Draft:Alex Kontorovich

Alex V. Kontorovich is an American mathematician who works in the areas of analytic number theory, automorphic forms and representation theory, L-functions, harmonic analysis, and homogeneous dynamics.

Bio and Career

Kontorovich earned a Bachelor's degree from Princeton University in 2002, and a PhD from Columbia University in 2007 where he studied under Dorian Goldfeld and Peter Sarnak. From 2007 to 2010 he was a Tamarkin Assistant Professor at Brown University. He was later an Assistant Professor at Stony Brook University and then Assistant Professor and Associate Professor at Yale University. Since 2014 he has been at Rutgers University, where he is currently a Distinguished Professor.

Kontorovich has held visiting positions at Harvard, ETH Zurich, and the Institute for Advanced Study. He served as the Editor-in-Chief for the journal Experimental Mathematics, and currently serves as Managing Editor for the Journal of the Association for Mathematical Research.

Work

In 2011, with Jean Bourgain, Kontorovich proved a conjecture from 1971 due to S. K. Zaremba, which stated that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant.

In 2008, he proved with Hee Oh a theorem about the fractal dimension of Apollonian circle packings (fractal dimension $\delta =1.30568\dots$, showing that number of circles with radius $r$ behaves asymptotically for large $r$ as $N(r)\sim C\cdot r^{\delta }$, where the constant $C$ depends on the first three circles that are mutually tangent to each other. In their analysis, they used number-theoretical and dynamic aspects of the problem, which were previously explored in particular by Jeffrey Lagarias, Peter Sarnak, Allan Wilks and Ronald Graham.

In his paper "From Apollonius to Zaremba: Local-global phenomena in thin orbits", Kontorovich suggested surprising connections between numerical-theoretical and geometric problems. The number-theoretical problem is the Zaremba problem mentioned above. The geometric problem is about so-called integral Soddy sphere packings (named after the chemist Frederick Soddy), generalizations of Apollonian circle packings in three dimensions, whereby the curvatures are integer. Kontorovich proved that sufficiently large natural numbers that meet certain congruence conditions of the problem can be represented as curvatures in such a sphere packing.

Kontorovich has also worked on the Collatz problem and developed stochastic models to predict the associated dynamics with Jeffrey Lagarias. Here and in a problem of the distribution of the values of L-functions, he showed with Steven J. Miller the validity of Benford's law.

Honors and Awards

Kontorovich is a Fellow of the American Mathematical Society. He was a Sloan Research Fellow from 2013 to 2015.

In 2014 Kontorovich received the Levi L. Conant Prize for the paper "From Apollonius to Zaremba: Local-global phenomena in thin orbits".

Selected Publications

• From Apollonius to Zaremba: Local-global phenomena in thin orbits. Bull. Amer. Math. Soc. 50 (2013), 187–228. Arxiv
• with Jean Bourgain: On the Local-Global Conjecture for Apollonian Gaskets. Inventiones Mathematicae 196 (2014), 589–650. Arxiv
• with Jean Bourgain: On Zaremba’s Conjecture. Annals of Mathematics 180 (2014), 137–196. Arxiv
• with Hee Oh: Apollonian Packings and Horospheres on Hyperbolic 3-manifolds. J. Amer. Math. Soc. 24 (2011), 603–648. Arxiv
• with Hee Oh: Almost Prime Pythagorean Triples in Thin Orbits. J. Reine Angew. Math. 667 (2012), 89–131. Arxiv

External links

• Homepage