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Last updated Oct 20, 2023
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The best mathematicians on Earth spent 50 years failing to crack 1 tangled knot. A graduate student solved it in under a week, and thought it was just her homework.
For half a century, the Conway knot had 1 stubborn question hanging over it: is it a "slice" of a knot in 4 dimensions, or not? Every tool the field invented broke on this 1 knot. Then in 2018 Lisa Piccirillo heard about it at a conference, sat down that evening, and just started playing with it.
She did not even take it seriously. She called it homework and refused to work on it during the day, because she did not think it counted as "real math."
Under a week later, she had the answer. When she casually mentioned it to a professor, he started yelling at her: why aren't you more excited, this is going straight to the top journal in the world.
Here is what is worth saving from how she did it:
Do not attack the monster head on. Everyone tried to measure the Conway knot directly and failed. She built a brand new knot that secretly shared the same deep 4D structure, then solved the easy twin instead.
Not knowing it was "impossible" was an advantage. She had not absorbed 50 years of "this can't be done," so she just tried.
Sideways beats stronger. She did not invent a more powerful hammer. She changed the angle.
The line that should stick with you: half the reason hard problems stay unsolved is that the experts already agreed they were too hard to touch. She wasn't smarter than everyone who came before. She just didn't flinch. That proof took her from student to MIT professor.
The full lecture where this exact knot comes up is free, sitting right here in your feed.
Save this. The next time everyone tells you something can't be done, you'll want to reread how she did it anyway.
Samuel Cohen, the Oxford mathematician who proved there is exactly one correct way to measure uncertainty, and nobody has found a way around it since:
"I used to think there could be many reasonable ways to measure uncertainty. Then I proved that only one survives three simple rules, and the moment I saw it collapse into a single formula, it changed how I look at every unknown since."
this is the exact proof sitting quietly underneath every signal a trading desk treats as informative, and almost nobody outside a math department has ever seen the three rules that force it into existence.
strip away the notation and the setup is simple. surprise should shrink continuously as an event gets more likely, and the surprise of two independent events happening together should just add up.
those two rules sound harmless. but they're strict enough that only one function in all of mathematics can satisfy them, and it happens to be a logarithm, not something anyone would guess by intuition alone.
nobody calling a signal "highly informative" out loud credits the proof that defines what informative even means.
zoom out to what this quietly powers today. every model comparing a live market to its own prediction, every benchmark measuring how far a forecast drifted from reality, is built on this exact same formula for surprise, just wearing a different name.
the industry sells "information edge" like it's a vibe, something a good trader just feels. but the entire concept has a precise, provable definition, and almost nobody using the term could derive it from scratch.
surprise was never just a feeling. it was the only function that could survive three simple rules, and everything else quietly collapsed under them.
Bookmark this legend and follow @mindarchx for more alpha.
Visualization of the Poincaré sphere.
Showing how every fully polarized state of light can be represented by a point on a unit sphere. Linear polarization lies on the equator, circular polarization at the poles, and elliptical polarization everywhere in between. The sphere maps polarization states; it is not the physical path of light.
Minda Presiden PAS
DAP MEMIMPIN PH MENYAMBUNG KERJA PENJAJAH HAPUSKAN MELAYU ISLAM
Tercetusnya kenyataan dan tindakan yang secara terbuka mencabar kedudukan Islam dalam Perlembagaan Persekutuan, memperlekehkan kedudukan Raja-raja Melayu yang dihilangkan kedaulatannya oleh penjajah, melemahkan kedudukan bahasa Melayu yang dinobatkan menjadi Bahasa Kebangsaan, membela golongan yang murtad dalam kalangan penganut Islam dan memberi tekanan kepada mereka yang memeluk Islam, serta menentang langkah pelaksanaan syariat oleh beberapa buah kerajaan negeri.
Malah, mereka juga meletakkan kalangan bukan Islam, khususnya daripada DAP dalam kementerian penting yang menjadi ancaman kepada Melayu dan Islam, serta merancang agenda politik seperti mengadakan pilihan raya Pihak Berkuasa Tempatan (PBT) dan lain-lain.
