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Maypole Dance of Braid Like Groups
a year ago
- Attended a May Day party featuring a traditional maypole dance.
- Maypole dance involves dancers holding ribbons that wind around a pole, creating decorative patterns.
- The maypole dance was analyzed mathematically, drawing parallels to knot theory and braid groups.
- Braid groups involve strings wound around each other, with specific rules for movement and interaction.
- The maypole dance introduces a variation of the braid group, termed the maypole braid group (M_n).
- M_n includes additional generators and relations to account for the circular nature of the maypole.
- New relations involve rotational movements and interactions between strands in a circular arrangement.
- The analysis highlights the intersection of cultural traditions and mathematical concepts.
When do girls fall behind in maths? Gigantic study pinpoints the moment
a year ago
- A large study in France reveals that the 'mathematical gender gap' begins in the first year of school.
- Boys and girls start with similar math scores, but boys pull ahead within four months, with a more significant gap after 12 months.
- The study suggests that environmental factors in school, not innate differences, are responsible for the gap.
- Recommendations to address the gap include reducing math anxiety, encouraging equal participation, and fostering curiosity outside the classroom.
- The study analyzed nearly three million children across different cohorts, socioeconomic groups, and regions, showing the gap's universality.
- The gap's emergence is linked to the start of formal education, not age, as shown by comparing children born just days apart in different school years.
- Researchers emphasize the need for broad policy interventions to address the gap, as it affects all groups universally.
The Hat, the Spectre and SAT Solvers (2024)
a year ago
- Introduction to aperiodic tilings with a single monotile (the Hat) and SAT solvers.
- Explanation of SAT solvers and their application in solving boolean algebra problems.
- Guide on using a SAT solver in WASM for practical applications like solving Sudoku puzzles.
- Application of SAT solvers to tiling problems, specifically using the Hat and Turtle shapes.
- Introduction of the Spectre, a true chiral aperiodic monotile, and its mathematical significance.
- Practical guide on using the Hats app to explore tiling patterns and solve tiling problems.
- References to original sources and further reading on aperiodic tilings and SAT solvers.
Think of a Number
a year ago
- Sam Altman's claims about AGI being imminent are criticized as irresponsible hype.
- Current AI can handle undergraduate mathematics but fails at PhD-level tasks, indicating no true AGI.
- The author proposes creating a secret database of hard number theory problems to test AI's mathematical understanding.
- Questions should require non-negative integer answers, be beyond undergraduate level, and not easily guessable or found online.
- The experiment aims to distinguish between AI's pattern-matching and genuine mathematical thinking.
- The author seeks collaboration from PhD-level number theorists to contribute challenging problems.
- AI companies will be invited to test their models against the database, with results made public.
- The project contrasts with FrontierMath by ensuring questions are not pre-exposed to AI models.
- Example questions would resemble those in FrontierMath but be harder and more uniformly difficult.
- The goal is to assess whether AI can truly think mathematically, beyond stochastic parroting.
Elliptic Curves as Art
a year ago
- Project focuses on visualizing elliptic curves.
- Website is currently under construction.
- Project by Nadir Hajouji and Steve Trettel.
Mathematicians discover a new way to find prime numbers
10 months ago
- Prime numbers are whole numbers greater than 1, divisible only by 1 and themselves.
- Mathematicians struggle to identify patterns in prime numbers, especially for large numbers.
- A new approach using integer partitions has been developed to detect prime numbers.
- The method involves solving Diophantine equations related to partition functions.
- This discovery provides infinitely many new ways to define prime numbers.
- The research could inspire further investigations into combinatorial functions and number theory.
- Open questions like the twin prime conjecture and Goldbach’s conjecture remain unsolved.
- The findings highlight the richness of connections in mathematics and stimulate fresh thinking.
Is Mathematics Mostly Chaos or Mostly Order?
10 months ago
- Mathematicians gathered in Finland to discuss new notions of infinity, exploring whether mathematics is more chaotic or ordered.
- Georg Cantor's work in the 1870s showed that infinities come in different sizes, leading to the concept of cardinal numbers.
