Top Mathematics discussions

NishMath - #math

@Trebor //
References: Trebor
Recent discussions in theoretical computer science and programming have touched upon diverse topics, ranging from type theory for SDG (Sustainable Development Goals) to the complexities encountered in programming. One thread explored the characteristics a type theory for SDG should possess, suggesting it should include a judgmentally commutative ring, possibly a Q-algebra, where neutral forms of type R are polynomials with other neutral forms as indeterminates. Participants believe such a system would have decidable typechecking.

A common sentiment shared among programmers, particularly those using languages with dependent types like Rust, is the initial hurdle of satisfying the compiler's requirements. Some have described the experience as an engaging puzzle that can involve spending considerable time to prove the validity of their code. The discussion also addressed the subjective nature of "complexity" in programming, suggesting it is a term often used to dismiss unfamiliar concepts rather than a concrete measure of inherent difficulty.

In related news, Microsoft’s Krysta Svore has announced geometric error-correcting codes as a potential advancement toward practical quantum computing. These codes utilize high-dimensional geometry to enhance performance, potentially leading to more efficient encoding and logical operations with fewer qubits. The approach builds on topological error correction, employing a mathematical method called Hermite normal form to reshape the grid, resulting in substantial reductions in qubit count and faster logical clock speeds. This geometric reshaping results in substantial reductions in qubit count. In one notable case, they achieved six logical qubits using just 96 physical qubits, which is a 16-to-1 ratio that would mark a significant improvement over standard two-dimensional codes.

Recommended read:
References :
  • Trebor: A type theory for SDG should contain a judgmentally commutative ring (or Q-algebra?), so the neutral forms of type R are polynomials whose indeterminates are other neutral forms. Seems to have decidable typechecking to me.

@martinescardo.github.io //
The mathematics community is buzzing with activity, including upcoming online events and ongoing discussions about research methodologies. A significant event to watch for is the online celebration marking the 40th anniversary of Elliptic Curve Cryptography (ECC) on August 11, 2025. This event will commemorate the foundational work of Victor Miller and Neal Koblitz in 1985. It is anticipated to be a very important event for those in the cryptography community and to those who work with elliptic curves.

The ECC celebration will feature personal reflections from Miller and Koblitz, alongside lectures by Dan Boneh and Kristin Lauter, who will explore ECC's broad impact on cryptography and its unforeseen applications. The history of ECC is used as a good example of how fundamental research can lead to unexpected and practical outcomes. This serves as a good way to promote blue skies research.

In other news, mathematicians are actively discussing the use of formal methods in their research. One Mathstodon user described using LaTeX and Agda in TypeTopology for writing papers and formalizing mathematical remarks. They found that formalizing remarks in a paper could reveal errors in thinking and improve results, even in meta-mathematical methodology. This shows how computational tools are increasingly being used to verify and explore mathematical ideas, highlighting the practical utility of pure math skills in applied contexts.

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@phys.org //
References: bigthink.com , phys.org
Recent research is challenging previous assumptions about the composition and structure of the smallest galaxies. Traditionally believed to be dominated by dark matter due to the expulsion of normal matter through stellar winds and radiation during star formation, new evidence suggests that supermassive black holes may play a more significant role than previously thought. A recent study indicates that Segue 1, known as the most dark matter-dominated galaxy, might harbor a supermassive black hole at its center, potentially altering our understanding of galactic dynamics in low-mass systems. This proposition offers an alternative explanation for the observed gravitational effects, suggesting that these central black holes could be anchoring these tiny galaxies.

The realm of statistical analysis is also undergoing significant advancements. Mathematician Tyron Lardy has pioneered a novel approach to hypothesis testing, utilizing e-values instead of the conventional p-values. E-values, representing 'expected value', provide greater flexibility, particularly during mid-study analysis when adjustments to data collection or analysis plans are necessary. Unlike p-values, which require conclusions to be drawn only after all data is gathered to maintain statistical validity, e-values remain statistically sound even with modifications to the research process. This advancement holds promise for fields like medicine and psychology, where complex situations often demand adaptable data handling techniques.

The development of e-values is based on the concept of betting, where the e-value signifies the potential earnings from such bets, offering quantifiable evidence against the initial assumption. This approach allows researchers to assess whether an assumption still holds true. While the general method for calculating optimal e-values can be intricate, its flexibility and robustness in handling data adjustments offer a valuable tool for scientific research, enhancing the reliability and adaptability of hypothesis testing in various disciplines.

Recommended read:
References :
  • bigthink.com: bigthink.com/starts-with-a-bang/supermassive-black-holes-tiniest-galaxies
  • phys.org: Smarter hypothesis testing with statistics: How e-values can improve scientific research

MAA@maa.org //
References: maa.org
The Mathematical Association of America (MAA) has announced the recipients of the 2025 awards for expository mathematical writing in MAA publications. The awards recognize outstanding contributions to mathematical literature. Jordan S. Ellenberg was awarded the Chauvenet Prize for his work "Geometry, Inference, Complexity, and Democracy," which appeared in the Bulletin (New Series) of the American Mathematical Society. Ellenberg's work explores the challenge of fairly dividing democratic polities into legislative districts, demonstrating the practical application of mathematics to societal issues. Ismar Volíc of Wellesley College, received the Euler Book Prize for his book "Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation," which brings complex topics, such as voting theory, apportionment, gerrymandering, and the Electoral College, to life.

