and brings us closer to fully understanding Anderson’s original model and designing materials with specific electronic properties. It’s a key step in understanding systems between order and randomness.
A new mathematical proof resolves a 35-year-old bet between Noga Alon and Peter Sarnak regarding the prevalence of optimal expander graphs, demonstrating that both mathematicians were partially incorrect. The proof, building on work in random matrix theory, reveals that approximately 69% of regular graphs are Ramanujan graphs.
Explores the role of conditional probability in understanding events and Bayes' theorem, with examples in regression analysis and everyday scenarios, demonstrating how our biological tissue runs probabilistic machinery.