- Set theorists have discovered a hierarchy of large cardinals, but new cardinals by Aguilera, Bagaria, and Lücke challenge this order.
- These new cardinals 'explode' when combined with smaller ones, suggesting more chaos in mathematics than previously thought.
- Kurt Gödel's incompleteness theorems imply that no mathematical system can be complete, requiring continuous addition of new axioms.
- Hugh Woodin's 'Ultimate L' program aims to build a model approximating the mathematical universe, assuming a structured hierarchy of large cardinals.
- The new cardinals, exacting and ultraexacting, may indicate that the mathematical universe is more chaotic and less definable than believed.
- Evidence suggests these new cardinals are consistent with ZFC, but their implications for the HOD conjecture remain debated.
- Some mathematicians welcome the potential chaos, seeing it as an opportunity for further exploration and discovery in set theory.
A new pyramid-like shape always lands the same side up
10 months ago
- Plato envisioned the cosmos as five geometric shapes, including the tetrahedron, which still holds mysteries today.
- John Conway and Richard Guy explored whether a uniform monostable tetrahedron could exist, concluding it wasn't possible with even weight distribution.
- Gábor Domokos and team proved a tetrahedron with uneven weight distribution can be monostable, leading to the creation of a physical model.
- The physical model required extreme precision, using lightweight carbon fiber and dense tungsten carbide to achieve the desired monostable property.
- The research has potential applications in designing self-righting spacecraft and highlights the importance of experimentation in mathematics.
Google DeepMind team up to solve the Navier-Stokes million-dollar problem
10 months ago
- Spanish mathematician Javier Gómez Serrano collaborates with Google DeepMind to solve the Navier-Stokes equations, a Millennium Prize Problem offering a $1 million reward.
- The Navier-Stokes equations, formulated in the 19th century, describe fluid motion and remain unsolved regarding their regularity or potential for sudden behavioral changes.
- A team of 20 researchers, including geophysicists and mathematicians, has been working discreetly for three years, leveraging AI to refine solutions and predict fluid behavior.
- Gómez Serrano's team uses machine learning neural networks to analyze singularities in fluid dynamics, with progress suggesting a solution may be imminent within five years.
- Other competing teams include Thomas Hou's group in California and Diego Córdoba's team in Madrid, each approaching the problem with different strategies.
- Gómez Serrano also contributed to AlphaEvolve, an AI system that solves complex mathematical problems with a 95% success rate, potentially revolutionizing mathematical research.
- Google DeepMind's Demis Hassabis predicts artificial general intelligence (AGI) could emerge by 2030, though Gómez Serrano remains cautiously optimistic about AI's future impact.
BusyBeaver(6) Is Quite Large
10 months ago
- The author reflects on the increasing difficulty of focusing on research due to the current state of the world, but finds solace in breakthroughs like those in Busy Beaver research.
- In 2022, it was discovered that BB(6) > 10^36,534, later improved to BB(6) > 15^10, showcasing the rapid growth of the Busy Beaver function.
- The Busy Beaver function BB(n) represents the maximum steps a Turing machine with n states can take before halting on an all-0 input tape.
- Recent improvements show BB(6) > 10,000,000^10 and later BB(6) > 2 tetrated to the 2 tetrated to the 2 tetrated to the 9, demonstrating its enormous size.
- The author speculates that BB(n) might become independent of ZFC axioms at n=7, 8, or 9, much earlier than previously thought (currently known at n=745).
- The author attended STOC’2025 in Prague, presenting a plenary lecture on 'The Status of Quantum Speedups,' with slides available online.
The Unreasonable Effectiveness of Mathematical Experiments
10 months ago
- Proposes redefining mathematics as an experimental science focused on patterns in computation.
- Highlights Voevodsky's crisis of confidence as an example of mathematical uncertainty.
- Argues that mathematical truth is about predicting computational outcomes, not eternal truths.
- Suggests that contradictions in mathematics indicate failed theories, not logical impossibilities.
- Explains the 'unreasonable effectiveness' of mathematics in physics through this framework.
- Posits that consistency in use, rather than deduction from axioms, signifies mathematical understanding.
Does Form Shape Function?
10 months ago
- L. Mahadevan explores the interplay between form and function in biological systems using mathematics and physics.
- Mahadevan's work includes studying the equilibrium shape of elastic Möbius strips and brain folding patterns.
- Brain folds (gyri and sulci) are formed due to differential growth rates between gray and white matter, similar to swelling gels.
