Structure and randomness : pages from year one of a mathematical blog

There Are Many Bits And Pieces Of Folklore In Mathematics That Are Passed Down From Advisor To Student, Or From Collaborator To Collaborator, But Which Are Too Fuzzy And Non-rigorous To Be Discussed In The Formal Literature. Traditionally, It Was A Matter Of Luck And Location As To Who Learned Such Folklore Mathematics. But Today, Such Bits And Pieces Can Be Communicated Effectively And Efficiently Via The Semiformal Medium Of Research Blogging. This Book Grew From Such A Blog. In 2007, Terry Tao Began A Mathematical Blog, As An Outgrowth Of His Own Website At Ucla. This Book Is Based On A Selection Of Articles From The First Year Of That Blog. These Articles Discuss A Wide Range Of Mathematics And Its Applications, Ranging From Expository Articles On Quantum Mechanics, Einstein's Equation E = Mc[superscript 2], Or Compressed Sensing, To Open Problems In Analysis, Combinatorics, Geometry, Number Theory, And Algebra, To Lecture Series On Random Matrices, Fourier Analysis, Or The Dichotomy Between Structure And Randomness That Is Present In Many Subfields Of Mathematics, To More Philosophical Discussions On Such Topics As The Interplay Between Finitary And Infinitary In Analysis. Some Selected Commentary From Readers Of The Blog Has Also Been Included At The End Of Each Article. While The Articles Vary Widely In Subject Matter And Level, They Should Be Broadly Accessible To Readers With A General Graduate Mathematics Background; The Focus In Many Articles Is On The 'big Picture' And On Informal Discussion, With Technical Details Largely Being Left To The Referenced Literature.--book Jacket. Ch. 1. Expository Articles -- 1.1. Quantum Mechanics And Tomb Raider -- 1.2. Compressed Sensing And Single-pixel Cameras -- 1.3. Soft Analysis, Hard Analysis, And The Finite Convergence Principle -- 1.4. Lebesgue Differentiation Theorem And The Szemeredi Regularity Lemma -- 1.5. Ultrafilters, Non-standard Analysis, And Epsilon Management -- 1.6. Dyadic Models -- 1.7. Math Doesn't Suck, And The Chayes-mckellar-winn Theorem -- 1.8. Nonfirstorderisability -- 1.9. Amplification, Arbitrage, And The Tensor Power Trick -- 1.10. Crossing Number Inequality -- 1.11. Ratner's Theorems -- 1.12. Unipotent Elements Of The Lorentz Group, And Conic Sections -- 1.13. Jordan Normal Form And The Euclidean Algorithm -- 1.14. John's Blowup Theorem For The Non-linear Wave Equation -- 1.15. Hilbert's Nullstellensatz -- 1.16. Hahn-banach Theorem, Menger's Theorem, And Belly's Theorem -- 1.17. Einstein's Derivation Of E = Mc[superscript 2] -- Ch. 2. Lectures -- 2.1. Simons Lecture Series: Structure And Randomness -- 2.2. Ostrowski Lecture: The Uniform Uncertainty Principle And Compressed Sensing -- 2.3. Milliman Lecture Series: Recent Developments In Arithmetic Combinatorics -- Ch. 3. Open Problems -- 3.1. Best Bounds For Cap Sets -- 3.2. Non-commutative Freiman Theorem -- 3.3. Mahler's Conjecture For Convex Bodies -- 3.4. Why Global Regularity For Navier-stokes Is Hard -- 3.5. Scarring For The Bunimovich Stadium -- 3.6. Triangle And Diamond Densities In Large Dense Graphs -- 3.7. What Is A Quantum Honeycomb? -- 3.8. Boundedness Of The Trilinear Hilbert Transform -- 3.9. Effective Skolem-mahler-lech Theorem -- 3.10. Parity Problem In Sieve Theory -- 3.11. Deterministic Rip Matrices -- 3.12. Non-linear Carleson Conjecture. Terence Tao. Includes Bibliographical References (p. 283-298).