Visions in Mathematics (eBook)

Copyright Page 5
Table of Contents 7
Foreword 
9 
ALGEBRAIC AND PROBABILISTIC METHODS IN DISCRETE MATHEMATICS 12
1 Introduction 
12 
2 Algebraic Techniques 
13 
2.1 Combinatorial Nullstellensatz 
13 
2.2 The dimension argument 18
3 Probabilistic Methods 20
4 The Algorithmic Aspects 
24 
References 
25 
CHALLENGES IN ANALYSIS 28
1 Digital Transcriptions of Functions, Libraries of Waveforms 29
2 Transcribing Dense Matrices for Efficient Computations 
33 
References 
37 
NONCOMMUTATIVE GEOMETRY YEAR 2000 38
1 Introduction 
39 
2 Geometry 
39 
3 Quantum Mechanics 42
4 Noncommutative Geometry 
45 
5 A Basic Example 
47 
6 Topology 53
7 Differential Topology 
55 
8 Calculus and Infinitesimals 
62 
9 Spectral Triples 
67 
10 Noncommutative 4-manifolds and the Instanton Algebra 
77 
11 Noncommutative Spectral Manifolds 
80 
12 Test with Space-time 
84 
13 Operator Theoretic Index Formula 86
14 Deffeomorphism Invariant Geometry 
88 
15 Characteristic Classes for Actions of Hopf Algebras 
90 
16 Hopf Algebras, Renormalization and the Riemann-Hilbert Problem 
92 
17 Number Theory 
102 
18 Appendix, the Cyclic Category 
110 
References 
111 
INTRODUCTION TO SYMPLECTIC FIELD THEORY 
117 
1 Symplectic and Analytic Setup 
120 
1.1 Contact preliminaries 
120 
1.2 Dynamics of Reeb vector fields 
123 
1.3 Splitting of a symplectic manifold along a contact hypersurface 
124 
1.4 Compatible almost complex structures 
126 
1.5 Holomorphic curves in symplectic cobordisms 
127 
1.6 Compactification of the moduli spaces MAg,r(r-,r+) 129
1.7 Dimension of the moduli spaces MAg,r(r-,r+) 135
1.8 Coherent orientation of the moduli spaces of holomorphic curves 
139 
1.8.1 Determinants 
139 
1.8.2 Cauchy-Riemann type operators on closed surfaces 
140 
1.8.3 A special class of Cauchy-Riemann type operators on punctured Riemann surfaces 
144 
1.8.4 Remark about the coherent orientation for asymptotic operators with symmetries 
148 
1.8.5 Coherent orientations of moduli spaces 
149 
1.9 First attempt of algebraization: Contact Floer homology 150
1.9.1 Recollection of finite-dimensional Floer theory 
150 
1.9.2 Floer homology for the action functional 
155 
1.9.3 Examples 
159 
1.9.4 Relative contact homology and contact non-squeezing theorems 
161 
2 Algebraic Formalism 
164 
2.1 Informal introduction 
164 
2.2 Contact manifolds 
170 
2.2.1 Evaluation maps 
170 
2.2.2 Correlators 
171 
2.2.3 Three differential algebras 172
2.3 Symplectic cobordisms 179
2.3.1 Evaluation maps and correlators 
179 
2.3.2 Potentials of symplectic cobordisms 
181 
2.4 Chain homotopy 
186 
2.5 Composition of cobordisms 
192 
2.6 Invariants of contact manifolds 
198 
2.7 A differential equation for potentials of symplectic cobordisms 
200 
2.8 Invariants of Legendrian knots 
204 
2.9 Remarks, examples, and further algebraic construction in SFT 
208 
2.9.1 Dealing with torsion elements in H1 
208 
2.9.2 Morse-Bott formalism 
209 
2.9.3 Computing rational Gromov-Witten invariants of CPn 218
2.9.4 Satellites 
222 
References 
226 
HOLOMORPHIC CURVES AND REAL THREE-DIMENSIONAL DYNAMICS 
231 
1 A Relationship Between Certain Vector Fields and a Holomorphic Curve Theory 
