Bob Hough's home page
Department of Mathematics
Stony Brook University
Stony Brook, NY 11794
e-mail: robert.hough at stonybrook.edu
I am an assistant professor of mathematics at Stony Brook University, with research interests in probability and analytic number theory. Previously I have been a post-doctoral fellow at the Institute for Advanced Study, Princeton, and a member of a research team led by Ben Green at the Mathematical Institute, Oxford and DPMMS, Cambridge. I completed my PhD in Mathematics at Stanford University in 2012 under the supervision of K. Soundararajan. I have also completed a masters degree in computer science at Stanford, with an emphasis in algorithms.
Areas of research interest:
- Probability, discrete mathematics, analytic number theory
Specific research projects:
Several years ago I solved an old problem of Erdős by showing that the least modulus of a distinct covering system of congruences cannot be arbitrarily large. Here are talks by Ben Green, and myself discussing the solution. I have an ongoing research project with Pace Nielsen at BYU studying covering systems.
- A number of years ago Persi Diaconis tricked me into studying a random walk on a group, and I have been doing so from time to time ever since. I am especially interested in the cut-off phenomenon, in which a Markov chain transitions to stationarity in a narrow window about its mixing time. These papers use (new) integral formulae for the characters of the symmetric and (with Yunjiang Jiang) orthogonal groups. This preprint with Persi solves an old problem about the mixing time of coordinates in finite nilpotent groups.
- I have studied several distribution problems in analytic number theory, including the
and extreme values of L-functions, and the distribution of shapes of fixed torsion ideal classes in imaginary quadratic fields. In a recent preprint, I partially generalize the Shintani zeta function to an object giving spectral information about the distribution of the shapes of cubic orders. This is part of an ongoing project with Frank Thorne at USC and Takashi Taniguchi at Kobe University.
Slides on mixing and sandpiles, from a talk at the University of Washington probability seminar.
An illustration of Leon Green's Theorem.
The right figure is an orbit on the Heisenberg nilmanifold and the left is the projected orbit on the abelianization. Green's Theorem states that the first orbit is asymptotically equidistributed if and only if the second one is.
Publications and preprints:
- The shape of quartic fields. Preprint .
- The shape of cubic fields. Research in Mathematics, submitted. Preprint .
- Covering systems with restricted divisibility. Preprint .
- Sandpiles on the square lattice. Preprint .
- Maass form twisted Shintani L-functions. Proc. AMS, to appear. Preprint.
- Mixing and cut-off in cycle walks. Electronic Journal of Probability, to appear. Preprint.
- with P. Diaconis. Random walk on unipotent matrix groups. Preprint.
- with Y. Jiang. Asymptotic mixing time analysis
of a random walk on the orthogonal group. Annals of Probability, to appear. Link.
- The angle of large values of L-functions. Journal of Number Theory, 167 (2016): 353--393. Link.
- The random k-cycle walk on the symmetric group. Probability Theory and Related Fields 165, no. 1 (2016): 447--482. Link.
- Solution of the minimum modulus problem for covering systems. Annals of Math 181, no. 1 (2015): 361--382. Link.
- The distribution of the logarithm of orthogonal and symplectic L-functions. Forum Math 26, no. 2 (2014): 523--546.
- Zero-density estimate for modular form L-functions in weight aspect. Acta Arith. 154 (2012), 187-216.
- The resonance method for large character sums. Mathematika 59, no. 01 (2013): 87--118.
- Equidistribution of bounded torsion CM points.
Journal d'Analyse Math, to appear..
- Summation of a random multiplicative function on numbers having few prime factors. Math. Proc. Camb. Phil. Soc., 150 (2011), pp. 193-214.
- Tesselation of a triangle by repeated barycentric subdivision. Elec. Comm. Prob., 14 (2009).
Older teaching material:
- At Cambridge I gave a Part III course in analytic number theory. Topics included the prime number theorem in arithmetic progressions, Linnik's theorem on the least prime in an arithmetic progression, and Selberg's theorem on the distribution of log zeta on the half-line.
At Stanford I was TA for the 3-term honors multivariable calculus sequence. Handouts from my sections are available
- Some practice problems for Stanford's analysis qual.
Information about Stanford's Polya problem solving seminar is
here and here.