In unpublished lecture notes, William A. Veech considered the
following potential property of the Möbius function:
"In any Furstenberg system of the Möbius function, the
zero-coordinate is orthogonal to any function measurable with
respect to the...
We construct examples showing that the correlation in the Mobius
disjointness conjecture can go to zero arbitrarily slowly. In fact,
our methods yield a more general result, where in lieu of μ(n) one
can put any bounded sequence such that the Cesàro...
I will discuss work in progress with Morgan Weiler on knot
filtered embedded contact homology (ECH) of open book
decompositions of S^3 along T(2,q) torus knots to deduce
information about the dynamics of symplectomorphisms of the genus
(q-1)/2 pages...
Behaviors of objects of algebraic interest, such as polynomials,
elliptic curves and number fields -- many of which are still
unknown -- fall into the field of arithmetic statistics.. By
bringing in Fourier analysis to interplay with a wide
variety...
We discuss some consequences of the existence of a Siegel zero
for various questions relating to the distribution of the prime
numbers, and in particular to conjectures of Hardy-Littlewood and
Chowla type. This is joint work with Joni Teravainen.
Abstract: Low-lying horocycles are known to equidistribute on
the modular curve. Here we consider the joint distribution of two
low-lying horocycles of different speeds in the product of two
modular curves and show equidistribution under certain...
This is a joint work with Reza Gheissari (Northwestern) and
Aukosh Jagannath (Waterloo), Outstanding paper award at NeurIPS
2022. We study the scaling limits of stochastic gradient descent
(SGD) with constant step-size in the high-dimensional regime...
Lagrangian Floer theory is a useful tool for studying the
structure of the homology of Lagrangian submanifolds. In some
cases, it can be used to detect more- we show it can detect the
framed bordism class of certain Lagrangians and in
particular...
I discuss the spectral and arithmetic side of the relative trace
formula of Kuznetsov type for congruence subgroups of SL(n, Z) with
applications to automorphic density theorems. A particular focus is
on properties of general Kloosterman sums as...