231 
2 The Behavior of a Finite Energy Map Near a Puncture 
238 
3 The Conley-Zehnder Index 
242 
4 Holomorphic Curves and More Dynamics 
246 
5 Finite Energy Foliations and Dynamics 
251 
6 About Possible Generalizations to Other Manifolds 253
References 
258 
PERSPECTIVES ON THE ANALYTIC THEORY OF L-FUNCTIONS 
262 
1 Introduction and Background 
262 
2 Fundamental Conjectures 
269 
3 Function Field Analogues 
274 
4 Dirichlet L-Functions GL(1)/Q 276
5 Special Values 
278 
6 Subconvexity and Equidistribution 
281 
7 GL(2) Tools 286
8 Symmetry and Attacks on GRH 
291 
References 
293 
COMBINATORICS WITH A GEOMETRIC FLAVOR 299
Introduction 
299 
1 Combinatorial Geometry: An Invitation to Tverberg's Theorem 300
1.1 Radon's theorem and order types (oriented matroids) 300
1.2 Tverberg's theorem 
301 
1.3 Topological versions 
302 
1.4 The dimension of Tverberg's points 
302 
1.5 Conditions for Tverberg partitions and graph colorings 
303 
1.5.1 Conditions for a Tverberg partition into 3 parts 
303 
1.5.2 Point configurations from graphs 
303 
1.5.3 The four color theorem 
304 
1.6 Other problems and connections 
305 
1.6.1 Halving hyperplanes and colored Tverberg's theorems 
305 
1.6.2 Eckhoff's partitions conjecture 
305 
1.7 Some links and references 
306 
2 Polytopes and Algebraic Combinatorics: How General is the Upper Bound Theorem? 
307 
2.1 Cyclic polytopes and upper bound theorem 
307 
2.1.1 Cyclic polytopes 
307 
2.1.2 The upper bound theorem 
307 
2.1.3 A stronger form of the UBT 
308 
2.2 Stanley-Reisner rings and their generic initial ideals (algebraic shifting) 308
2.3 How general is the upper bound theorem? 
309 
2.3.1 Witt spaces 
309 
2.3.2 Embeddability 
310 
2.3.3 An upper bound conjecture for j-sets 
310 
2.4 Duality and h-numbers 
310 
2.4.1 The Dehn-Sommerville relations 
310 
2.4.2 The Cohen-Macaulay property 
311 
2.4.3 Partial unimodality and the Braden-MacPherson theorem 
311 
2.4.4 Other duality relations 
312 
2.5 Neighborliness 313
2.5.1 Neighborly polytopes and spheres 
313 
2.5.2 Triangulations of manifolds 
313 
2.5.3 Neighborly embedded manifolds 
314 
2.6 Other problems and connections 
315 
2.6.1 Clique complexes and spheres 315
2.6.2 Cubical upper bound theorems 
315 
2.7 Some links and references 
315 
3 Extremal and Probabilistic Combinatorics: the Discrete Cube and Influence of Variables 
316 
3.1 Influence of variables on Boolean functions 
316 
3.1.1 The discrete cube 
316 
3.1.2 Influence of variables 
317 
3.1.3 Russo's lemma and threshold intervals 
319 
3.1.4 Fourier-Walsh expansion 
319 
3.1.5 Noise sensitivity 
319 
3.2 Other general propertis of subsets of the discrete cube 
320 
3.2.1 Discrete isoperimetric inequalities 
320 
3.2.2 FKG and Shearer's lemma 
321 
3.3 Advanced theorems on influences 
321 
3.4 Two basic problems 
322 
3.5 Examples 
322 
3.5.1 Weighted majority 322
3.5.2 Majority of majorities, tribes, runs 
322 
3.5.3 Recursive majorities 
323 
3.5.4 Random subsets of Mn and error correcting codes 
323 
3.5.5 Cliques in graphs 
323 
3.5.6 General graph properties 
323 
3.5.7 Random formulas, the 3-SAT problem 324
3.5.8 Crossing events in percolation 
324 
