Lupen

sharing what I learn

[Lupen @0xLupenn]

MIT FILMED THE BELOVED DENIS AUROUX - THE PROFESSOR WHOSE CALCULUS COURSE HAS BEEN STUDIED BY ENGINEERS ACROSS THE WORLD WHO WENT ON TO BUILD BILLION DOLLAR PRODUCTS - PROVE THAT THE DIVERGENCE THEOREM TURNS A SURFACE INTEGRAL THAT TAKES AN HOUR INTO ONE THAT TAKES THIRTY SECONDS

This is Denis Auroux, MIT, 18.02 Multivariable Calculus, Fall 2007. His students consistently call him the clearest mathematical mind they have ever learned from. He opens with one claim - the divergence theorem is not a new idea. It is Green's theorem promoted from two dimensions to three and it turns a surface integral that would take hours into a triple integral that takes minutes.

He starts with the statement. Take a closed surface in three-dimensional space - a sphere, a cube, any shape that has a clear inside and outside. The flux of a vector field through that surface equals the integral of the divergence of the field over the entire enclosed volume. Divergence is just the partial of the x component with respect to x plus the partial of y with respect to y plus the partial of z with respect to z. Three partial derivatives, one number at each point, integrated over a solid region.

Then why this is useful. Computing flux directly requires parameterizing the surface, finding the normal vector, dotting it with the field, and integrating over two variables. For complicated surfaces this is a nightmare. The divergence theorem replaces that surface integral with a volume integral over the inside. For many fields the divergence is a simple constant and the volume integral is just that constant times the volume of the region. A calculation that would take an hour takes thirty seconds.

Then the gravitational field. The field is minus one over rho squared in the radial direction where rho is the distance from the origin. Compute the flux through a sphere of radius a directly and you get a clean answer after some work. Apply the divergence theorem naively and you get a problem - the divergence of this field is zero everywhere except at the origin where it is undefined. The divergence theorem only applies to smooth fields. At the origin the field blows up. The fix is to cut out a tiny sphere around the origin and apply the theorem to the region between the two spheres.

Then the electric field version. This is exactly what Gauss's law in electromagnetism says. The flux of the electric field through any closed surface equals the total charge enclosed divided by epsilon zero. The divergence theorem is why Gauss's law works. The charge at the origin is the singularity. The surface captures its contribution no matter how you draw it.

Watch the moment he computes the flux of xi plus yj plus zk through a sphere of radius a using the divergence theorem in five seconds - divergence is three, volume is four thirds pi a cubed, flux is four pi a cubed - and then notes this would have taken a full page of calculation using the direct method.

A physics undergraduate I know rewatched this lecture before their electromagnetism final. Said it was the first time Gauss's law felt like a mathematical theorem rather than an experimental fact someone had decided to call a law.

Free on YouTube, MIT OpenCourseWare, Creative Commons license.

bookmark this and watch later - after this lecture every closed surface you see will feel like a shortcut to an integral over its inside

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

MIT FILMED GILBERT STRANG - THE PROFESSOR WHO EXPLAINS MATHEMATICS MORE CLEARLY THAN ANYONE ALIVE - SHOW IN 25 MINUTES WHY EVERY QUANT ON WALL STREET AND EVERY DATA SCIENTIST AT GOOGLE USES LOG SCALES TO SE WHAT ORDINARY GRAPHS COMPLETELY HIDE

This is Gilbert Strang, MIT, a standalone lecture on logarithms and log graphs. He opens by asking which functions grow faster than others and then shows what happens to each one when x equals one thousand.

He starts with the comparison. Linear gives ten to the third. Quadratic gives ten to the sixth. The national debt lives around ten to the ninth. Two to the thousandth is ten to the three hundredth. E to the thousandth is ten to the four thirty-four. One thousand factorial is roughly ten to the two thousand five hundred sixty-six. One thousand to the thousandth is ten to the three thousandth. These numbers are completely unwriteable. Their logarithms are three, three hundred, four thirty-four, one thousand, two thousand five hundred sixty-six, three thousand. Reasonable numbers you can actually read and compare.

Then log scales. Mark a line not with zero one two three but with one, ten, one hundred, one thousand. Every step multiplies by ten. Halfway between one and ten is not five and a half - it is the square root of ten. Zero cannot appear on a log scale because its logarithm is negative infinity.

Then the money application. You measure data from an experiment and suspect a power law y equals a times x to the n but you do not know n. Plot on ordinary paper and you cannot tell n equals one point five from n equals two. A quant who gets that wrong prices a derivative incorrectly and loses money. Plot on log-log paper and the equation becomes a straight line whose slope is exactly n. Get a ruler and read it off. That is how growth rates are found in practice.

