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Programming Language Theory has a public relations problem
10 months ago
- Programming Language Theory (PLT) is misunderstood by outsiders, seen as cryptic and impractical.
- PLT is appreciated as a pure mathematical art, requiring education to fully understand.
- Theoretical work in PLT is often overhyped for practical applications, leading to disappointment.
- PLT lacks the interconnectedness seen in other branches of mathematics, making it harder to build upon previous work.
- High levels of abstraction in PLT make it difficult for newcomers to grasp.
- PLT is inherently hard due to the complexity and vast design space of programming languages.
- Solutions include being honest about PLT's beauty, creating better introductory materials, and simplifying proofs.
Generating (almost) equally-spaced points along a parabola
10 months ago
- The article discusses generating equally-spaced points along a parabola for coin placement in an endless runner game.
- Initial approaches of sampling at regular time or horizontal intervals resulted in uneven coin spacing.
- A mathematical approach using arc length was considered but deemed impractical due to complexity.
- A simplified method approximating infinitesimal changes in arc length provided a more aesthetically pleasing and performant solution.
- The final approach balances mathematical approximation with practical implementation, improving visual consistency without heavy computation.
Mathematician has solved the Kakeya conjecture
10 months ago
- The Kakeya conjecture, posed by Soichi Kakeya in 1917, asks for the smallest surface area needed to rotate a needle to point in the opposite direction.
- Hong Wang and Joshua Zahl solved the Kakeya conjecture in three dimensions, a major mathematical achievement of the 21st century.
- The solution involves complex calculations and spans 127 pages, with only the authors fully understanding the intricate reasoning.
- Wang's work connects the Kakeya conjecture to the restriction conjecture in harmonic analysis, which has applications in medical imaging and digital file compression.
- The restriction conjecture deals with the behavior of the Fourier transform on curved surfaces like spheres.
- Wang's approach uses wave packets and overlapping parallelepipeds, described as 'baroque suprematism' by mathematician Antonio Córdoba.
- Luis Vega, a Spanish mathematician, indirectly inspired Wang's focus on the restriction conjecture.
- Wang is a humble figure, avoiding speculation about winning the Fields Medal, despite her groundbreaking work.
Breakthrough proof bringing mathematics closer to a grand unified theory
10 months ago
- The Langlands programme, a grand unified theory of mathematics, has seen a major breakthrough with the proof of the geometric Langlands conjecture.
- A team of nine mathematicians, led by Dennis Gaitsgory and Sam Raskin, published five papers totaling nearly 1,000 pages to achieve this proof.
- The proof opens new research avenues and bridges different areas of mathematics, earning Gaitsgory the $3-million Breakthrough Prize in Mathematics.
- The Langlands programme connects number theory and harmonic analysis, with the geometric Langlands conjecture focusing on Riemann surfaces and complex manifolds.
- The geometric Langlands conjecture links fundamental groups and sheaves, offering insights that may help advance the arithmetic Langlands conjecture.
- The proof has implications for local versions of the Langlands conjectures and has inspired connections to quantum physics, particularly S-duality in gauge theories.
- Edward Witten and Anton Kapustin showed that geometric Langlands symmetry arises naturally in certain quantum field theories, linking pure mathematics to physics.
The (Unfinished) PDE Coffee Table Book
10 months ago
- A project at Oxford University (2000-2001) aimed to create a book with 100 two-page spreads on partial differential equations (PDEs), featuring color illustrations.
- 34 out of the planned 100 pages were completed before the project stalled.
- The completed pages cover various PDEs, including the Laplace equation, Heat equation, Wave equation, and others, each with a brief description.
- The project is currently inactive with no plans for completion, but the finished pages are made available for educational purposes.
- Copyright is held by Lloyd N. Trefethen (2011), with all publication rights reserved.
Gemini with Deep Think officially achieves gold-medal standard at the IMO
9 months ago
- Gemini Deep Think achieved gold-medal performance at IMO 2025 by solving 5 out of 6 problems.
- The model operated end-to-end in natural language, producing rigorous proofs within the 4.5-hour competition limit.
- Advanced reasoning techniques like parallel thinking and reinforcement learning were used to enhance problem-solving.
