### AIM Five-Year Fellows

The American Institute of Mathematics is pleased to announce
that recipients of the AIM Five-Year Fellowships for the
year 2000 are Henry Cohn of Harvard University and Vadim Kaloshin of
Princeton University. They were chosen from a pool of more than 120
applicants. As part of the Fellowship each will receive 60 months of
full-time research support.

Henry Cohn received his S.B. in mathematics from MIT in 1995.
He is finishing his PhD this year under the direction of Noam Elkies;
his thesis will be entitled "New bounds on sphere packings."
In his thesis, Henry develops new techniques which improve
upper estimates on the packing density of spheres in Euclidean
spaces of dimensions 4 through 36. His bounds in dimensions 8 and 24
are only 0.0001% and 0.07% higher than the densities of known packings
(E_8 and the Leech lattice). Henry has already published several
papers in combinatorics and number theory; notable are his work
with N. Elkies and J. Propp, "Local Statistics for Random Domino
Tilings of the Aztec Diamond," (Duke Mathematical Journal 85 (1996),
and his paper with R. Kenyon and J. Propp entitled "A variational principle for
domino tilings" which has been accepted for publication in the Journal
of the American Mathematical Society. He is also interested in
the question of the irrationality of the Riemann zeta-function for
arguments which are odd integers that are 5 or more.

Vadim Kaloshin received his B.S. from Moscow State in 1994.
He is finishing his PhD this year in the area of dynamical systems
under the direction of John Mather. His thesis is entitled
"Growth of number of periodic orbits of generic diffeomorphisms."
As part of his thesis, Vadim gives a new, more elementary, proof
of the Artin-Mazur result that "every smooth invertible self-map of a
compact manifold can be approximated by one for which the number
of periodic points of period p is less than an exponential function
of p." This proof has been published in the Annals of Mathematics,
vol. 150 (1999). As another part of his thesis (to appear in
Communications of Mathematical Physics) he shows that the number
of periodic points of period p can grow arbitrarily fast with p
for a generic set of smooth invertible self-maps of a compact manifold. Other notable work includes a paper with B. Hunt
(Nonlinearity, 1999) where it is shown that the Hausdorff dimension
of a fractal set in a Banach space is not necessarily preserved
under projection to finite dimensional Euclidean space (whereas
the Hausdorff dimension is preserved under projection from one finite
dimensional space to another).