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• A mad's day work

• A country of which nothing is known but the name Grothendieck and motives

Allyn Jackson: As if Summoned from the Void, Part I

Allyn Jackson: As if Summoned From the Void, Part II

Piotr Pragasz: Notes on the Life and Work of Grothendieck

Bloch, Beilinson, Illusie: Reminiscences

Articles

• Steve Lansburg. The generalist.

• Winfried Scharlau. Who is Alexander Grothendieck?

• Colin McLarty. The rising sea: Grothendieck on simplicity and generality.

• Pierre Cartier. A country of which nothing is known but the name Grothendieck and “motives”

• Martin Schwartz. The importance of stupidity in scientific research

• Henri Poincare. Intuition and logic in mathematics.

Books

• Francis Su - Mathematics for human flourishing

• David Richeson - Euler's gem - the polyhedron formula and the birth of topology

• Donal O'Shea - The Poincare conjecture

• Matt Parker - Humble Pi_A comedy of maths errors

• Mario Livio - Is God a mathematician?

• K. Houston - How to think like a mathematician

• Steven Krantz - A mathematician's survival guide_Graduate school and early career development

• William Byers - Deep thinking_What mathematics can teach us about the mind

• Robin Wilson - Euler pioneering equation_The most beautiful theorem in mathematics

• William Byers - How mathematicians think_Using ambiguity, contradiction, and paradox to create mathematics

• Eugenia Cheng - Beyond infinity_An expedition to the outer limits of the mathematical universe

• Eugenia Cheng - The art of logic in an illogical world

• Eugenia Cheng - The joy of abstraction - an exploration of math, life, and category theory

• Eugenia Cheng - Is math real? - how simple questions lead us to mathematics' deepest truths

• Eugenia Cheng - Cakes, custard & category theory- easy recipes for understanding complex mathematics

• Bruce Schechter - My brain is open - the mathematical journey of Paul Erdos

• Paul R. Halmos - I want to be a mathematician - an automathography

• Jeffery Weeks - The shape of spaces

• David Bessis - Mathematica_A secret world of intuition and curiosity

• Grothendieck. Harvest and Sowing

• Gorhtendieck in Vietnam

• Thurston. On proof and progress in mathematics.

Algebra

• Tensor product

• Constructing the tensor product of modules

• How to conquer tensorphobia

• Viewing matrices and probability as graphs 

• Matrices as tensor networks

• Direct product vs Cartesian product (finite direct sum = finite direct product) 

• The integral domain hierarchy - 1

• The integral domain hierarchy - 2

Analysis

Articles

• Four flavors of continuity

Tools

• Note: Francis Su - 

• Video: Francis Su - 

Category

• Category theory: link - link - 

• Functor: link - 

• Natural transformation: link - 

• Yoneda lemma: link - 

• Limit and colimit: link - 

• Adjunction: link - 

• Groupoid: link - [Emily Riehl. CT_P35]

• The philosophy of Yoneda Lemma

• Understanding Yoneda Lemma

• Some perspectives on Yoneda lemma

• Computational category theory

• Group elements, categorically

• Commutative diagrams explained

• A diagram is a functor

Qoutes:

• It frames a possible template for any mathematical theory: the theory should have nouns and verbs, i.e., objects, and morphisms, and there should be an explicit notion of composition related to the morphisms; the theory should, in brief, be packaged by a category. Barry Mazur - link

Geometry

• Mathematicians transcend the geometric theory of motion

• A fight to fix geometry's foundations

• How physics found a geometric structure for math to play with

• New geometric perspective cracks old problem about rectangles

• Momental proof settles geometric Langlands conjecture

Graph

• Deep  Neural Networks Help to Explain Living Brains

• Icosian Game: A Graph Theory Game

• In highly connected networks, there's always a loop

Objects

• Traditional and figure-eight Klein bottle

• Triangles

• Trisecting the angle

• Introducing the Klein Bottle

• Inside the Klein Bottle

• Wrapping Spheres Around Spheres

• Seifert Surfaces for Knots and Links

• The Shape of Inner Space

• Manifold Destiny

• Smooth Manifolds

• Hidden Dimensions

• Platonic Solids in All Dimensions

• Spheres within spheres: A journey through many dimensions

• Spheres within spheres: Simplicity and failure

• The Shape and Fate of the Universe

• What is the Geometry of the Universe?