- Termite mounds function like lungs, using passive ventilation driven by diurnal temperature changes to regulate airflow.
- Social insects like termites build complex structures without centralized planning, showcasing swarm intelligence.
- Mahadevan investigates how organisms learn geometry and probability, suggesting human perception of geometry is statistical.
- His research spans from biological morphogenesis to robotics, emphasizing the importance of embodiment and environment interaction.
How Does Graph Theory Shape Our World?
10 months ago
- Graph theory, founded by Leonhard Euler in the 18th century, studies relationships between objects (nodes) and their connections (edges).
- Maria Chudnovsky, a mathematician at Princeton, solved the decades-old 'perfect graph' problem and applied graph theory to real-world scenarios like wedding seating arrangements.
- Graph theory has applications in social networks, transportation systems, and map coloring, exemplified by the Four Color Theorem.
- Perfect graphs are those where the chromatic number equals the size of the largest clique, simplifying coloring problems.
- Chudnovsky's proof of the Strong Perfect Graph Theorem involved breaking complex graphs into manageable parts, leveraging visual and abstract reasoning.
- Mathematics serves as a universal language, transcending cultural and linguistic barriers, fostering clear and logical thinking.
- Graph theory's visual and discrete nature appeals to those who think in terms of relationships and structures, making it a powerful tool in both theoretical and practical domains.
The world of Voronoi diagrams (2021)
10 months ago
- Voronoi diagrams, also known as Dirichlet tessellation or Thiessen polygons, are natural patterns found in various fields like cartography, biology, and architecture.
- A Voronoi diagram divides a plane into cells around points, where each cell contains the area closest to its point.
- These diagrams are common in nature, seen in structures like onion skins, jackfruit shells, and giraffe coats due to efficient space usage and uniform growth patterns.
- Voronoi patterns are used in human-made structures, such as the 'Water cube' in Beijing and crackled glazes in Song dynasty ceramics.
- They are also popular in graphic arts for creating abstract patterns and backgrounds.
- Voronoi diagrams can be generalized to n-dimensional spaces and use different distance metrics, like Euclidean or Manhattan distances.
- They are closely related to the k-nearest neighbors algorithm and Delaunay triangulation, which is useful in modeling surfaces and objects.
- Lloyd’s algorithm, used in k-means clustering, iteratively improves Voronoi diagrams by moving points to cell centroids, leading to more uniform cells.
- Efficient algorithms for constructing Voronoi diagrams include the Sweep line algorithm and methods based on Delaunay triangulation.
- Voronoi diagrams have diverse applications, from modeling forest trees to planning robot paths and procedural map generation in games.
Hannah Cairo has solved the Mizohata-Takeuchi conjecture
10 months ago
- Hannah Cairo, a 17-year-old, refuted the Mizohata-Takeuchi conjecture, a problem in harmonic analysis unsolved for 40 years.
- She constructed a counterexample using fractals and careful arrangement, challenging a widely believed conjecture.
- Cairo took advanced math classes at UC Berkeley while still in high school, after moving from the Bahamas.
- The conjecture is part of harmonic analysis, which breaks down functions into simpler components like sinusoidal waves.
- Cairo's work was presented at the prestigious El Escorial Meetings, her first international scientific trip.
- She enjoys teaching and public speaking, aiming to help others and share her love for mathematics.
- Cairo will begin her Ph.D. at the University of Maryland under the supervision of Professor Ruixiang Zhang.
- Her journey into advanced mathematics began with self-study and participation in online math programs during the pandemic.
Million Times Million
10 months ago
- The author grew up using the long scale system for naming large numbers, where a million times a million is a billion.
- The long scale system uses prefixes (bi-, tri-, etc.) to indicate how many times a million is multiplied by itself, making it intuitive.
- Upon entering university and the tech world, the author discovered the short scale system, where a million times a million is a trillion, causing initial confusion.
- The short scale counts additional thousands multiplied onto one thousand, with prefixes representing the number of these multiplications.
- The short scale is now the dominant convention in technology, computing, finance, and most English-speaking contexts.
- Despite adapting to the short scale, the author retains nostalgia for the long scale's elegance and simplicity.
Product of Additive Inverses
10 months ago
