Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the corresponding nonlinear group of morphims of affine three space.[-]

Peter Sarnak is a South African-born mathematician with dual South-African and American nationalities. He has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is an editor of the Annals of Mathematics. He is known for his work in analytic number theory. Sarnak is also on the permanent faculty at the School of Mathematics of the Institute for Advanced Study. He also sits on the Board of Adjudicators and the selection committee for the Mathematics award, given under the auspices of the Shaw Prize.

Sarnak graduated University of the Witwatersrand (B.Sc. 1975) and Stanford University (Ph.D. 1980), under the direction of Paul Cohen. Sarnak's highly cited work (with A. Lubotzky and R. Philips) applied deep results in number theory to Ramanujan graphs, with connections to combinatorics and computer science.

Peter Sarnak was awarded the Polya Prize of Society of Industrial & Applied Mathematics in 1998, the Ostrowski Prize in 2001, the Levi L. Conant Prize in 2003, the Frank Nelson Cole Prize in Number Theory in 2005 and a Lester R. Ford Award in 2012. He is the recipient of the 2014 Wolf Prize in Mathematics.

He was also elected as member of the National Academy of Sciences (USA) and Fellow of the Royal Society (UK) in 2002. He was awarded an honorary doctorate by the Hebrew University of Jerusalem in 2010. He was also awarded an honorary doctorate by the University of Chicago in 2015.[-]