Topology

Articles

• Open sets are everything

• The Fundamental Group

• Topology 101: The Hole Truth

• How Mathematicians Use Homology to Make Sense

• Mathematicians Settle Erdos Coloring Conjecture

• A Proof That Some Spaces Can't be Cut

• Disentangling Loops with an Ambient Isotopy

• How Tadayuki Watanabe Disproved Major Conjecture About Spheres

• What Shape is the Universe? A New Study Suggests We've Got It All Wrong

• If Our Universe is Expanding, Then What is It Expanding to?

• Dividing Walls: Topology and Topography I

• Dividing Walls: Topology and Topography II

• Exotic Spheres or Why 4-Dimensional Space is a Crazy Place

• Maths in a Minute: Simplices - The Atoms of Topology

• The Mathematical Shapes in Your Brain

• The Shape of Things to Come I

• The Shape of Things to Come II

• The Shape of Things to Come III

• Euler's Polyhedron Formula

• Tinkertoy Models Produce New Geometric Insights

• *Mathematicians Find Structure in Biased Polynomial

• Mathematicians Answer Old Question About Old Graphs

• New Proof Reveals That Graphs With No Pentagons Are Fundamentally Different

• The Hidden Twist to Making a Mobius Strip

• Mathematicians Solve Decades-Old Classification Problem

• *In Topology, When Are Two Shapes the Same?

• New Math Book Rescues Landmark Topology Proof

• Euler's Formula

• Euler-Poincare Formula

• Groups, Geometry and Gromov

• A Journey to Define Dimension 

• Graduate Student Solves Decades-Old Conway Knot Problem

• How Close is Close?

• Sphere Packing Solved in Higher Dimensions 

• An Ancient Geometry Problem Falls to New Mathematical Techniques

• Art Gallery Problems

• How to Cut Cake Fairly and Finally Eat It Too

• All Is Not Fair in Cake-Cutting and Math

• Graph Theory: Euler's Formula for Planar Graphs

• Father-Son Team Solves Geometry Problem With Infinite Folds

• Jinyoung Park and Huy Tuan Pham Prove the Kahn-Kalai Conjecture

• Elegant Six-Page Proof Reveal the Emergence of Random Structure

• Grigori Perelman: The Mad Monk of Math

• The Development of Combinatorics

• The New Math of Wrinkling

• Father-Son Team Solves Geometry Problem With Infinite Folds

• A Number Theorist Who Connects Math to Other Creative Pursuits

• What a Math Party Game Tells Us About Graph Theory

• Surfaces Beyond Imagination Are Discovered After Decades-Long Search

• Quantum Field Theory Pries Open Mathematical Puzzle (Intuitive Holes in Topology)

• Ask Ethan: What would happen if you traveled in a straight line forever?

• Mathematicians Identify Threshold at Which Shapes Give Way

• An old conjecture falls, making spheres a lot more complicated

• The birth of non-Euclidean geometry  (video explained)

• Mathematicians marvel at 'crazy' cut through four dimensions

• To pack spheres tightly, mathematicians throw them at random

• A new agenda for low-dimensional topology (unsolved problems K1, K2, K3)

• Computational topology

• Homology theory - a primer

• Combinatorial g-conjecture for interval subdivisions, communications in algebra

Tools

• Note: Bradley - Tyrone Cutler - Following Peter May - Peter May - Chinese Researchers - Alexander Suciu - Blumberg - Maxim - Alex Nelson - Fushida-Hardy - Bert Guillou - Petersen-Berglund - Drew Heard - Oscar Randal-Williams - Stefan Schwede - Resource for homotopy theory - Andrew Putnam - Aareyan Manzoor - Rezk - Notre Dame - 

• Video: Anthony Bosman - Vidit Nanda - Loring Tu - Gabriel Islambouli - Zanachan -

Top - References

• Fibrartion: link - [Fib of posets: Steve Awodey. CC_Eg 9.26] 

• Spectrum: [AT - Davis, Kirk. Ch 7, P168_good motivation]

• Cobordism:  [AT - Paul Kirk]

• Operad: link - 02 - 

• Groupoid: [Eugenia. TJoA - P393] [Steve Awodey. CT - P168]

• Classifying space:  link - link -  link - link -

• (Infty, n) - category: [Eugenia. TJoA - 394]

• Pushout and adjunction spaces

• Postnikov tower

Physics

• Something from nothing? A vacuum can yield flashes of light

• Empty space is not empty

• Are virtual particles really constantly popping in and out of existence? Or are they merely a mathematical bookkeeping device for quantum mechanics?