Then exponential growth. Plot y equals b times ten to the cx on semi-log paper - log scale vertically, ordinary scale horizontally - and you get a straight line whose slope is c. Every epidemiologist tracking a pandemic uses this. Every investor tracking compound returns uses this.

Watch the moment he says if you took the logarithm of the national debt it would not look too serious - it would just be around nine moving toward ten.

A quantitative analyst I know rewatched this lecture before starting a derivatives role. Said it was the first time log-log plots felt like a professional tool rather than a classroom curiosity.

Free on YouTube, MIT OpenCourseWare, Creative Commons license.

bookmark this and watch later - after this lecture every graph on ordinary paper will feel like it is hiding something a log scale would show you in one glance

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

THIS HARVARD ALGEBRA LECTURE FROM THE ERA WHEN ABSTRACT ALGEBRA WAS STILL BEING SHAPED INTO ITS MODERN FORM EXPLAINS IN ONE HOUR WHAT MOST TEXTBOOKS SPEND THREE CHAPTERS OBSCURING - THAT ISOMORPHISM MEANS EXACTLY THE SAME MULTIPLICATION TABLE WITH DIFFERENT LABELS

This is a Harvard abstract algebra lecture, part of the legendary course taught by Benedict Gross and his colleagues. He opens with one claim - groups are not just abstract objects, they are symmetries of something concrete, and every abstract fact about them has a geometric picture behind it.

He starts with two groups that look completely different. The first is the fourth roots of unity in the complex numbers - one, i, minus one, minus i under multiplication. The second is a cyclic permutation of four letters inside the symmetric group. Write out the multiplication table for both. They are identical up to renaming the elements. These are not just similar groups. They are the same group wearing different clothes.

Then isomorphisms. An isomorphism is a bijection between two groups that preserves multiplication. It is a relabeling that makes two multiplication tables identical. The map sending i to the k-th power to the corresponding permutation is an explicit isomorphism. Every cyclic group of order n is isomorphic to every other cyclic group of order n. There is essentially only one such group.

Then the Klein four-group. Four elements, each one squaring to the identity, any two multiplying to the third. It appears as a subgroup of permutations and simultaneously as a set of diagonal matrices with entries plus or minus one. The two realizations look nothing alike. They are isomorphic. The question of whether the Klein four-group is isomorphic to the cyclic group of order four is settled in one line - the cyclic group has an element of order four, the Klein four-group does not.

Then homomorphisms. Drop the requirement that the map be bijective but keep the condition that it preserves multiplication. The determinant is a homomorphism from invertible matrices to nonzero real numbers. The map sending even integers to the identity and odd integers to the single transposition in S2 is a homomorphism. Every isomorphism is a homomorphism but not every homomorphism is an isomorphism.

Watch the moment he asks whether there is always a homomorphism between any two groups. The answer is yes - the trivial one that sends everything to the identity. It is the most boring possible map. It is also a perfectly valid homomorphism.

A graduate student I know rewatched this lecture before their qualifying exam on group theory. Said it was the first time isomorphism felt like a precise version of something they had always sensed but never been able to state.

Free on YouTube, Harvard, full course available.

bookmark this and watch later - after this lecture every time you see two different structures with the same rules you will know they are secretly the same object

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

THE MIT PROFESSOR WHOSE CALCULUS STUDENTS SAY NO ONE HAS EVER MADE THEM FEEL MORE CAPABLE INTRODUCES LINE INTEGRALS BY PROVING THAT SOMETIMES THE GEOMETRIC INSIGHT SOLVES IN THREE LINES WHAT THE CALCULATION TAKES HALF A BLACKBOARD TO FINISH

This is Denis Auroux, MIT, 18.02 Multivariable Calculus, Fall 2007. His students consistently rank him among the most beloved teachers in the mathematics department. He opens with one claim - vector fields and line integrals are completely different from double integrals and it actually helps to forget everything from last week.

He starts with vector fields. At every point in the plane you have a vector. Wind maps are vector fields. Gravitational fields are vector fields. The field xi plus yj points radially outward from the origin and grows with distance. The field minus yi plus xj rotates everything counterclockwise at unit angular velocity. A particle of fluid in that field traces a perfect circle and returns to its starting point in exactly two pi units of time.

Then work. When a force pushes a particle along a trajectory the work done is the force dotted with the displacement. For a straight line and constant force this is simple multiplication. For a curved trajectory where the force changes at every point you cut the path into infinitely many tiny pieces, dot the force with each tiny displacement, and sum. That sum is the line integral.

Then computation. Parameterize the curve with a single variable. Express x and y in terms of that variable. Replace dx and dy with their derivatives times dt. The two-dimensional integral collapses into a single ordinary integral you already know how to solve.