Awarded individuals are reciving either the Chauvenet Prize, the Euler Book Prize, the Daniel Solow Author’s Award, the George Pólya Awards, the Paul R. Halmos–Lester R. Ford Awards, the Trevor Evans Award, or the Carl B. Allendoerfer Awards. Ellenberg's article, drawn from his 2020 Current Events Bulletin lecture, showcases how mathematical approaches can measure fairness in democratic processes. Volíc's book makes complex topics accessible to readers, highlighting the crucial role of mathematics in collective decision-making, and providing essential insights without political bias. Both works exemplify clear and engaging writing, effectively communicating intricate mathematical ideas to a wider audience.

As summer approaches, Denise Gaskins is offering discounts on her math game books at the Playful Math Store. This presents an opportunity for families and educators to enhance mathematical learning through playful activities. Gaskins' "Math You Can Play" series offers math games sorted by topics traditionally taught at various age levels, with teaching tips and advice aimed at parents and teachers. Her new series, "Tabletop Math Games Collection," also covers the same mathematical topics. These books are designed for direct use by players of all ages, making them ideal for spontaneous math play. These books are available in both physical and digital formats, providing flexibility for use in math centers, homeschool co-op classes, or at home.

Recommended read:
References :
  • maa.org: Mathematicians Awarded for 2025 Expository Mathematical Writing in MAA Publications

@Math Blog //
References: medium.com , Math Blog
Percentages are a fundamental concept in mathematics, representing a fraction with a denominator of 100. The term "percent" comes from the Latin phrase "per centum," meaning "by the hundred". A percentage is denoted by the symbol %, and is used to express a part of a whole. For example, if a student scores 65 percent on a test, it means they obtained 65 marks for every 100 marks. Understanding percentages is crucial as they frequently appear in daily life, from calculating discounts to understanding statistics.

Percentages offer a standardized way to compare different quantities or proportions. To convert a fraction to a percentage, the goal is to express the fraction with a denominator of 100. If David secures 475 marks out of 500, this can be converted to a percentage by dividing both the numerator and the denominator by 5, resulting in 95/100, or 95%. Conversely, 9% is equivalent to 9/100. Visual representations can also aid in understanding percentages, such as imagining a battery made up of 100 small cells, where each cell represents 1%. If all cells are charged then the battery is at 100%.

In addition to understanding percentages, other mathematical concepts like linear regression are important in more advanced applications. Linear regression is a fundamental machine learning model used to find correlations between variables and make predictions. For instance, it can be used to predict ice cream sales based on temperature data. The model identifies a relationship between the input feature (temperature) and the target feature (ice cream sales) and uses a general line to make predictions. The equation of this line, f(x) = mx + b, where 'm' is the slope and 'b' is the y-intercept, helps in understanding how changes in the input feature affect the predicted output.

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  • medium.com: Medium article on The Secrets of Linear Regression Uncovered
  • Math Blog: Explanation of Digital SAT Math Problems

@Math Blog //
Mathematical concepts and their applications are gaining increased attention, with recent explorations into diverse areas. A blog post discusses the fundamental differences between mathematical and statistical reasoning, using the example of predicting days with the fewest noninduced births. Researchers are also delving into methods for eliminating parameters in parametric equations. A podcast delves into the philosophy of mathematics and set theory, examining the nature of mathematics itself.

The article "Eliminating the Parameter in Parametric Equations" provides a guide for expressing relationships between variables `x` and `y` when they are defined in terms of a parameter `t`. It explains the process of removing the parameter to obtain a direct equation between `x` and `y`, showcasing examples and solutions. Furthermore, there is a discussion on Charlotte Mason's approach to mathematics using living books as a method of teaching.

Python's dominance in AI and machine learning is a significant development. An article explores the factors behind this, highlighting Python's readability, extensive libraries like NumPy, Pandas, Scikit-learn, TensorFlow, and PyTorch, and the role of AI hype in its rise. The Church of Logic podcast also featured a discussion with Joel David Hamkins on the philosophy of mathematics and set theory, particularly exploring differing perspectives on the nature of mathematics.

Recommended read:
References :
  • Math Blog: This webpage explains how to eliminate the parameter in parametric equations.
  • denisegaskins.com: An article discussing Charlotte Mason's approach to mathematics using living books.

@www.newtonproject.sussex.ac.uk //
References: Xi'an's Og , Pat'sBlog , Pat'sBlog ...
Recent blog posts are delving into a variety of mathematical topics, offering insights and explorations across different areas of the field. These posts cover historical aspects of mathematics, examine specific mathematical concepts, and explore the connections between mathematics and other disciplines. This collection of diverse content aims to provide readers with a broader understanding and appreciation of mathematics.

