Lecture 10

Toda's Theorem: PH in BPP^⊕P in P^Gap-P=P^PP

Lecture Notes by Darrell Hoy

Lecture 9

Toda-Ogihara: For every f in Gap^PH there is a g in Gap P s.t. f(x)=g(x,r) with high probability

Lecture Notes by Aedan Maher

Lecture 8

#P^NP in #coNP which randomly reduces to GapP.

Lecture 7

Mumuley-Vazirani-Vazirani Isolation Lemma
Valiant-Vazirani as a corollary

Lecture Notes: Madhav Suresh

Lecture 6

PP closed under union and intersection

Scribe Notes by Georgios Askalidis

Lecture 5

SPP is low for Gap-P and the other Gap Classes
Beginning of PP closed under union

Lecture Notes by Prem Seetharaman

PP is Closed under Intersection by Beigel, Reingold and Spielman
PP Is Closed under Truth-Table Reductions by Fortnow and Reingold

Lecture 4

#P and Gap Classes

Lecture Notes by Geoff Hill

Graph Automorphism in SPP (Kobler, Schoning, Turan)
Graph Isomorphism in SPP (Arvind-Kurer)