In this lecture, we will review recent works regarding
spectral statistics of the normalized adjacency
matrices of random $d$-regular graphs on $N$ vertices.
Denote their eigenvalues by $\lambda_1=d/\sqrt{d-1}\geq
\la_2\geq\la_3\cdots\geq \la_N$...
Ratner's landmark equidstribution results for unipotent flows
have had dramatic applications in many mathematical areas. Recently
there has been considerable progress in the long sought for goal of
getting effective equidistribution results for...
(Joint with Samuel Grushevsky, Gabriele Mondello, Riccardo
Salvati Manni) We determine the maximal dimension of a compact
subvariety of the moduli space of principally polarized abelian
varieties Ag for any value of g. For g<16 the dimension is g−1,
while for g≥16, it is determined by the larged dimensional compact
Shimura subvariety, which we determine. Our methods rely on
deforming the boundary using special varieties, and functional
transcendence theory.
To study the asymptotic behavior of orbits of a dynamical
system, one can look at orbit closures or invariant measures. When
the underlying system has a homogeneous structure, usually coming
from a Lie group, with appropriate assumptions a wide...
the special values of Dirichlet L-functions have long been a
source of fascination and frustration. From Euler's solution in
1734 of the Basel problem to Apery's proof in 1978 that
zeta(3...
Cohen, Lenstra, and Martinet have given highly influential
conjectures on the distribution of class groups of number fields,
the finite abelian groups that control the factorization in number
fields. Malle, using tabulation of class groups of number...
The Higher order Fourier uniformity conjecture asserts that on
most short intervals, the Mobius function is asymptotically uniform
in the sense of Gowers; in particular, its normalized Fourier
coefficients decay to zero. This conjecture is known
to...
What is the densest lattice sphere packing in the d-dimensional
Euclidean space? In this talk we will investigate this question as
dimension d goes to infinity and we will focus on the lower bounds
for the best packing density, or in other words on...
Discrete subgroups of PSL(2,C) are called Kleinian groups and
they are fundamental groups of complete oriented hyperbolic
3-manifolds/orbifolds. Except for countably many conjugacy classes,
all Kleinian groups have infinite co-volume in PSL(2, C).