Then the geometric shortcut. The line integral equals the integral of the tangential component of the force along the curve. If the force is always perpendicular to the curve the work is zero with no calculation needed. The field xi plus yj on a circle of radius a gives zero immediately because the radial force is always perpendicular to the circular path. The field minus yi plus xj on the same circle gives two pi a squared immediately because the force always points exactly along the path with magnitude a.

Watch the moment he computes the same integral two ways side by side. The geometric method takes three lines. The parametric method takes half a blackboard. Same answer. The insight about perpendicularity or parallelism does in seconds what algebra does in minutes.

A mechanical engineering student I know rewatched this lecture before a problem set on conservative force fields. Said it was the first time the dot product felt like it was measuring something physical rather than producing a number.

Free on YouTube, MIT OpenCourseWare, Creative Commons license.

bookmark this and watch later - after this lecture every curved path through a force field will feel like an integral waiting to be simplified

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

THE PROFESSOR MIT STUDENTS HAVE CALLED THE MOST BELOVED TEACHER IN THE HISTORY OF THE DEPARTMENT OPENS THIS LECTURE WITH A SINGLE CLAIM - THAT UNIVERSITIES ARE TEACHING YOU THE WRONG MATHEMATICS AND THE RIGHT ONE HAS BEEN HIDING IN PLAIN SIGHT THE ENTIRE TIME

This is Gilbert Strang, MIT, 18.085 Computational Science and Engineering, review lecture. He opens with one provocation - too much calculus. Semester after semester of derivatives and integrals and you run out of time before you reach the mathematics that anyone who actually computes is using every single day.

He starts with two vectors. Not on the same line, going in different directions. Take all their combinations and you fill a plane. Add a third vector that falls outside that plane and you fill all of three-dimensional space. Add a third vector that happens to lie in the same plane as the first two and you are stuck there forever. That is the entire question of linear algebra stated in one picture.

Then two matrices built from the same vectors. The first has an inverse. Input goes in, output comes out, and the inverse matrix brings you back every single time. The second matrix is singular. Three equations whose left sides always add to zero, which means the right sides must add to zero too or there is no solution. One matrix is a perfect map. The other collapses an entire dimension and you can never recover it.

Then the word subspace. A plane through the origin is a subspace. A line through the origin is a subspace. The origin alone is a subspace. The whole space is a subspace. Every object linear algebra studies is built from taking all combinations of some set of vectors and asking what you get. The answer to that question is the entire subject.

Watch the moment he says that differentiation and integration are just one matrix and its inverse and the fundamental theorem of calculus is nothing more than that statement written in the language of functions instead of vectors.

A software engineer I know rewatched this lecture before their first job at a data infrastructure company. Said it was the first time matrices felt like objects with a shape and a personality rather than a grid of numbers to multiply.

Free on YouTube, MIT OpenCourseWare, Creative Commons license.

bookmark this and watch later - after this lecture every matrix you ever see will feel like a question about which directions it keeps and which ones it destroys

[Potato @Potatxxl]

https://t.co/7hL5a8r784

[Lupen @0xLupenn]

EINSTEIN WROTE THE FORMULA THAT MAKES GPS IMPOSSIBLE TO BUILD - AND A MIT CALCULUS LECTURE FROM 2007 SHOWS THE ONE APPROXIMATION THAT MADE ENGINEERS IGNORE EINSTEIN AND BUILD IT ANYWAY

This is David Jerison, MIT, 18.01 Single Variable Calculus, Fall 2007. He opens with one claim - there is a single formula that turns any curved function into a straight line. The rest of the lecture is what you can do once you have it.

He starts with the logarithm. At the point x equals one, the curve and its tangent line touch. Move slightly left or right and they separate. But close enough to that point, ln of x and x minus one are the same object for every practical purpose. One is a curve that bends toward infinity. The other is a line you compute in your head.

Then the catalog. Sine of x becomes x. Cosine becomes one. E to the x becomes one plus x. Every hard function replaced by something a child could evaluate. Near zero, the curve and the line are indistinguishable. That gap between globally different and locally identical is where calculus lives.

Then the GPS satellite. Special relativity says a clock in orbit runs differently than a clock on the ground because the satellite moves at four kilometers per second. The formula contains a square root that looks impossible to work with. The engineers applied exactly this approximation, the square root became one plus one half u, and the transmitter offset fell out in one line.

Watch the moment he multiplies two approximations together and throws away the x squared term without computing it. Not because it is zero. Because it was already lost the moment they chose to approximate.

A GPS engineer I know showed this lecture to a junior colleague who kept asking why approximations were allowed in production code. Said it was the first time negligible felt like a mathematical fact rather than an excuse.

