Pustam | पुस्तम | পুস্তম🇳🇵

Brilliant Minds Born in Budapest 🇭🇺

Budapest 🇭🇺 is home to barely two million people today, yet its intellectual legacy has shaped the modern world far beyond its size.

So many extraordinary scientists emerged from Budapest in the early twentieth century that fellow physicists jokingly invented the "Martians" hypothesis: the only explanation, they said, was that these brilliant Hungarians were actually visitors from Mars, hiding in plain sight, speaking a strange language (Hungarian), and quietly transforming science.

Behind the joke lies an astonishing reality. Budapest produced an unparalleled concentration of scientific talent. From John von Neumann, whose ideas underpin modern computing, to Paul Erdős, one of the most prolific mathematicians in history; from Nobel laureates Eugene Wigner, Dennis Gabor, and Albert Szent-Györgyi to visionaries like Leo Szilard, Edward Teller, Theodore von Kármán, George Pólya, László Lovász, Dénes Kőnig, Rózsa Péter, Béla Bollobás, and Peter Lax, these minds revolutionized mathematics, physics, computer science, aerospace engineering, medicine, and beyond.

What made Budapest so exceptional? Outstanding schools, a rich mathematical tradition, a flourishing intellectual culture, and a generation that inspired one another to pursue excellence. Many later emigrated to the United States and Western Europe, where their ideas reshaped universities, research institutions, and modern technology.

The "Martians" were never aliens, of course. They were simply some of the greatest scientific minds humanity has ever produced, and remarkably, many of them were born in the same city.

#Budapest #Hungary #Martians #TheMartians #Science #STEM #Mathematics #Physics

Hilbert wakes up 1,000 years later.

“Has the Riemann hypothesis been proven?”

“Yes. But there’s something else you should know…”

It wasn’t proven by a human.

Mathematics is the canary in the coal mine of any education system: It's a delicate pyramid that crumbles if the base is not strong.

Using IMO results, here are the countries that improved most from 2006 to 2026 and those that declined the most. Using rank percentile within each year’s field:

Most improved
🇸🇦 Saudi Arabia: 2.2% → 69.8% (+67.6pp)
🇲🇾 Malaysia: 22.5% → 83.6% (+61.1pp)
🇰🇬 Kyrgyzstan: 14.6% → 70.7% (+56.1pp)
🇧🇩 Bangladesh: 11.2% → 67.2% (+56.0pp)
🇮🇳 India: 61.8% → 94.8% (+33.0pp)

Biggest declines
🇲🇩 Moldova: 91.0% → 55.2% (−35.8pp)
🇮🇹 Italy: 87.6% → 68.1% (−19.5pp)
🇫🇮 Finland: 57.3% → 39.7% (−17.6pp)
🇩🇪 Germany: 96.6% → 83.6% (−13.0pp)
🇹🇼 Taiwan: 89.9% → 80.2% (−9.7pp)

#Mathematics #MathEducation #Education #STEM #IMO #InternationalMathematicalOlympiad #MathOlympiad #EducationPolicy #Learning #AcademicExcellence #DataVisualization #EducationResearch #GlobalEducation #SchoolEducation #STEMEducation #ProblemSolving #Innovation #FutureSkills #StudentSuccess #EducationalDevelopment #MathematicsMatters #Statistics #GlobalRankings #India #SaudiArabia #Malaysia #Bangladesh #Kyrgyzstan #Finland #Germany

Green's Function ✍️

It explains how mathematicians and physicists solve complex equations by first understanding the response to the simplest possible disturbance. Imagine tapping the surface of a calm pond with a single pebble. Ripples spread outward in every direction, carrying information about how the water reacts to that one tiny push. Once this basic response is known, the effect of any larger or more complicated disturbance can be built by combining countless such ripples.

A Green's function acts as this fundamental response. It describes how a system reacts to a single point source located at a specific position. Whether the system is carrying heat, transmitting sound, propagating light, or generating electric fields, the Green's function captures exactly how that influence spreads through space.

Instead of solving the entire problem at once, scientists break a complicated source into many tiny point sources. Each point produces its own response, and all of these responses are added together. This principle of superposition transforms an otherwise difficult equation into a collection of simple, manageable solutions.