• The Mystery at the Heart of Physics That Only Math Can Solve

• What is Time?

Mathematicians

• A Mathematician Who Fled to Freedom but Still Stares Down Doubts

• A Path Less Taken to the Peak of the Math World

• Learning From A Master: A Conversation with Claire Voisin

• The Mathematician Who Delights in Building Bridge

• Emmy Murphy Is a Mathematician Who Finds Beauty in Flexibility

• The Value of Mathematical Story Telling: Our Perspective on Giving Talks

• How to be successful as a research mathematician? Follow your gut

• Grothendieck

• Sullivan - Simplicity is the point

Number

• Prime in Arithmetic Progressions - The Genius of Dirichlet

• How Many Number Exist? Infinity Proof Moves Math Closer to an Answer

• Think of a Number. How Do Math Magicians Know What It Is?

• Mathematicians Measure Infinity and Find They are Equal

• How Infinite Series Reveal The Unity of Mathematics

• How Godel's Proof Works

• Mathematicians Clear Hurdle in Quest to Decode Primes

• Prime Gap Grows After Decades - Long Lull

• How the square root of 2 became a number

• Sensational proof delivers new insights into prime numbers

Philosophy

• The experience of mathematical beauty and its neural correlates

• Math & beauty & brain area

• The Future of Mathematics

• Is There Romance in Mathematics?

• Who knows two?

Miscellaneous

• The Simple Math Problem We Still Can't Solve 

• *Mathematician Proves Huge Result on Dangerous Problem

• Computer Search Settles 90-Year_Old Math Problem 

• Computer Scientists Attempt to Corner the Collatz Conjecture

• To Win This Game, Learn to Avoid Math Patterns 

• Solving Crime With Maths: Bloodstain Pattern Analysis

• Triangular Number Pattern

• Mathematicians Explore Mirror Link Between Two Geometric Worlds

• Three Decades Later, Mystery Numbers Explained

• Euler's 243-Year-Old 'Impossible' Puzzle Get a Quantum Solution

• Mathematicians Prove 30-Year-Old Andre-Oort Conjecture

• Why Triangles Are Easy And Tetrahedra Are Hard

• Mathematicians Outwit Hidden Number Conspiracy  

• Why e, the Transcendental Math Constant, Is Just the Best

• Banach-Tarski and the Paradox of Infinite Cloning

• To Settle Infinity Dispute, a New Law of Logic 

• Galois Groups and the Symmetries of the Polynomials

• Mathematicians Find a New Class of Digitally Delicate Primes

• Matrix Multiplication Inches Closer to Mythic Goal

• 3-D Fractals Offers Clues to Complex System

• Machine Learning Become a Mathematical Collaborator

• A Long - Sought Proof, Found and Almost Lost

• How Rational Math Catches Slippery Irrational Numbers 

• Simple Set Game Proof Stuns Mathematicians

• Mathematicians catch a Pattern by Figuring Out How to Avoid It

• Mathematician Find Wrinkle in Famed Fluid Equations 

• Mathematicians as a Team Sport

• Mathematicians Begin to Tame Wild Sunflower Problem 

• In Computer We Trust?

• Elegant Six-Page Proof Reveals the Emergence of Random Structure

• For His Sporting Approach to Math, A Fields Medal

• *Einstein Shape: This holy grail of tiles will revolutionize material science

• Why Mathematicians Re-Prove What They Already Know

• Why the 'Sleeping Beauty Problem' Is Keeping Mathematicians Awake

• What can you do with an Einstein?

Fun Times 

More Reading

Good advice from Tao

Bhavana - The Mathematics Magazine

Medium

The more sleep you lose, the faster you die.