Free on YouTube, MIT OpenCourseWare, Creative Commons license.

bookmark this and watch later - after this lecture every time your phone finds you in two seconds you will know which equation made that possible

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

THE MIT PROFESSOR WHO TAUGHT MATHEMATICS EVERYWHERE FROM DEATH ROW TO THE WORLD'S TOP UNIVERSITIES OPENED HIS ALGEBRA COURSE WITH ONE CLAIM THAT CHANGES EVERYTHING - ALGEBRA IS JUST ARITHMETIC RUN BACKWARDS AND YOU ALREADY KNOW HOW TO DO IT

This is Herb Gross, Classic Algebra, Gateways to Algebra, lecture 1. The same professor whose Calculus Revisited series MIT placed on OpenCourseWare and engineers have been rewatching for over 50 years. He opens with one claim - algebra is not a new subject. It is the last chapter of arithmetic.

He starts with the checkerboard. A six by six board with twelve checkers. Place four in the corners. Arrange the remaining eight so no row, column or diagonal has more than two. Trial and error could take hours. Logic takes thirty seconds. The top row already has two checkers so nothing goes there. The first column is full. Eight squares remain. Eight checkers remain. Either the solution is those eight squares or there is no solution at all.

Then feet and inches. Five feet is sixty inches. Arithmetic: start with five, multiply by twelve, get sixty. One direction, one calculation, a calculator handles it. Now flip it. Sixty inches is how many feet. The relationship between feet and inches has not changed. What changed is which end you start from. Algebra is working backwards from the answer to find the input. Division undoes multiplication. That is the entire idea.

Then the formula. Instead of writing multiply the number of feet by twelve to get the number of inches, write I equals twelve F. One line. Same meaning. If you are given F that is arithmetic. If you are given I that is algebra. The only difference is which direction you travel through the same relationship.

Watch the moment he shows that multiplying by twelve appears in dozens of unrelated problems. Feet to inches. Years to months. Hours to dozens. The calculation is always identical. The only thing that changes is what the letters mean. Mathematics names the pattern once and uses it everywhere.

A parent I know rewatched this lecture alongside their child who was failing algebra. Said it was the first time the word algebra stopped being frightening and started being a description of something they both already understood.

Free on YouTube, MIT OpenCourseWare.

bookmark this and watch later - after this lecture every algebra problem you see will feel like an arithmetic problem you already know how to solve run in the opposite direction

[Potato @Potatxxl]

https://t.co/7hL5a8r784

[Lupen @0xLupenn]

A FIELDS MEDAL WINNER FROM MIT WHO REWROTE THE FOUNDATIONS OF TOPOLOGY STOOD AT A BLACKBOARD AND EXPLAINED WHY EVERY SMOOTH SURFACE ON EARTH CAN BE DESCRIBED BY A SINGLE MATRIX - AND WHY THAT IDEA CONNECTS DIFFERENTIAL GEOMETRY TO THE DEEPEST UNSOLVED PROBLEMS IN MANIFOLD THEORY

This is John Milnor, MIT, one of the most decorated mathematicians of the twentieth century - Fields Medal, Wolf Prize, Abel Prize, and the man who discovered that the seven-dimensional sphere admits twenty-eight distinct smooth structures that are topologically identical but geometrically incompatible. He opens with one claim - Grassmann manifolds are not just abstract objects. They are the universal space that classifies every vector bundle that has ever existed.

He starts with the Grassmann manifold. Fix a Euclidean space of dimension k. Consider all n-dimensional subspaces passing through the origin. That collection is the Grassmannian. To make it a smooth manifold he represents each subspace as a matrix - symmetric, idempotent, with trace equal to n. Each such matrix is an orthogonal projection onto exactly one n-dimensional subspace. The abstract geometry collapses into linear algebra.

Then the Gauss map. Given any smooth manifold embedded in Euclidean space, assign to each point the tangent plane at that point, translated to the origin. That assignment is a map from the manifold into the Grassmannian. Gauss defined this for surfaces in three-space. Milnor generalizes it to manifolds of any dimension in any ambient space.

Then the cobordism problem. When is a compact oriented manifold the boundary of a higher-dimensional manifold? The answer comes from the Gauss map. Push the fundamental homology class of the manifold through the Gauss map into the homology of the Grassmannian. The result is a collection of integers called Pontryagin numbers and integers modulo two called Stiefel-Whitney numbers. The manifold is a boundary if and only if all of them are zero.

Then the classification theorem. Any vector bundle over a finite complex maps into the Grassmannian in a way that is unique up to homotopy. The Grassmannian is the universal model. Classifying vector bundles is the same problem as classifying maps into this one space. The geometry reduces to homotopy theory.