Green's functions have become one of the most powerful tools in mathematics and physics. They are used to solve differential equations, predict electromagnetic and gravitational fields, analyse quantum systems, model heat flow, and understand wave propagation. By revealing how a system responds to the smallest possible input, they provide a universal framework for understanding even the most complex physical phenomena.

I wasn't ready for it, lol.

The book: Introduction to Linear Algebra by Gilbert Strang, the sixth edition

Niels Abel was born 224 years ago, on August 5, 1802. At just 21, he proved that no general algebraic formula can solve equations of degree 5, a problem that had challenged mathematicians for roughly 250 years.

Short on money, Abel paid the printer himself and condensed his proof into just six pages to keep costs down. He sent a copy to Gauss, but never received a response.

Abel died of tuberculosis at just 26. Two days later, a letter arrived announcing that he had been appointed professor in Berlin.

A breakthrough recognized too late.

#NielsAbel #Mathematics #MathHistory #HistoryOfMathematics #Mathematician #Algebra #AbstractAlgebra #QuinticEquation #Quintic #GaloisTheory #MathematicalHistory #MathFacts #ScienceHistory #STEMHistory #STEM #MathematicalBreakthrough #MathematicalGenius #Norway #NorwegianHistory #CarlFriedrichGauss #Equations #NumberTheory #PureMathematics #MathEducation #ScienceCommunication #OnThisDay #August5 #History #MathematicsLovers #MathTwitter

Erdős problems resolved by year, 1930–2026.

The hierarchy of mathematical spaces

These diagrams illustrate the hierarchical relationships and nesting of various mathematical spaces used in functional analysis and geometry. They demonstrate how specific structures, such as Hilbert and Banach spaces, are specialized subsets of broader categories like normed linear spaces and metric spaces. The visuals highlight the essential properties required for each classification, ranging from basic topological sets to complex systems involving inner products and completeness. By organizing these concepts into flowcharts and Venn diagrams, the sources clarify how adding constraints like distance, magnitude, or orthogonality transforms one type of space into another. Ultimately, the collection provides a comprehensive map of how abstract vector spaces relate to concrete examples like Euclidean space.

#educationalcontent #mathematics #functionalanalysis #mathematicalspaces

Neural networks train by constructing a computational graph in the forward pass, chaining basic operations such as x × y × z into composite functions of the inputs.

Derivatives of the output with respect to every input are then obtained in the backward pass by applying the chain rule at each node, where local gradients multiply: ∂f/∂y = ∂f/∂(x × y) × ∂(x × y)/∂y and ∇ₓz = ∇ₓy · ∇ᵧz, with intermediate values stored along the graph.

This process updates the parameters of convolutional networks that classify medical scans to flag early-stage tumors in hospital imaging systems.

#NeuralNetworks #NeuralNets #DL #ML #DeepLearning #MachineLearning #AI #NNs #CNN

Some scientists whose lives ended before the age of 40:

🇫🇷 Évariste Galois — 20
🇳🇴 Niels Henrik Abel — 26
🇮🇳 Srinivasa Ramanujan — 32
🇬🇧 William Kingdon Clifford — 33
🇫🇷 Blaise Pascal — 39
🇩🇪 Bernhard Riemann — 39
🇫🇷 Sadi Carnot — 36
🇬🇧 Ada Lovelace — 36
🇩🇪 Heinrich Hertz — 36
🇷🇺 Alexander Friedmann — 37
🇫🇷 Augustin-Jean Fresnel — 39
🇬🇧 Rosalind Franklin — 37
🇮🇹 Ettore Majorana — 31

Fields Medals by country, from 1936 to 2026 🥇🏅

China 🇨🇳 enters the top tier with 2 new medalists in 2026!

#FieldsMedal #FieldsMedalists #Medalists #Medal #Medalist #Math #Mathematics #EastAsia #China

Analysis of the xG (expected goals) created and conceded per game for each R16 team, along with Germany and the Netherlands (R32).

The statistics suggest that 🇪🇸 Spain's World Cup victory was driven primarily by defensive dominance rather than overwhelming attacking output. Clearly, they had the best balance between attacking output and defensive solidity (both efficient and defensive) throughout the tournament. They were technically the most complete team and thoroughly deserved to be crowned World Champions.🌎🥇🏆

• Lowest xG conceded in the tournament: 0.30 per match.

• Highest xG difference among all semi-finalists: +1.65.