3.5.9 First passage percolation 
324 
3.5.10 Boolean functions expressed by bounded depth Boolean circuits 
325 
3.5.11 Determinants, eigenvalues 
325 
3.5.12 Signed combination of vectors 
325 
3.5.13 Linear objective functions 
325 
3.6 Some links and references 
326 
4 Enumerative Combinatorics and Probability: Counting Trees and Random Trees 
326 
4.1 Kirchhoff, Cayley, Kasteleyn and Tutte 
326 
4.2 Random spanning trees and loop erased random walk 
327 
4.3 Random spanning trees II 
327 
4.4 Random spanning trees III 
328 
4.5 Higher dimensions 329
4.6 Haiman's diagonal harmonics 
330 
4.7 Some links and references 
330 
5. Optimization: How Good is the Simplex Algorithm? 331
5.1 The simplex method 
331 
5.2 The combinatorics of linear programming 
331 
5.3 Some classes of pivot rules 
333 
5.4 Can geometry help? 
334 
5.5 Can geometry help? II: How to distinguish geometric objective functions 
335 
5.6 Some links and references 
336 
References 
337 
TOPICS IN ASYMPTOTIC GEOMETRIC ANALYSIS 349
1 About the Subject 
349 
2 Essay on Asymtotic Theory 
352 
2.1 Entropy and volume behavior in high dimension 
352 
2.1.1 Technical remarks 
353 
2.2 "Isomorphic" geometry 
353 
2.2.1 Remarks 
355 
2.3 More asymptotic ideology (examples of isomorphic study) 
356 
2.3.1 Example of phase transition Local form
356 
2.3.2 
357 
2.3.3 
357 
2.3.4 
358 
2.3.5 "Outside" of the isomorphic phase transition 
359 
2.4 Approximation what we expected from our old intuition and reality of the new one
359 
2.5 General references 
360 
3. Isomorphic Form of Isoperimetric problems Concentration Phenomenon
360 
3.1 The standard form 
360 
3.2 Metric G-spaces (X, p) 
364 
3.3 Probability spaces (X,u) 
365 
3.4 Functional point of view 
365 
4 Concluding Remarks 
366 
4.1 
366 
4.2 Speculations 
367 
5 Some Open Problems of Asymptotic Geometic Analysis 
367 
References 
370 
QUANTUM INFORMATION THEORY: RESULTS AND OPEN PROBLEMS 
373 
1 Introduction 
373 
2 Shannon Theory 
374 
3 Quantum Mechanics 
375 
4 Von Neumann Entropy 
380 
5 Source Coding 
381 
6 Accessible Information 
383 
7 The Classical Capacity of a Quantum Channel 
386 
8 Quantum Teleportation and Superdense Coding 
389 
9 Other Results from Quantum Information Theory 
391 
References 
394 
UNIVERSALITY, PHASE TRANSITIONS AND STATISTICAL MECHANICS 
396 
1 Introduction 
396 
2 Random Walk and Self-avoiding Walk 
398 
3 Ising Model 
400 
4 Lattice Field Models and Anharmonic Oscillators 
406 
5 Random Schrodinger, Random Matrices and Supersymmetry 
408 
References 
412 
HOW CLASSICAL PHYSICS HELPS MATHEMATICS 
416 
1 Introduction 
416 
2 n-wave Equations and n-orthogonal Coordinate Systems 
419 
3 Theory of Surfaces as a Chapter of Theory of Solitons 
425 
4 Long-time Asymptotics in the Hamiltonian PDE Equation 
430 
5 Briefly on Collapses 
434 
References 
436 
ADDENDUM: DISCUSSIONS AT THE DEAD SEA 27-27 August, 1999 
437 
Introdution 
437 
DISCUSSION on MATHEMATICAL PHYSICS 
439 
DISCUSSION on GEOMETRY 
451 
DISCUSSION on MATHEMATICS in the REAL WORLD 471
DISCUSSION on COMPUTER SCIENCE and DISCRETE MATHEMATICS 
506 
REFLECTIONS ON THE DEVELOPMENT OF MATHEMATICS IN THE 20TH CENTURY 
534