Watch the moment he introduces three parallel versions of the tangent bundle - smooth, piecewise linear, and topological - and shows that the question of whether a topological manifold can be smoothed is equivalent to asking whether a certain map can be lifted from the topological classifying space to the smooth one. A geometric question becomes a lifting problem. He then mentions that Novikov had just announced results from the Soviet Union on the topological case - completely unknown the month before.

A topology PhD student I know rewatched this lecture before their qualifying exam on characteristic classes. Said it was the first time the Grassmannian felt like the answer to a question rather than a space someone had invented for its own sake.

Free on YouTube, filmed in the 1960s.

bookmark this and watch later - after this lecture every smooth surface you see will feel like a map into a Grassmannian waiting to be classified

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

A MATHEMATICIAN WHO WON THE FIELDS MEDAL AND REWROTE THE FOUNDATIONS OF TOPOLOGY STOOD AT A BLACKBOARD AND EXPLAINED WHY EVERY SMOOTH SURFACE ON EARTH CAN BE DESCRIBED BY A SINGLE MATRIX - AND WHY THAT IDEA CONNECTS DIFFERENTIAL GEOMETRY TO THE DEEPEST UNSOLVED PROBLEMS IN MANIFOLD THEORY

This is John Milnor, one of the most decorated mathematicians of the twentieth century - Fields Medal, Wolf Prize, Abel Prize, and the man who discovered that the seven-dimensional sphere admits twenty-eight distinct smooth structures that are topologically identical but geometrically incompatible. He opens with one claim - Grassmann manifolds are not just abstract objects. They are the universal space that classifies every vector bundle that has ever existed.

He starts with the Grassmann manifold. Fix a Euclidean space of dimension k. Consider all n-dimensional subspaces passing through the origin. That collection is the Grassmannian. To make it a smooth manifold he represents each subspace as a matrix - symmetric, idempotent, with trace equal to n. Symmetric means it has an orthonormal basis of eigenvectors. Idempotent means every eigenvalue is zero or one. Trace n means exactly n eigenvalues equal one. Each such matrix is an orthogonal projection onto exactly one n-dimensional subspace. The abstract geometry collapses into linear algebra.

Then the Gauss map. Given any smooth manifold embedded in Euclidean space, assign to each point the tangent plane at that point, translated to the origin. That assignment is a map from the manifold into the Grassmannian. Gauss defined this for surfaces in three-space. Milnor generalizes it to manifolds of any dimension in any ambient space. The tangent plane at each point becomes a point in the Grassmannian.

Then the cobordism problem. When is a compact oriented manifold the boundary of a higher-dimensional manifold? The answer comes from the Gauss map. Push the fundamental homology class of the manifold through the Gauss map into the homology of the Grassmannian. The result is a collection of integers called Pontryagin numbers and integers modulo two called Stiefel-Whitney numbers. The manifold is a boundary if and only if all of them are zero.

Then the classification theorem. Any vector bundle over a finite complex maps into the Grassmannian in a way that is unique up to homotopy. The Grassmannian is the universal model. Classifying vector bundles is the same problem as classifying maps into this one space. The geometry reduces to homotopy theory.

Watch the moment he introduces three parallel versions of the tangent bundle - smooth, piecewise linear, and topological - and shows that the question of whether a topological manifold can be smoothed is equivalent to asking whether a certain map into the topological classifying space can be lifted to the smooth one. A geometric question becomes a lifting problem in homotopy theory. He then mentions that Novikov had just announced progress on the topological case from the Soviet Union - results that had been completely unknown the month before.

A topology PhD student I know rewatched this lecture before their qualifying exam on characteristic classes. Said it was the first time the Grassmannian felt like the answer to a question rather than a space someone had invented for its own sake.

Free on YouTube, filmed in the 1960s.

bookmark this and watch later - after this lecture every smooth surface you see will feel like a map into a Grassmannian waiting to be classified

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

MIT FILMED THE LECTURE WHERE THEIR MOST LEGENDARY LINEAR ALGEBRA PROFESSOR EXPLAINS WHY EVERY GPS SIGNAL EVERY MEDICAL SCAN AND EVERY AI MODEL ON EARTH IS SECRETLY SOLVING A PROBLEM THAT HAS NO SOLUTION

This is Gilbert Strang, MIT, 18.06 Linear Algebra, lecture 15. The professor whose textbooks sit in over 3,000 universities across 135 countries and whose course has been watched by millions of engineers and mathematicians worldwide. He opens with one claim - when equations have no solution, you do not give up. You solve the closest problem that can be solved.

He starts with a vector b and a line through a. The projection of b onto that line is the point closest to b that actually lives on the line. The key is a right angle. The error - the gap between b and its projection - must be perpendicular to a. That one geometric fact produces the entire formula. The projection is a times a transpose b divided by a transpose a. One fraction. The numerator is a matrix. The denominator is a number.