• Held both 🇫🇷 France (0.31 xG) and 🇦🇷 Argentina (0.22 xG), the other finalists and one of the strongest attacking teams, to almost no chance in the knockout rounds.

While 🇫🇷 France were the most prolific attacking side and 🇧🇷 Brazil (eliminated in R16) led the tournament in xG created, 🇪🇸 La Roja consistently controlled matches at both ends of the pitch. Their ability to suppress opponents' chance creation was the defining statistical feature of the 2026 FIFA World Cup.

#xG #ExpectedGoals #FootballAnalytics #FootballStats #SoccerAnalytics #FIFAWorldCup #WorldCup2026 #Spain #LaRoja #SpainNationalTeam #WorldChampions #Football #Soccer #MatchAnalysis #DataAnalysis #SportsAnalytics #TacticalAnalysis #FootballTactics #KnockoutStage #DefensiveMasterclass #DefensiveSolidity #EliteDefense #AttackVsDefense #SpainFootball #France #Argentina #Brazil #WorldCupStats #FootballData #SoccerStats

Mathematics behind Artemis II

On May 20, 2026, OpenAI made an announcement that shook the mathematical world. An internal AI model, one not available to the public, had come up with a counterexample to the “unit distance” problem, a conjecture made in 1946 by Paul Erdős, the prolific, itinerant Hungarian mathematician. Erdős posed thousands of questions, but this one was special: It was both simple to explain and mathematically deep. It was the first historically significant proof to come from an AI model. Though the model’s result wasn’t definitive, human mathematicians would substantially improve on it within weeks, it was innovative, bringing in ideas from a distant branch of math that no one had successfully applied to this problem before. And it was influential: Within a few days, related techniques were used to solve other important problems. Then on August 1, OpenAI announced that an unreleased model named Astra made 10 additional mathematical advances, including finding solutions to three more problems posed by Erdős. Many mathematicians have hailed developments such as these as a phase transition in the mathematical capability of AI models. These models are “changing dramatically the way mathematical research is being done,” said Noga Alon of Princeton University, who has solved dozens of Erdős problems over his decades-long career.

https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803

#AI #Erdős #Erdos #ErdősProblems #PaulErdős #OpenAI #Revolution #ArtificialIntelligence #QuantaMagazine #Maths #Math #NumberTheory #MathRevolution

🌪️ Turbulence: The Greatest Unsolved Problem in Classical Physics

Despite centuries of research, turbulence remains one of the deepest mysteries in physics. From the chaotic wake behind an aircraft to swirling hurricanes, ocean currents, combustion, blood flow, and even the birth of stars, turbulent motion is everywhere, yet predicting it precisely remains extraordinarily difficult.

This challenge has fascinated some of history's greatest scientific minds. Horace Lamb famously remarked that, upon reaching heaven, he hoped to finally understand quantum electrodynamics and turbulence, adding that he was "rather optimistic" about the former. Richard Feynman likewise described turbulence as one of the most important unsolved problems in classical physics. A quote often attributed to Werner Heisenberg expresses a similar sentiment, although its authenticity remains uncertain.

The difficulty lies in turbulence's nonlinear nature: tiny changes can produce dramatically different outcomes, with countless interacting vortices spanning an enormous range of scales. While the Navier–Stokes equations describe fluid motion, obtaining complete analytical solutions for turbulent flows remains one of the greatest challenges in physics, mathematics, and computational science.

Understanding turbulence isn't merely an academic pursuit, it has profound implications for aviation, climate modelling, renewable energy, engineering, astrophysics, medicine, and space exploration. Every advance brings us closer to more efficient aircraft, more accurate weather forecasts, cleaner energy systems, and deeper insights into the natural world.

Sometimes, the most familiar phenomena are also the most mysterious.

#Turbulence

This is unironically the mid-term future of research math.

The next generations of models will have taste and the ability to invent new theories. Then math will be fully automated.

Lobachevsky's integral formula
\[\displaystyle
\begin{aligned}
\int_{0}^{\infty} \frac{\sin^{2}x}{x^{2}}\,f(x)\,dx \\[0.5em]
= \int_{0}^{\infty} \frac{\sin x}{x}\,f(x)\,dx \\[0.5em]
= \int_{0}^{\pi/2} f(x)\,dx
\end{aligned}
\]