- This marks a significant improvement over last year's silver-medal performance by AlphaProof and AlphaGeometry.
- Google plans to release a version of the Deep Think model to trusted testers before wider availability.
- The achievement highlights AI's growing potential in advanced mathematical reasoning and problem-solving.
Human teens beat AI at an international math competition
9 months ago
- AI models achieved gold-level scores at the International Mathematics Olympiad (IMO) for the first time.
- Google and OpenAI's AI programs solved 5 out of 6 IMO problems, earning gold medals.
- Complex mathematics remains a significant challenge for AI due to its reliance on token prediction rather than logical reasoning.
- AI's performance at the IMO showed major improvements, with models explaining their solutions clearly.
- Despite progress, AI still falls short of the perfect scores achieved by some human competitors.
- Concerns persist about the massive energy and resource consumption required for AI advancements.
Tom Lehrer, song satirist and mathematician, dies at 97
9 months ago
- Tom Lehrer, renowned song satirist and mathematician, passed away at 97.
- He was known for his witty and erudite songs that satirized marriage, politics, racism, and the Cold War.
- Lehrer abandoned his music career to return to teaching math at Harvard and other universities.
- In 2020, he granted the public permission to use his lyrics freely, waiving copyright fees.
- His songs included 'Poisoning Pigeons in the Park,' 'The Old Dope Peddler,' and 'The Vatican Rag,' which mocked various societal norms and institutions.
- Lehrer was influenced by musical greats like Gilbert and Sullivan and Stephen Sondheim.
- He was uncomfortable with public performances and preferred teaching over touring.
- Lehrer contributed songs to the 1964 TV show 'That Was the Week That Was' and the children's educational show 'The Electric Company.'
- Born in 1928 in New York City, Lehrer skipped two grades and entered Harvard at 15.
- He taught math at the University of California, Santa Cruz, well into his late 70s.
Zigzag Number Spiral
9 months ago
- The article discusses a zigzag number spiral grid where numbers are arranged in a spiral-like manner, reversing direction at the edges.
- The goal is to find a closed-form expression for the number at the mth row and nth column, denoted as f(m, n).
- The spiral touches the left edge on even-numbered rows and the top edge on odd-numbered columns, leading to specific patterns.
- Formulas for edge numbers are derived: f(m, 1) = m² if m is even, and (m-1)² + 1 if m is odd; similarly for f(1, n).
- A piecewise formula for f(m, n) is developed based on whether m or n is greater and their parity (odd/even).
- The piecewise formula is simplified into a single closed-form expression: f(m, n) = (max(m, n))² - max(m, n) + 1 + (-1)^max(m, n) * (m - n).
- An alternative approach focuses on diagonal numbers, leading to the same closed-form expression.
- The diagonal analysis shows f(n, n) = n² - n + 1 for all positive integers n, simplifying the general formula.
Math Is Quietly in Crisis over NSF Funding Cuts
9 months ago
- Mathematics research relies on conferences, workshops, and institutes for collaboration and problem-solving.
- The Trump administration's NSF has significantly cut funding for mathematics, impacting conferences, fellowships, and early-career researchers.
- The American Mathematical Society has offered $1 million in backstop grants to mitigate funding losses, but it's not a long-term solution.
- Funding cuts are eroding trust and creating uncertainty, with long-term consequences for research programs and young mathematicians.
- Young mathematicians are considering leaving academia for private sector jobs or more supportive countries due to diminished opportunities.
- Mathematics plays a crucial role in innovation, national security, and other scientific fields, making funding cuts shortsighted.
At 17, Hannah Cairo Solved a Major Math Mystery
9 months ago
- Hannah Cairo, a 17-year-old homeschooled student from the Bahamas, solved the 40-year-old Mizohata-Takeuchi conjecture in mathematics.
- Cairo's proof showed the conjecture to be false, defying mathematicians' expectations and altering the landscape of harmonic analysis.
- She was largely self-taught, using online resources and graduate-level textbooks, and later participated in advanced math programs like the Berkeley Math Circle.
- Cairo's work has significant implications for harmonic analysis, including disproving Stein's conjecture, which was thought to connect key questions in the field.
- Despite her achievements, Cairo faced challenges in gaining admission to graduate programs due to her lack of formal education credentials, but she will start a PhD at the University of Maryland.