Then the projection matrix. Every projection is carried out by a matrix P that multiplies b and lands on the line. That matrix has two properties that define all projections everywhere. First, P transpose equals P - the matrix is symmetric. Second, P squared equals P - project twice and you stay where you are. The first projection moves you onto the line. The second leaves you there.

Then higher dimensions. Instead of projecting onto a line, project onto a plane. Instead of one vector a, you have a matrix A with two columns spanning the plane. The error vector must now be perpendicular to both columns - perpendicular to the entire plane. That condition produces one matrix equation. A transpose times the error equals zero. The error lives in the null space of A transpose, which is exactly perpendicular to the column space of A. The four fundamental subspaces appear without being called.

Then the formula for n dimensions. The projection matrix is A times A transpose A inverse times A transpose. Strang warns the room not to simplify it by canceling A inverse against A - A is not square, it has no inverse, and the formula cannot be pulled apart. The only case where it simplifies to the identity is when A spans the whole space and the projection is b itself.

Watch the moment he applies the formula to fitting a line through three points that no line can pass through. Three equations, two unknowns, no solution. He writes the matrix, writes the right-hand side, and shows that multiplying both sides by A transpose produces an equation that always has a solution. That equation is the foundation of least squares, GPS triangulation, and every regression model trained in 2026.

A data scientist I know rewatched this lecture before building a recommendation system from scratch. Said it was the first time the normal equations felt like geometry rather than a formula someone handed down to memorize.

Free on YouTube, MIT OpenCourseWare, filmed over 20 years ago.

bookmark this and watch later - after this lecture every unsolvable equation you meet will feel like a projection waiting to be found

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

MIT SENT THE SAME KIND OF GUY WHO TAUGHT FOURIER HIMSELF - A PROFESSOR SO SHARP HE CAN MAKE ANY SQUARE ROOT IN EXISTENCE DISAPPEAR IN ONE SUBSTITUTION - AND HE DID IT IN FRONT OF 200 STUDENTS IN 45 MINUTES WHILE FILLING IN FOR SOMEONE ELSE

This is Professor Miller, MIT 18.01, lecture 27. Jerison called in sick. Miller walks in cold, draws a circle, and opens with one claim - every square root of the form a squared minus y squared is that circle. Not a metaphor. Geometrically, that is what is happening.

He opens with odd powers. Any integral where sine or cosine appears to an odd power splits in one move - one factor becomes the differential of the other. Substitute u equals sine x, the cosine dx disappears into du, and the whole thing collapses into a polynomial. Three examples, 8 minutes, no slowdown.

Then even powers. Nothing to grab onto. He pulls the half angle formula - cosine squared is 1 plus cosine of 2x, all divided by 2. The square vanishes at the cost of doubling the angle. He runs this twice on sine squared times cosine squared and gets the same answer both times via two completely different routes. Seems personally offended that even exponents exist.

Then the circle. A slice cut from a disc of radius a up to height b - find the area. The integral is the square root of a squared minus y squared, from 0 to b. He stares at it, draws the same circle from the start, substitutes y equals a sine theta, and the square root is gone in one line. The thing that had no solution is now a problem from 20 minutes ago.

Watch the moment he checks the final answer on the diagram. Two terms - the first is a squared times theta over 2, which is exactly the sector area. The second is b times the square root of a squared minus b squared divided by 2, which is the triangle area. 45 minutes of calculus and 10 seconds of geometry give the exact same number. He calls it "enough cause for celebration" and walks out.

A physics PhD student I know rewatched this before their first mechanics exam. Said it was the first time trig substitution felt like cheating rather than suffering.

Free on YouTube, MIT OpenCourseWare, Creative Commons license.

bookmark this and watch later - after this lecture every square root with a constant minus something inside will feel like a circle that hasn't been drawn yet

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

AN OXFORD PROFESSOR OF MATHEMATICAL MODELING GAVE A PUBLIC LECTURE ON THE 20TH ANNIVERSARY OF ANDREW WILES PROVING FERMAT'S LAST THEOREM - IN THE BUILDING NAMED AFTER WILES - AND EXPLAINED WHY EVERY WRONG MODEL TELLS YOU MORE THAN A CORRECT ONE

This is Alain Goriely, Oxford Professor of Mathematical Modeling, public lecture at the Andrew Wiles Building, September 2014. He opens with Lord Kelvin - if you cannot express something in numbers your knowledge is of a meager and unsatisfactory kind. Then he immediately shows why that is not enough, because more complex models are often worse.

The climate modeling section is worth the entire lecture. The simplest possible model - energy in from the sun minus energy reflected - predicts Earth's average temperature at minus 18 degrees. The actual temperature is plus 15. The model is wrong by 33 degrees. But the error tells you exactly what is missing - the atmosphere and the greenhouse effect. One wrong calculation and you understand something fundamental about the planet.

