In these notes we introduce Levin’s theory of average-case complexity. This theory is still in its infancy: in these notes we will introduce the notion of “distributional problems,” discuss various formalizations of the notion of “algorithms that are efficient on average,” introduce a reducibility that preserves efficient average-case solvability. Next time we will prove that there is a problem...

3 September 9, 2014 9 3.1 Direct products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Commutative diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Sub-things . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 Let’s play Guess the BS! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.5 Kernels . . . . . . . . ....

1. Surfaces Intuitively, a surface is a subset of R3 that “looks like” a plane when you zoom in on it, in the same way that the graph of a smooth function looks like a line when you zoom in on it. Surfaces are more complex objects than curves, and to appropriately describe them we will require a more complex definition than we had for curves. In this section we will gradually build the definiti...

This is Harvard College’s Math 145b, instructed by Peter Koellner. The formal name for this class is “Set Theory II”. This class deals with large cardinals and their inner forms. The permanent URL is http://www.mit.edu/~evanchen/coursework.html, along with all my other course notes. If you received a Dropbox link to this document, please be aware that the Dropbox link will expire after the end ...

In the last lecture we described the construction of the zigzag product of graphs and completed the analysis of its expansion. This was the final building block for our construction of arbitrarily large d-regular expanders, for a constant d. In particular, we had d = 372 and achieved λ2 ≤ λ < d/2. Notice that, if we needed a larger expansion, we could always recur to graph powering. Given G, wi...