Show HN: Mathpad – Physical keypad for typing 100+ math symbols anywhere
9 months ago
- Mathpad is a specialized keypad designed for typing mathematical equations efficiently.
- It provides over 100 mathematical symbols and Greek letters at your fingertips.
- Compatible with Windows, macOS, and Linux, supporting various output modes like LaTeX and Unicode.
- Fully open-source hardware and software, built on QMK firmware with hot-swappable Gateron KS-33 switches.
- Designed for students, educators, scientists, and engineers to simplify math typing in documents, chats, and code.
- Production-ready with a transparent roadmap: components ordered post-campaign, assembly, testing, and global distribution via Crowd Supply.
- Limitations include Latin-based keyboard layout support only and a required Windows application for functionality.
- Risk mitigation strategies in place for supply chain delays and assembly damage.
- Produced by Summa Cogni in the UK, sold and shipped by Crowd Supply.
Terrence Tao Loses Funding
9 months ago
- Terence Tao is a renowned mathematician known for his exceptional intelligence and contributions to math.
- He started showing mathematical prowess at a very young age, excelling in calculus by age 7 and winning the International Mathematical Olympiad at 13.
- Tao's work includes crucial insights into prime numbers, random-number generation, and improvements in MRI technology.
- He was awarded the Fields Medal in 2006 and was featured in National Geographic's 'Genius' issue in 2017.
- Tao is admired for mentoring young scholars and sharing credit generously, earning him the nickname 'Mozart of Math.'
- Recent funding cuts by the Trump administration have jeopardized his research and raised concerns about the future of STEM in America.
Mathematician Finds Easier Way to Solve Quadratic Equations
9 months ago
- A mathematician at Carnegie Mellon University, Dr. Po-Shen Loh, developed a simpler method to solve quadratic equations.
- The new method avoids traditional techniques like completing the square and focuses on averaging roots for easier understanding.
- Quadratic equations are polynomials with x² terms, often taught with simple examples, but real-life applications can be complex.
- Dr. Loh's approach involves representing roots as equidistant values from their average, simplifying the solving process.
- This method reduces guesswork and is more intuitive, helping students grasp quadratic equations better.
- Understanding quadratic equations is foundational for advanced math topics like precalculus.
- Dr. Loh aims to bridge the gap between classroom examples and real-world applications of quadratic equations.
Show HN: Prime Number Grid Visualizer
9 months ago
- Prime Grid is a tool that displays prime numbers in a left-to-right, top-to-bottom grid layout.
- It can be used to identify visual patterns, perform mathematical analysis, or explore potential hidden codes.
- Users can adjust the grid by clicking into the column input and holding the up arrow to see cascading effects.
- Created by Danny Duplex, who humorously claims to be solving the world's least important problems.
Derivatives, Gradients, Jacobians and Hessians
9 months ago
- Derivatives are fundamental in calculus, indicating how a function changes at each point.
- Derivatives are used for optimization, such as finding minima or maxima on a graph.
- Gradient descent is an iterative optimization method inspired by derivatives, adjusting step size to find minima.
- Gradients extend derivatives to higher-dimensional functions, providing vectors that indicate the steepest ascent or descent.
- The Jacobian matrix combines gradients of functions with multiple outputs, describing how space is warped at a point.
- The Hessian matrix consists of second derivatives, useful for understanding the curvature of functions in optimization.
- Hessian matrices are computationally intensive but valuable in optimization, with quasi-Newton methods offering alternatives.
The Dogs in Pursuit Problem
9 months ago
- Four dogs start at the corners of a square with side length 1, each running towards the dog immediately anti-clockwise to it.
- All dogs run at the same speed, spiraling towards the center until they collide.
- The problem asks how far each dog travels before collision.
- Bonus: Extend the problem to equilateral triangles and regular pentagons.
- The puzzle is inspired by pursuit problems, which often involve calculus.
- Steven Strogatz, a mathematician, highlights the Infinity Principle in calculus—breaking complex motions into simpler parts.
- Pursuit problems are visually engaging and historically significant in calculus.
- The spiral paths of the dogs resemble mathematical art or optical illusions.