Then Lorenz and chaos. Three equations, two nonlinear terms, and the discovery that two solutions starting at imperceptibly different initial conditions diverge completely within weeks. Mathematically proven. Two weeks is the hard limit on weather prediction and it will never extend to a year no matter how good the computers become. The butterfly was not a metaphor - it was a theorem.

Then perovskite solar cells, brain swelling from trauma, and seashell spine formation - three completely different systems all modeled using the same principle. Identify the minimum number of physical processes needed. Write the simplest possible equations. Find where the model fails. Learn from the failure.

Watch the moment he shows that opening the skull in a brain trauma patient can be predicted mathematically - the damage propagates by compressing neighboring tissue which then loses oxygen and swells in turn, and the model shows exactly when a craniotomy stops the cascade.

A postdoc I know watched this lecture before switching from pure mathematics to mathematical biology. Said it was the first time modeling felt like a creative discipline rather than applied calculation.

Free on YouTube, recorded at Oxford University.

bookmark this and watch later - after this lecture every wrong model will feel like useful information rather than a failure

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

THE GENIUS MIT PROFESSOR WHOSE EXPLANATIONS ARE SO SHARP AND CLEAR THAT HE SHOWS IN 35 MINUTES WHY ARCHIMEDES - ONE OF THE GREATEST MATHEMATICIANS WHO EVER LIVED - STILL MISSED THE ONE CONNECTION THAT MAKES INTEGRATION OBVIOUS

This is Gilbert Strang, MIT, the professor whose textbooks sit in over 3,000 universities across 135 countries. He opens with one sentence - we know the slope at every point and we want to find the height. Everything else follows from that.

He starts with numbers. Heights 0, 1, 4, 9, 16. Slopes 1, 3, 5, 7. Going from heights to slopes is subtraction. Going from slopes back to heights is running addition - accumulate all the changes and you recover where you ended up. The fundamental theorem of calculus is just that observation, pushed to the limit where the steps become infinitely small.

Then the limiting process. Chop the interval into 4 pieces, approximate the slope as constant in each piece, multiply each slope by the width of its piece. That gives the area of a rectangle. Add all the rectangles. Now cut the pieces in half - the rectangles get thinner, the approximation improves. In the limit, the sum of infinitely many infinitely thin rectangles becomes the area under the curve.

Watch the moment he finds the area under a straight sloping line two ways simultaneously - first by geometry using a triangle, getting exactly 1, and then by finding the function whose derivative gives the slope, substituting x equals 1, and getting exactly 1. Two completely different methods. Same answer. That is the fundamental theorem of calculus landing in real time.

A student I know watched this lecture the night before their first calculus exam. Said it was the first time integration felt like something you could derive yourself rather than something you had to believe.

Free on YouTube, MIT OpenCourseWare, Gilbert Strang at a chalkboard.

bookmark this and watch later - after this lecture the integral sign will feel less like a symbol and more like a promise that the area is waiting to be found

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

https://t.co/87P80B6arY

[Lupen @0xLupenn]

The hardest problem on the hardest math test in the US has a solution that fits in two sentences. DeepMind's AlphaProof scored silver at the IMO last year solving problems just like this one.

Choose 4 random points on a sphere - what's the probability the tetrahedron they form contains the center? The answer is 1/8, and the trick that gets you there is the same reframing move that unlocks most hard problems in probability.

15 minutes. Bookmark & watch today. The problem-solving lesson is more useful than the answer.

[Eron @0xEronn]

https://t.co/3OCTdTK7yf

[Lupen @0xLupenn]

As someone who has read every transformer paper that drops, this attention mechanism breakdown is the closest thing to a "why GPT actually works" explainer I've ever seen released for free.

GPT-3 runs 96 attention heads in parallel across 96 layers, with 58 billion parameters devoted just to attention. This video shows exactly what each one is doing to your words before the model predicts the next token.

20 minutes. Bookmark & watch today. The math behind every ChatGPT response is in here.

[Eron @0xEronn]

https://t.co/3OCTdTK7yf

[Lupen @0xLupenn]

Babylonians listed Pythagorean triples in 1800 BC - more than a thousand years before Pythagoras. LLMs still struggle to generate new ones without hallucinating.

There are infinitely many right triangles with whole number sides, and they all come from one trick: squaring complex numbers. Every triple ever discovered fits somewhere in a single diagram.

15 minutes. Bookmark & watch today. The geometry behind it connects directly to Fermat's last theorem.

[Eron @0xEronn]

https://t.co/3OCTdTK7yf

[Lupen @0xLupenn]

The algorithm behind ChatGPT's training also solves the oldest trick question in coding interviews. Most engineers get it wrong because they think about it as a search problem.

