Exercises for the Pcmi Summer School

Weak immersions of surfaces with L 2 - bounded second fundamental form Lecture 3 PCMI Graduate Summer School 2013

Weak immersions of surfaces with L 2 - bounded second fundamental form Lecture 2 PCMI Graduate Summer School 2013 Tristan Rivière

Triangles, Ellipses, and Cubic Polynomials

1. INTRODUCTION. Discussions that led to this paper began during an electronic version of the Secondary School Teachers Program, a part of the Park City Mathematics Institute (PCMI), offered at the University of Cincinnati in the summer of 2006 for local high school mathematics teachers. The authors were working with the teachers on daily problem sets provided by PCMI; complex numbers and polyn...

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Advanced Course on Computational Complexity Lecture 14: Lower Bounds for Tree Resolution

The Resolution proof system can be defined as a restriction of the Sequent Calculus that allows only variables to appear as formulas in the sequents and that uses only the cut rule. (This is not the usual definition of Resolution but it is equivalent to it.) What can we do with such limited proofs? One thing we cannot do is start with the axioms and build larger formulas because we cannot intro...

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 5: The Landscape of Complexity Classes

The inclusions L ⊆ NL and P ⊆ NP are trivial since deterministic machines can be viewed as special kinds of nondeterministic machines. The equality PSPACE = NPSPACE is a consequence of Savitch’s Theorem and the inclusion NL ⊆ P can also be obtained by that same idea of reducing to graph reachability. The fact that EXP contains PSPACE, and therefore NPSPACE, was pointed out just before the proof...