The blog posts include diverse mathematical items. For example, one post references Gemma Frisius' "Arithmeticae Practicae Methodus Facilis" (1540) and its entry in *MAA Mathematical Treasures. Another commemorates April 13 as "On This Day in Math," highlighting mathematical facts associated with the number 103. This includes its unique properties as a prime number and its presence in Ramanujan's mathematical explorations. Furthermore, the blog explores historical events like the coining of the word "microscope" in 1620 and Lord Brouncker's published mathematical result in 1668.

From statistical physics to number theory, these blogs showcase the versatility and interdisciplinary nature of mathematical thought. One blog even mentions using statistical physics concepts to analyze election results. These blog postings aim to engage readers with a range of mathematical subjects, from historical figures and publications to contemporary applications and connections.

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  • Xi'an's Og: Blog post about various mathematical topics.
  • Pat'sBlog: Blog post about various mathematical topics.
  • Stats Chat: Blog post about various mathematical topics.
  • Pat'sBlog: Blog post discussing the history of mathematical induction and the origin of the term.
  • Pat'sBlog: Pandigital Primes

@primes.utm.edu //
This week saw a flurry of mathematical activity, highlighted by the 238th Carnival of Mathematics, organized by Aperiodical. The carnival showcases a variety of submissions and mathematical art, focusing on the number 238 itself. Noteworthy facts about 238 include that it is 2 × 7 × 17, the sum of the first 13 primes, and a "triprime". The carnival also encourages exploration beyond pure mathematics, with community members contributing insights linking the number to uranium isotopes, birth minutes, and even hexadecimal representations. It also shines a light on #MathArtMarch, with examples of crochet, coding, and painting from around the world.

Continuing the daily exploration of numbers, several interesting facts and events were highlighted for April 6th, 7th, 8th and 10th. The number 96, the 96th day of the year, was examined for its unique properties, such as being the smallest number expressible as the difference of two squares in four different ways. Events like Euler's first paper on partitions (April 7th, 1741) and Al-Biruni's observation of a solar eclipse in 1019 were also noted, linking mathematical concepts to historical contexts. Also, the number 97 has been noted as the 97th day of the year, where 97 is the largest prime that we can ever find that is less than the sum of square of its digits.

In recreational mathematics, a "Salute" game for reinforcing multiplication and division was featured, emphasizing the inverse relationship between these operations. Additionally, the concept of "truncatable primes" was explored through a game where players create strings of prime numbers by adding digits to either end of a number. The number 91 was discussed as the 91st day of the year where 10 n + 91 and 10 n + 93 are twin primes for n = 1, 2, 3 and 4. Finally, highlighting mathematics beyond academia, James Abram Garfield, a former Congressman and mathematician, was mentioned for his original proof of the Pythagorean Theorem, illustrating the interdisciplinary nature of mathematics.

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Greg Bock@The Quantum Insider //
References: The Quantum Insider
Quantum computing has taken a significant leap forward with Phasecraft's development of a novel quantum simulation method called THRIFT (Trotter Heuristic Resource Improved Formulas for Time-dynamics). This breakthrough, detailed in a recent *Nature Communications* publication, drastically improves simulation efficiency and lowers computational costs, bringing real-world quantum applications closer to reality. THRIFT optimizes quantum simulations by prioritizing interactions with different energy scales within quantum systems, streamlining their implementation into smaller, more manageable steps.

This approach allows for larger and longer simulations to be executed without the need for increased quantum circuit size, thereby reducing computational resources and costs. In benchmarking tests using the 1D transverse-field Ising model, a widely used benchmark in quantum physics, THRIFT achieved a tenfold improvement in both simulation estimates and circuit complexities, enabling simulations that are ten times larger and run ten times longer compared to traditional methods. This development holds immense promise for advancements in materials science and drug discovery.

Separately, mathematicians have achieved a breakthrough in understanding and modeling melting ice and other similar phenomena through a new proof that resolves long-standing issues related to singularities. A powerful mathematical technique used to model melting ice and other phenomena had been hampered by “nightmare scenarios.” A new proof has removed that obstacle. This new proof addresses concerns about "nightmare scenarios" that previously hindered the analysis of these processes, ensuring that singularities do not impede the continued evolution of the surface being modeled. The resolution, described in Quanta Magazine, allows mathematicians to more effectively assess the surface's evolution even after a singularity appears.

Finally, researchers at Cornell University have introduced a novel data representation method inspired by quantum mechanics that tackles the challenge of handling big, noisy data sets. This quantum statistical approach simplifies large data sets and filters out noise, allowing for more efficient analysis than traditional methods. By borrowing mathematical structures from quantum mechanics, this technique enables a more concise representation of complex data, potentially revolutionizing innovation in data-rich fields such as healthcare and epigenetics where traditional methods have proven insufficient.

Recommended read:
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  • The Quantum Insider: Press RELEASE — In a breakthrough that puts us a step closer to real-world quantum applications, Phasecraft – the quantum algorithms company – has developed a novel approach to quantum simulation that significantly improves efficiency while cutting computational costs. The method, known as THRIFT (Trotter Heuristic Resource Improved Formulas for Time-dynamics), optimizes the quantum.