Dynamic programming breaks hard problems into smaller ones and builds the answer bottom-up. The longest increasing subsequence, box stacking, sequence alignment - all the same 5-step pattern under the hood.

22 minutes. Bookmark & watch today. The framework works on any DP problem you'll ever see.

[Eron @0xEronn]

Every LLM you used today was trained by an algorithm a Frenchman named Trémaux invented in the 1800s to escape mazes.

Backprop isn't calculus. It's a reversed DFS post-order. GPT-3 is a graph with 175 billion parameters, and PyTorch walks it that exact way on every training step.

20 minutes. Bookmark & watch today.

[Lupen @0xLupenn]

THE MOST LEGENDARY PROFESSOR IN MIT HISTORY USED A SIMPLE DRIVING PROBLEM TO SHOW THE EXACT MATHEMATICAL THINKING THAT HEDGE FUNDS AND ENGINEERING FIRMS PAY MILLIONS FOR - AND ALMOST NOBODY WATCHES IT

This is Gilbert Strang, MIT, the professor whose textbooks sit in over 3,000 universities across 135 countries. He opens with three generations of one function - position, speed, acceleration - and shows that the second derivative is simply the bending of a curve. Positive bending means concave up. Negative bending means concave down.

Then maximum and minimum. At a maximum the slope is zero and the bending is negative - the curve is arching downward. At a minimum the slope is zero and the bending is positive - the curve is pushing upward. Two numbers tell you everything about the shape of any function near any point.

Then inflection points - the moment bending changes sign. The curve stops curving one way and starts curving the other. Second derivative passes through zero. He shows it on sine x at exactly 180 degrees and on x cubed minus x squared at x equals one third.

Then the driving problem. Home is 3 miles from the highway. The highway is twice as fast as city streets. Where do you join it? He names the unknown x, writes the time function, takes the derivative, sets it to zero, squares both sides to kill the square root, and solves. The answer is a over root 3. Which is exactly 30 degrees.

Watch the moment he connects the 30 degree answer back to the ratio of speeds he chose at the start. The sine of 30 degrees is one half. The speed ratio was one to two. The answer was never a coincidence.

A civil engineer I know used this exact method to optimize road junction placement for a highway project. Said it was the first time a calculus lecture solved something he actually needed at work.

Free on YouTube, MIT OpenCourseWare, Gilbert Strang at a chalkboard.

bookmark this and watch later - after this lecture every optimization problem will feel like a driving route waiting to be solved

[the lich @thelichhh]

https://t.co/aqsY18NvVz

[Lupen @0xLupenn]

A FIELDS MEDALIST WHO WROTE OVER 50 MATHEMATICAL TEXTS THAT HAVE TAUGHT GENERATIONS OF MATHEMATICIANS WORLDWIDE SPENT AN HOUR EXPLAINING THE GEOMETRY THAT RUNS UNDERNEATH EVERY GRADIENT DESCENT STEP - AND THE FILM SAT FORGOTTEN FOR DECADES

This is John Milnor, Princeton, lecture 1 on differential topology. He won the Fields Medal in 1962 - mathematics' highest honour - at 31 years old. His books on topology, K-theory and dynamical systems have been the standard texts in graduate programs across the world for 60 years. As an undergraduate he solved an open problem in topology under the impression it was a homework exercise.

He opens with one definition - a smooth mapping is one where all partial derivatives exist and are continuous. Not one derivative. Not three. All of them, forever. He explains why in one sentence: if you only required three derivatives, every time you differentiated you would lose one. Infinite differentiability costs nothing and buys you everything.

Then manifolds. A smooth manifold is a set that locally looks like flat Euclidean space - the way the surface of the Earth looks flat when you stand on it, even though it curves globally. Every latent space in a neural network, every loss surface, every paper with the word "manifold" in the abstract is built on this definition.

He then splits all of topology into 4 categories and shows that the same sphere looks completely different depending on which category you work in. Two have clean answers. One has no satisfactory answer at all. This is the man who proved there are 28 different ways to put a smooth structure on a 7-dimensional sphere - structures that are topologically identical but geometrically distinct.

Watch the moment he defines homotopy equivalence. Two spaces are equivalent if you can continuously deform one into the other. All of Euclidean space turns out to be equivalent to a single point.

A topologist I know shows this lecture to every PhD student before their first seminar. Says it is the only place where the 4 branches of topology are explained so clearly that you understand why they are different subjects.

Free on YouTube, filmed in 1965, one fixed camera, chalk only.

bookmark this and watch later - after this lecture the word "smooth" will mean something permanently different

[the lich @thelichhh]

https://t.co/aqsY18NvVz