A Brown University undergraduate, Noah Solomon, disproved a long-standing conjecture in data science known as the "conjecture of Kahan." This conjecture, which had puzzled researchers for 40 years, stated that certain algorithms used for floating-point computations could only produce a limited number of outputs. Solomon developed a novel geometric approach to the problem, discovering a counterexample that demonstrates these algorithms can actually produce infinitely many outputs under specific conditions. His work has significant implications for numerical analysis and computer science, as it clarifies the behavior of these fundamental algorithms and opens new avenues for research into improving their accuracy and reliability.
This post presents a simplified, self-contained proof of a key lemma used in proving the Fundamental Theorem of Galois Theory. This lemma establishes a bijection between intermediate fields of a Galois extension and subgroups of its Galois group. The core idea involves demonstrating that for a finite Galois extension K/F and an intermediate field E, the fixed field of the automorphism group fixing E (denoted as Inv(Gal(K/E)) is E itself. The proof leverages the linear independence of field automorphisms and constructs a polynomial whose roots distinguish elements within and outside of E, thereby connecting the field structure to the group structure. This direct approach avoids more complex machinery sometimes used in other proofs, making the fundamental theorem's core connection more accessible.
Hacker News users discuss the linked blog post explaining a lemma used in the proof of the Fundamental Theorem of Galois Theory. Several commenters appreciate the clear explanation of a complex topic, with one pointing out how helpful the visualization and step-by-step breakdown of the proof is. Another commenter highlights the author's effective use of simple examples to illustrate the core concepts. Some discussion revolves around different approaches to teaching and understanding Galois theory, including alternative proofs and the role of intuition versus rigor. One user mentions the value of seeing multiple perspectives on the same concept to solidify understanding. The overall sentiment is positive, praising the author's pedagogical approach to demystifying a challenging area of mathematics.
Mathematicians have finally proven the Kakeya conjecture, a century-old problem concerning the smallest area required to rotate a unit line segment 180 degrees in a plane. The collaborative work, spearheaded by Nets Katz and Joshua Zahl, builds upon previous partial solutions and introduces a novel geometric argument. While their proof technically addresses the finite field version of the conjecture, it's considered a significant breakthrough with strong implications for the original Euclidean plane problem. The techniques developed for this proof are anticipated to have far-reaching consequences across various mathematical fields, including harmonic analysis and additive combinatorics.
HN commenters generally express excitement and appreciation for the breakthrough proof of the Kakeya conjecture, with several noting its accessibility even to non-mathematicians. Some discuss the implications of the proof and its reliance on additive combinatorics, a relatively new field. A few commenters delve into the history of the problem and the contributions of various mathematicians. The top comment highlights the fascinating connection between the conjecture and seemingly disparate areas like harmonic analysis and extractors for randomness. Others discuss the "once-in-a-century" claim, questioning its accuracy while acknowledging the significance of the achievement. A recurring theme is the beauty and elegance of the proof, reflecting a shared sense of awe at the power of mathematical reasoning.
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https://news.ycombinator.com/item?id=43378256
Hacker News commenters generally expressed excitement and praise for the undergraduate student's achievement. Several questioned the "40-year-old conjecture" framing, pointing out that the problem, while known, wasn't a major focus of active research. Some highlighted the importance of the mentor's role and the collaborative nature of research. Others delved into the technical details, discussing the specific implications of the findings for dimensionality reduction techniques like PCA and the difference between theoretical and practical significance in this context. A few commenters also noted the unusual amount of media attention for this type of result, speculating about the reasons behind it. A recurring theme was the refreshing nature of seeing an undergraduate making such a contribution.
The Hacker News post titled "Undergraduate Upends a 40-Year-Old Data Science Conjecture" has generated a number of comments discussing the Wired article about Miles Edwards's work on the Conjecture.
Several commenters express admiration for Edwards's achievement. One notes the impressive nature of disproving a conjecture at the undergraduate level, highlighting the rarity of such accomplishments. Another emphasizes the significance of finding a counterexample in a widely accepted theory.
Some comments delve into the specifics of the conjecture and Edwards's work. One commenter discusses the implications for k-means clustering, suggesting that while Lloyd's algorithm is still practically useful, the conjecture's disproof raises theoretical questions. Another commenter, claiming expertise in the area, points out that the conjecture was already known to be false in high dimensions and clarifies that Edwards's work focuses on the previously unexplored low-dimensional case. This commenter further details that Edwards's counterexample used only six points and five clusters in two dimensions.
There's discussion on the practical implications of the discovery. A commenter questions the real-world impact, arguing that constant factors are often more important than asymptotic complexity in practice, particularly in machine learning. Another echoes this sentiment, suggesting that the theoretical breakthrough might not translate into significant improvements in everyday clustering applications.
One commenter expresses skepticism about the Wired article's portrayal of Edwards's discovery as "upending" the field, arguing that such framing is overblown and misleading.
Finally, some comments provide additional context, including links to Edwards's paper and his advisor's blog post. This supplementary material allows interested readers to delve deeper into the technical details of the work.