- The article promotes Strogatz's book 'Infinite Powers' and the author's 'Football School' series.
Gap – A System for Computational Discrete Algebra
8 months ago
- GAP is a system for computational discrete algebra, focusing on Computational Group Theory.
- It includes a programming language, a library of algebraic algorithms, and large data libraries of algebraic objects.
- GAP is freely distributed, allowing users to study, modify, or extend it for specific uses.
- The current version is GAP 4.14.0, released on 05 December 2024, available on the install page.
- Contributions to GAP are welcome via GitHub, with guidance available for coding and package development.
- GAP has been developed through international cooperation and user contributions since 1986.
- GAP was awarded the ACM/SIGSAM Richard Dimick Jenks Memorial Prize in July 2008.
'Will you leave US for China?' It depends, mathematician Terence Tao says
8 months ago
- Mathematician Terence Tao discusses the possibility of leaving the US for China due to funding cuts under the Trump administration.
- Tao describes the current situation as 'very fluid and unstable,' more so than in the past 30 years.
- He states that future decisions depend on whether the current administration changes its course.
- Tao, known as the 'Mozart of maths,' is at a crossroads between geopolitics and scientific freedom.
- Born in Australia to Chinese parents, Tao has spent most of his life in the US, achieving tenure at UCLA at 24 and winning the Fields Medal at 31.
The Most Important Machine Learning Equations: A Comprehensive Guide
8 months ago
- Machine learning (ML) is a field driven by mathematics, with core equations essential for building models and optimizing algorithms.
- Probability and information theory provide the foundation for reasoning about uncertainty and measuring differences between distributions.
- Bayes’ Theorem is a cornerstone of probabilistic reasoning, used in tasks like classification and inference.
- Entropy measures uncertainty in a probability distribution and is fundamental in decision trees and information gain calculations.
- Joint and conditional probability are building blocks of Bayesian methods and probabilistic models.
- Kullback-Leibler Divergence (KLD) measures how much one probability distribution diverges from another, used in variational autoencoders (VAEs).
- Cross-entropy quantifies the difference between true and predicted distributions, widely used as a loss function in classification.
- Linear algebra powers transformations and structures in ML models, with linear transformations being core operations in neural networks.
- Eigenvalues and eigenvectors describe how a matrix scales and rotates space, crucial for understanding data variance in PCA.
- Singular Value Decomposition (SVD) breaks down a matrix into orthogonal matrices and a diagonal matrix, revealing intrinsic data structure.
- Gradient descent updates parameters by moving opposite to the gradient of the loss function, scaled by a learning rate.
- Backpropagation applies the chain rule to compute gradients of the loss with respect to weights in neural networks.
- Mean Squared Error (MSE) calculates the average squared difference between true and predicted values, common in regression tasks.
- The diffusion process describes a forward diffusion process where data is gradually noised over time, key in generative AI.
- Convolution combines two functions by sliding one over the other, extracting features in data like images, core to CNNs.
- Softmax converts raw scores into probabilities, ideal for multi-class classification in neural network outputs.
- Attention computes a weighted sum of values based on the similarity between queries and keys, powering transformers in NLP.

Hasty Briefsbeta

#mathematics

Programming Language Theory has a public relations problem

Generating (almost) equally-spaced points along a parabola

Mathematician has solved the Kakeya conjecture

Breakthrough proof bringing mathematics closer to a grand unified theory

The (Unfinished) PDE Coffee Table Book

Gemini with Deep Think officially achieves gold-medal standard at the IMO

Human teens beat AI at an international math competition

Tom Lehrer, song satirist and mathematician, dies at 97

Zigzag Number Spiral

Math Is Quietly in Crisis over NSF Funding Cuts

At 17, Hannah Cairo Solved a Major Math Mystery

Show HN: Mathpad – Physical keypad for typing 100+ math symbols anywhere

Terrence Tao Loses Funding

Mathematician Finds Easier Way to Solve Quadratic Equations

Show HN: Prime Number Grid Visualizer

Derivatives, Gradients, Jacobians and Hessians

The Dogs in Pursuit Problem

Gap – A System for Computational Discrete Algebra

'Will you leave US for China?' It depends, mathematician Terence Tao says

The Most Important Machine Learning Equations: A Comprehensive Guide