Pilihan raya PBT di Malaysia adalah proses memilih ahli majlis bandar atau perbandaran, yang telah dimansuhkan pada tahun 1960-an dan kini dilantik secara rasmi. Namun, ia menimbulkan perdebatan tentang isu demokrasi serta keutamaan pentadbiran. Pilihan raya ini kemudiannya dibatalkan kerana masih wujud kesan penyusunan masyarakat majmuk yang dilaksanakan oleh penjajah yang pernah dilihat menjadi sebab meletupnya Peristiwa 13 Mei 1969, sejarah hitam yang tidak boleh diulang lagi. Kini, ia menjadi agenda DAP untuk menguasai bandar yang didominasi oleh kaum Cina.
PH bersama DAP menggalakkan penggunaan bahasa Inggeris dan bahasa kaum bagi menyingkirkan bahasa Melayu yang telah menjadi bahasa perantaraan masyarakat majmuk di Tanah Melayu.
DAP juga meletakkan calon Islam di kawasan Melayu sehingga berwajah pakaian Islam, di samping calon bukan Islam oleh parti-parti lain dalam komponen Pakatan Harapan.
Akhirnya, DAP yang mendominasi Pakatan Harapan secara terbuka itu mendedahkan perjuangan sekularismenya yang ekstrem dan konsep Malaysian Malaysia yang diperjuangkannya. Konsep ini mengandungi teori luar tabii yang menentang fitrah ketetapan Allah tentang kewujudan bangsa.
Baca selanjutnya:
https://t.co/1pcX9XgsoL
SHE SOLVED A 50-YEAR-OLD MATH PROBLEM IN UNDER A WEEK BECAUSE SHE DIDN'T THINK IT COUNTED AS REAL WORK
lisa piccirillo, 2018, graduate student at UT Austin. she hears about the Conway knot at a conference and starts poking at it in the evenings.
her own words: she wouldn't let herself work on it during the day, because she didn't consider it to be real math. she thought of it like homework.
the problem had stood since the 1960s. of every knot with twelve or fewer crossings, mathematicians had determined "sliceness" for all of them except one. that one had a sculpture of itself on a gate at Cambridge's Isaac Newton Institute.
she cracked it in under a week.
the method is the elegant part. she built a different knot that shares the same four-dimensional trace as Conway's, proved hers wasn't slice, and since trace siblings share slice status, Conway's isn't either.
then she mentioned it casually to Cameron Gordon, a senior topologist. he started shouting "why aren't you more excited?" and said it was going to the Annals immediately.
it did. she was at MIT fourteen months out of grad school.
and the reason her field exists at all is genuinely strange.
four dimensions is the only one that breaks. flat n-dimensional space has exactly one smooth structure for every value of n. except four, where there are uncountably infinitely many.
not three. not five. not eleven. four.
nobody fully knows why.
In 1928, Paul Dirac combined quantum mechanics with Einstein's special relativity to describe particles moving at nearly the speed of light. His equation did more than explain the electron. It also predicted the existence of the positron, the electron's antimatter twin, which was discovered by Carl Anderson in 1932.
The Dirac equation is written as:
(iħγ^μ∂_μ − mc)ψ = 0
The gamma matrices make the equation consistent with relativity while naturally describing spin 1/2 particles. Today, the Dirac equation remains a foundation of quantum field theory and modern particle physics.
Ten million people have watched an MIT professor accidentally destroy the executive coaching industry.
He filmed the lecture once in January 2018 and died eighteen months later.
Executive coaches charge fifteen thousand dollars a session to teach a third of what he covered in one hour for free.
His name was Patrick Winston. He ran the MIT Artificial Intelligence Laboratory from 1972 to 1997 and wrote the AI textbook every computer science major in the world read for thirty years.
Every January for four decades, he gave a lecture called "How to Speak."
His entire framework fits on a napkin.
Do not read. Be in the image. Keep images simple. Eliminate clutter. Start with an empathetic connection. End with a punch line the audience can repeat over dinner. Never open with a joke. Never end with "thank you."
That last rule alone has probably cost the executive coaching industry a hundred million dollars.
"Your success in life will be determined largely by your ability to speak, your ability to write, and the quality of your ideas. In that order."
That is the actual opening line of the lecture. Winston believed it strongly enough to spend fifty years teaching computer scientists how to talk.
Founders spend $80,000 on an MBA and then hire a communications coach to teach them the same material Winston filmed once for free. Engineers write brilliant code and lose promotions to teammates who watched this lecture on the train.
The lecture is free on MIT OpenCourseWare. The textbook is free on his page.
Winston died in 2019. Almost none of the ten million viewers have actually implemented the four rules on the napkin.
The napkin is free. The willingness to actually use it in your next meeting is the entire edge.