An Irregular Mind

Cover13; -- Table of Contents -- Foreword -- List Of Publications Of Endre Szemeredi -- Universality, Tolerance, Chaos And Order -- 1. Introduction -- 2. The Strong Chromatic Number And Universal Graphs -- 3. Random Universal Fault Tolerant Graphs -- 4. Universal Graphs And Products Of Expanders -- 4.1. A Graph-Decomposition Result -- 4.2. A Sketch of the Universality of Gk, N -- 5. A Ramsey Type Problem -- 6. Balanced Homomorphisms And Subgraph Containment Problems -- 7. Concluding Remarks And Open Problems -- References -- Super-Uniformity Of The Typical Billiard Path -- 1. What Is Super-Uniformity? -- 1. Introduction. -- 2. Super-Uniformity of the Typical Billiard Path in the Unit Square. -- 2. Can We Beat The Monte Carlo Method? (I) -- 3. Can We Beat The Monte Carlo Method? (II) -- 1. Regular Sampling Is Adaptive. -- 2. A Surprising Way to "Beat" the Monte Carlo Method: Switching From Point Samples to Curves, Surfaces, and So on. -- 3. Summary. -- 4. Super-Uniformity: Proof Of Theorem 1 -- 5. Proof Of Theorem 2 -- 6. Proof Of Theorem 3 -- 7. Proof Of Theorem 4 -- 8. Proof Of Proposition 1.1 -- 9. Proof Of Proposition 2.1 -- 10. Proof Of Proposition 2.2 -- 11. Proof Of Proposition 3.1 -- 12. More On Super-Uniformity: Proof Of Theorem 5 -- References -- Percolation On Self-Dual Polygon Configurations -- 1. Introduction -- 2. The Model And Results -- 3. A Generalization Of Harris'S Lemma -- 3.1. High Probability Unions of Upsets -- 4. Colourings, Hypergraphs And Crossings -- 4.1. How Crossing Probabilities Vary -- 5. A Rectangle-Crossing Lemma -- 5.1. Bond Percolation on 7I} -- 5.2. A Rectangle-Crossing Lemma for Hyperlattices -- 5.3. A Stronger Rectangle-Crossing Lemma -- 6. Self-Duality And Rectangle Crossings -- 7. From Rectangle Crossings To Percolation -- 8. On The Critical Surface -- References -- On Exponential Sums In Finite Fields -- O. Introduction -- 1. A Sum-Product Property -- 2. Preliminary Estimates (1) -- 3. Preliminary Estimates (2) -- 4. Further Assumptions -- 5. Preliminary Estimates (3) -- 6. Estimation Of Trilinear Sums -- 7. Convolution Of Product Densities -- 8. The General Case -- References -- An Estimate Of Incomplete Mixed Character Sums -- Notation And Convention -- References -- Crossings Between Curves With Many Tangencies -- 1. Introduction -- 2. Levels -- Proof Of Theorem 1 -- 3. Constructive Upper Bound -- Proof Of Theorem 2 -- 4. Concluding Remarks -- References -- An Arithmetic Regularity Lemma, An Associated Counting Lemma, And Applications -- 1. Introduction -- 2. Proof Of The Arithmetic Regularity Lemma -- 3. Proof Of The Counting Lemma -- 4. Generalised Von Neumann Type Theorems -- 5. On A Conjecture Of Bergelson, Host, And Kra -- 6. Proof Of Szemeredi'S Theorem -- Appendix A. Properties Of Polynomial Sequences -- Appendix B.A Multiparameter Equidistribution Result -- Appendix C. The Baker-Campbell-Hausdorff Formula -- References -- Yet Another Proof Of Szemeredi's Theorem -- 1. I Ntroduction -- 2. Nilsequenc.