- Negative times negative equals positive is a fundamental rule in mathematics, applicable beyond numbers to algebraic structures like rings.
- The article provides an illustration using the distributive property to show why the product of two negatives must be positive.
- Rings are algebraic structures defined by seven axioms, including associativity, commutativity of addition, and distributivity of multiplication over addition.
- Closure under addition and multiplication is inherent in the definition of binary operations in rings.
- Theorems prove key properties in rings: the inverse of an inverse is the original element, multiplication by zero yields zero, and multiplication by additive inverses follows specific rules.
- The product of additive inverses in any ring results in the product of the original elements, formalizing the rule that negative times negative equals positive.
- These theorems generalize familiar arithmetic rules to a wide range of algebraic systems, demonstrating the power of abstract algebra.
New Sphere-Packing Record Stems from an Unexpected Source
10 months ago
- The sphere-packing problem involves arranging balls in a box as efficiently as possible, with applications in cryptography and communication.
- Johannes Kepler conjectured the optimal 3D sphere packing in the 17th century, but it took 400 years to prove.
- Mathematicians still don't know the optimal packing in higher dimensions, except for dimensions 8 and 24.
- Boaz Klartag recently set a new sphere-packing record using an old technique abandoned decades earlier.
- Klartag's method uses convex geometry to improve ellipsoid-based packing, achieving significant efficiency gains in high dimensions.
- His work revives debates on whether optimal packings should be ordered (lattice-based) or disordered.
- Klartag's result suggests symmetry and order might be key to optimal high-dimensional packings.
- The breakthrough has potential implications for cryptography and communications, sparking excitement among researchers.
- Klartag aims to reconnect convex geometry and lattice theory, fields that have grown apart over time.
Infinity is not a big number
10 months ago
- The text discusses incomprehensibly large numbers like Knuth arrows, Ackermann numbers, Graham's number, and TREE(3), which defy human conception.
- A Buddhist analogy is used to illustrate the vastness of these numbers, comparing them to the improbability of a turtle surfacing inside a floating tube in the ocean.
- The sum of all natural numbers (1 + 2 + 3 + ...) is mentioned as equating to -1/12, a result derived by mathematicians like Euler and Ramanujan and used in quantum physics.
- The author reflects on the nature of infinity, realizing it's not just an unreachably large number but a concept beyond current understanding, evoking a sense of wonder.
- There's optimism that humanity may one day comprehend infinity, suggesting it's not inherently unknowable, just beyond our current grasp.
Inverse Triangle Inequality
10 months ago
- The Inverse Triangle Inequality is a fundamental concept in mathematics, emphasizing the efficiency of direct paths over detours.
- This inequality is versatile, applicable in various contexts like Manhattan distance and Levenshtein distance between DNA strings.
- In software engineering, the principle is adapted to advocate for smaller, incremental changes rather than large, complex ones.
- Smaller commits and pull requests are recommended to speed up the review process and focus discussions on meaningful changes.
- Refactoring can be broken down into simpler steps, such as threading parameters first before changing logic, to manage complexity.
- Smaller, more frequent releases reduce stress and risk, making it easier to handle issues and upgrades.
- The 'Extra Mile' principle suggests taking time to explore solutions thoroughly to avoid local optima and ensure long-term code quality.
- Combining exploratory work with incremental refinements can lead to better outcomes by allowing for immediate rewrites and different perspectives.

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#mathematics

Maypole Dance of Braid Like Groups

When do girls fall behind in maths? Gigantic study pinpoints the moment

The Hat, the Spectre and SAT Solvers (2024)

Think of a Number

Elliptic Curves as Art

Mathematicians discover a new way to find prime numbers

Is Mathematics Mostly Chaos or Mostly Order?

A new pyramid-like shape always lands the same side up

Google DeepMind team up to solve the Navier-Stokes million-dollar problem

BusyBeaver(6) Is Quite Large

The Unreasonable Effectiveness of Mathematical Experiments

Does Form Shape Function?

How Does Graph Theory Shape Our World?

The world of Voronoi diagrams (2021)

Hannah Cairo has solved the Mizohata-Takeuchi conjecture

Million Times Million

Product of Additive Inverses

New Sphere-Packing Record Stems from an Unexpected Source

Infinity is not a big number

Inverse Triangle Inequality