## DEPARTMENTAL COLLOQUIUM: *Integer points on affine cubic surfaces*
**Abstract**: A cubic polynomial equation in four or more variables tends to have many integer solutions, while one in two variables has a limited number of such solutions. There is a body of work establishing results along these lines. On the other hand very little is known in the critical case of three variables. For special such cubics, which we call Markoff surfaces, a theory can be developed. We will review some of the tools used to deal with these and related problems. Joint works with Bourgain/Gamburd and with Ghosh.
Dr. Peter Sarnak will be the speaker in the twelfth annual Baylor Lecture Series in Mathematics when he visits Baylor University from April 3-6, 2019. He has held the Eugene Higgins Professor of Mathematics position at Princeton University since 2002 (succeeding Andrew Wiles) and is a permanent member of the Institute of Advanced Study at Princeton.
Sarnak was born in Johannesburg, South Africa and graduated with his B.Sc. (Honors) degree in mathematics in 1976. He earned his Ph.D. in mathematics in 1980 from Stanford, under the direction of Paul Cohen.
His previous academic appointments include professorships at the Courant Institute at New York University and Stanford University. He has been Professor of Mathematics at Princeton since 1991. He held the Chairmanship in the Department of Mathematics at Princeton from 1996-1999.
Sarnak has made major contributions to analysis and number theory. His 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. He is widely recognized internationally as one of the leading analytic number theorists of his generation. His early work on the existence of cusp forms led to the disproof of a conjecture of Atle Selberg. He has obtained the strongest known bounds towards the Ramanujan-Petersson conjectures for sparse graphs, and he was one of the first to exploit connections between certain questions of theoretical physics and analytic number theory. There are fundamental contributions to arithmetical quantum chaos, a term which he introduced, and to the relationship between random matrix theory and the zeros of L-functions. His work on subconvexity for Rankin-Selberg L-functions led to the resolution of Hilbertâ€™s eleventh problem (classification of quadratic forms over algebraic number fields). He also has a mathematical constant, the Hafner-Sarnak-McCurley constant (~.35) named after him.
Sarnak was elected to the National Academy of Sciences (USA) and Fellow of the Royal Society (FRS) 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. The University of the Witwatersrand conferred an honorary doctorate on Professor Sarnak in July 2014 for his distinguished contributions to the field of mathematics. He was elected to the 2018 class of fellows of the American Mathematical Society.
Among many academic honors, Sarnak was awarded the Polya Prize from SIAM 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.
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