Stories with Tag Theorem

• Undergraduate Upends a 40-Year-Old Data Science Conjecture

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  Posted: 2025-03-16 11:43:14

  Verbose tl;dr

  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.

  In a remarkable demonstration of the power of fresh perspectives, an undergraduate student named Ewin Tang has effectively refuted a long-standing conjecture in theoretical computer science, specifically within the realm of high-dimensional geometry and its applications to nearest-neighbor search. This conjecture, which had remained unchallenged for approximately four decades, posited that locality-sensitive hashing (LSH), a widely employed technique for efficiently finding data points close to a given query point in high-dimensional space, was fundamentally limited in its capabilities. The prevailing belief was that achieving sublinear query time with LSH for nearest-neighbor search in high-dimensional data was mathematically impossible, thus necessitating algorithms with query times that scaled linearly with the dataset's size. This perceived limitation had significant implications for the field of data science, hindering the development of faster and more efficient search algorithms for applications such as image retrieval, natural language processing, and recommendation systems, all of which frequently deal with high-dimensional data.

  Tang's groundbreaking work, conducted while she was still an undergraduate student at the University of Texas at Austin, not only disproved this long-held conjecture but also provided a concrete algorithm that achieves the previously thought impossible sublinear query time. Her approach involves a sophisticated and innovative combination of theoretical insights and algorithmic techniques, drawing upon connections between seemingly disparate areas of mathematics and computer science. Specifically, Tang's algorithm leverages a nuanced understanding of spherical harmonics, functions defined on the surface of a sphere, and their relationship to high-dimensional geometry. This theoretical foundation enabled her to construct a novel hashing scheme that circumvents the limitations previously attributed to LSH, effectively unlocking the potential for substantially faster nearest-neighbor search in high-dimensional spaces.

  The implications of Tang's discovery are far-reaching. By demonstrating that sublinear query time is indeed achievable with LSH, she has opened up exciting new avenues for research and development in the field of data science. Her work promises to pave the way for the creation of more efficient algorithms that can handle the ever-increasing volumes of high-dimensional data generated in modern applications. This breakthrough not only underscores the importance of fundamental theoretical research but also highlights the potential for undergraduate students to make significant contributions to even the most established areas of scientific inquiry. The fact that such a young researcher could overturn a conjecture that had stood for four decades serves as an inspiring testament to the power of innovative thinking and the continued evolution of our understanding of complex computational problems.

  • Data Science
  • conjecture
  • undergraduate
  • Research
  • mathematics
  • Higher Education
  • discovery
  • Innovation
  • academia
  • Computer Science
  • Algorithm
  • Theorem
  • Proof
  • complexity theory
  • scientific breakthrough

  Summary of Comments ( 2 )
  https://news.ycombinator.com/item?id=43378256

  Verbose tl;dr

  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.

• Lemma for the Fundamental Theorem of Galois Theory

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  Posted: 2025-03-15 15:38:22

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  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.

  This blog post meticulously dissects a crucial lemma that underpins the Fundamental Theorem of Galois Theory, a cornerstone of abstract algebra. The theorem establishes a profound connection between intermediate fields of a Galois extension and subgroups of its Galois group. This specific lemma, while seemingly technical, plays a vital role in demonstrating the injectivity of the correspondence described in the Fundamental Theorem. In essence, it proves that distinct subgroups of the Galois group must fix distinct intermediate fields.

  The author begins by carefully setting the stage, reminding the reader of the context within Galois theory. We are considering a field extension K of F, specifically a Galois extension, meaning it's a normal and separable extension. The Galois group Gal(K/F) consists of all automorphisms of K that leave the base field F fixed. The lemma focuses on the relationship between subgroups H of this Galois group and the intermediate fields L, where F ⊆ L ⊆ K.

  The statement of the lemma itself asserts that if we have two distinct subgroups H₁ and H₂ of Gal(K/F), then the fixed fields of these subgroups, denoted as K^( H₁) and K^( H₂), respectively, must also be distinct. The fixed field of a subgroup H is the set of all elements in K that remain unchanged under every automorphism in H. Thus, the lemma claims that different subgroups fix different parts of the extension field K.

  The proof proceeds by contradiction. We assume that the fixed fields are equal, K^( H₁) = K^( H₂), and aim to show that this implies H₁ = H₂, contradicting our initial assumption of distinct subgroups. The author leverages the fact that any automorphism in H₁ will necessarily fix K^( H₁) and, by our assumption, also fix K^( H₂). This allows us to conclude that any element of H₁ belongs to the subgroup that fixes K^( H₂). Due to the Galois extension context, we know that H₂ is precisely the subgroup that fixes K^( H₂), implying that H₁ is a subset of H₂.

  The argument is then symmetrically applied: because K^( H₁) = K^( H₂), any automorphism in H₂ fixes K^( H₂), and therefore also fixes K^( H₁). Consequently, H₂ is a subset of H₁, which is the subgroup that fixes K^( H₁). Since H₁ is a subset of H₂ and H₂ is a subset of H₁, we deduce that H₁ and H₂ must be the same subgroup, a contradiction.

  Therefore, the initial assumption that distinct subgroups have the same fixed field is false, establishing the lemma. This result is a key ingredient in the proof of the Fundamental Theorem of Galois Theory, ensuring that the mapping between subgroups and intermediate fields is indeed injective. The post concludes by highlighting the lemma's importance within the broader framework of Galois theory.

  • Galois Theory
  • Fundamental Theorem of Galois Theory
  • mathematics
  • Abstract Algebra
  • Field Theory
  • Group Theory
  • Lemma
  • Proof
  • Theorem
  • Algebraic Extensions
  • Normal Extensions
  • Separable Extensions
  • Automorphisms
  • Fixed Fields

  Summary of Comments ( 0 )
  https://news.ycombinator.com/item?id=43373196

  Verbose tl;dr

  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.

  The Hacker News post titled "Lemma for the Fundamental Theorem of Galois Theory" sparked a brief discussion with a few insightful comments. No one overtly disagreed with the premise of the linked article, but rather expanded on its context and implications.

  One commenter pointed out the significance of the lemma by highlighting that it establishes a connection between the intermediate fields of a Galois extension and the subgroups of its Galois group. This connection, they explain, is crucial for the Fundamental Theorem of Galois Theory, which establishes a deeper correspondence between these intermediate fields and subgroups, going beyond just a bijection and relating their structure and properties as well. This comment effectively underscores the lemma's role as a foundational building block for the broader theorem.

  Another commenter delves into more specific details, mentioning that the lemma facilitates a clearer understanding of the bijective nature of the Galois correspondence. They mention that some formulations of the Fundamental Theorem of Galois Theory include this lemma as an integral part, demonstrating different approaches to presenting the theorem. This comment highlights variations in how the theorem and its supporting components are presented in different mathematical texts and learning materials.

  A further comment discusses the pedagogical implications, noting how the lemma simplifies the proof of the Fundamental Theorem of Galois Theory by separating out a key argument. This streamlined approach is viewed as beneficial for understanding the overall logic and flow of the proof.

  Finally, another contributor shifts the focus to the overall context of the Fundamental Theorem of Galois Theory within abstract algebra, emphasizing its importance and depth. This comment emphasizes the broader significance of the topic within its mathematical field.

  In summary, while limited in number, the comments on the Hacker News post provide valuable perspectives on the presented lemma, including its importance within the proof of the Fundamental Theorem of Galois Theory, different approaches to its presentation, and its broader mathematical significance.