Introduction to the probabilistic methods.

• Lecture 1: Linear Algebra.
• Lecture 2: Review of Probability Theory.
• Lecture 3: Basic ideas of the Probabilistic Method (Last updated on Monday Sep 1, 2003).

• Lecture 4: Some useful inequalities and Asymptotic Estimates (Last updated on Monday Sep 25, 2003). This is not really a lecture. It is more like a note which shall be used in several lectures that follow. The inequalities & estimates are essential tools in the applications of the probabilistic methods. I will update this note as we go along.
• Lecture 5: Linearity of expectation (Scribed by Dazhen Pan, Last updated on Monday Sep 8, 2003).

• Lecture 6: Alterations (Scribed by Anusha R. Iyer, Last updated on Monday Sep 15, 2003).

• Lecture 7: Local Lemma (Scribed by Guang Xu, Last updated on Monday Sep 29, 2003).

Martingale and tight concentrations. (I have to skip Correlation Inequalities, since it would take too much time introducing posets. If you would like to know more on posets, please let me know.)

• Lecture 8: Martingale (Scribed by Qi Duan, Last updated on Tuesday Oct 14, 2003). This is very much incomplete. I will add more things later if I have the time. (Wonder when?)

Martingales and Tight concentrations

Martingales and Tight Concentrations

Pseudo randomness, Expander Graphs, Algebraic Graph Theory

• Lecture 9: Positive Definite Matrices, Variational Characterizations, Interlacing of Eigenvalues, Sylvester's Law of Inertia, Majorizations: clearly we won't have time to cover everything here. I just put this up for you as a reference, mostly. There are various other things I wanted to add to this note, especially the part on the Horn Conjecture. If you are interested in this conjecture, you could do an independent study on it with me.
• Lecture 10: Introduction to algebraic graph theory
• Lecture 11: Expanders

Topics: Circuit complexity

Topics: Derandomization