Bob Hough's home page

Department of Mathematics
Stony Brook University
Stony Brook, NY 11794
e-mail: robert.hough at
My cv.

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:

Specific research projects:

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:

  1. The shape of quartic fields. Preprint .
  2. The shape of cubic fields. Research in Mathematics, submitted. Preprint .
  3. Covering systems with restricted divisibility. Preprint .
  4. Sandpiles on the square lattice. Preprint .
  5. Maass form twisted Shintani L-functions. Proc. AMS, to appear. Preprint.
  6. Mixing and cut-off in cycle walks. Electronic Journal of Probability, to appear. Preprint.
  7. with P. Diaconis. Random walk on unipotent matrix groups. Preprint.
  8. with Y. Jiang. Asymptotic mixing time analysis of a random walk on the orthogonal group. Annals of Probability, to appear. Link.
  9. The angle of large values of L-functions. Journal of Number Theory, 167 (2016): 353--393. Link.
  10. The random k-cycle walk on the symmetric group. Probability Theory and Related Fields 165, no. 1 (2016): 447--482. Link.
  11. Solution of the minimum modulus problem for covering systems. Annals of Math 181, no. 1 (2015): 361--382. Link.
  12. The distribution of the logarithm of orthogonal and symplectic L-functions. Forum Math 26, no. 2 (2014): 523--546. Link. Errata.
  13. Zero-density estimate for modular form L-functions in weight aspect. Acta Arith. 154 (2012), 187-216. Link.
  14. The resonance method for large character sums. Mathematika 59, no. 01 (2013): 87--118. Link.
  15. Equidistribution of bounded torsion CM points. Journal d'Analyse Math, to appear..
  16. Summation of a random multiplicative function on numbers having few prime factors. Math. Proc. Camb. Phil. Soc., 150 (2011), pp. 193-214. Link .
  17. Tesselation of a triangle by repeated barycentric subdivision. Elec. Comm. Prob., 14 (2009). Link .

Older teaching material: