Lecture notes are courtesy of MIT students and are used with permission.

SES # | TOPICS | LECTURE NOTES |
---|---|---|

Spectral Graph Theory | ||

1 | Linear algebra review, adjacency and Laplacian matrices associated with a graph, example Laplacians | (PDF) |

2 | Properties of the Laplacian, positive semidefinite matricies, spectra of common graphs, connection to the continuous Laplacian | (PDF) |

3 | Courant-Fischer and Rayleigh quotients, graph cutting, Cheerger's Inequality | (PDF) |

4 | (Lazy) random walks, their stationary distribution and l2-convergence, normalized Laplacian, conductance, Monte Carlo methods | (PDF) |

5 | Monte Carlo methods continued, approximate DNF counting, approximating the permanent of 0-1 matrices | (PDF) |

6 | Diameters and eigenvalues, expander graphs | (PDF) |

7 | Nonblocking routing networks, local and almost-linear time clustering and partitioning, Lovasz-Simonovits Theorem | (PDF) |

8 | Local and almost-linear time clustering and partitioning (cont.), PageRank, introduction to sparsification | (PDF) |

9 | Sparsification (combinatorial and spectral), effective resistance, matrix pseudoinverses and tail bounds | (PDF) |

10 | Spectral sparsification (cont.), introduction to convex geometry | (PDF) |

Convex Geometry | ||

11 | Polar of a convex body, separating hyperplanes, norms and convex bodies, Banach-Mazur distance, Fritz John's theorem | (PDF) |

12 | Separating hyperplanes (cont.), Banach-Mazur distance, Fritz John's theorem, Brunn-Minkowski inequality | (PDF) |

13 | Brunn-Minkowski inequality (cont.), Brunn's theorem, isoperimetric inequality, Grunbaum's theorem | (PDF) |

14 | Approximating the volume of a convex body | (PDF) |

15 | Random sampling from a convex body (cont.), grid walk, introduction to concentration of measure | (PDF) |

16 | Concentration of measure and the isoperimetric inequality, Johnson-Lindenstrauss theorem | (PDF) |

17 | Johnson-Lindenstrauss theorem (cont.), Dvoretsky's theorem | (PDF) |

Lattices and Basis Reduction | ||

18 | Lattices, fundamental parallelepiped and dual of a lattice, shortest vectors, Blichfield's theorem | (PDF) |

19 | Minkowski's theorem, shortest/closest vector problem, lattice basis reduction, Gauss' algorithm | (PDF) |

20 | LLL algorithm for lattice basis reduction, application to integer programming | (PDF) |

Iterative Methods for Linear Algebra | ||

21 | Iterative methods to solve linear systems, steepest descent | (PDF) |

22 | Convergence analysis of steepest descent and conjugate gradients | (PDF) |

23 | Preconditioning on Laplacians, ultra-sparsifiers | (PDF) |

Multiplicative Weights | ||

24 | Multiplicative weights | (PDF) |

25 | Multiplicative weights and applications to zero-sum games, linear programming, boosting, and approximation algorithms | (PDF) |