I will describe recent work in progress on
logarithmic--exponential preparation theorems in analytically
generated sharply o-minimal structures. Our results imply the sharp
o-minimality of ℝexp
as well as a uniform version of Wilkie’s conjecture on...
Globally valued fields form a generalisation of global fields
that fits into the context of first order (continuous) logic. I
will describe these structures, and outline how they are connected
to various parts of arithmetic geometry: Arakelov...
Let SO(3,R) be the 3D-rotation group equipped with the
real-manifold topology and the normalized Haar measure \mu.
Confirming a conjecture by Breuillard and Green, we show that if A
is an open subset of SO(3,R) with sufficiently small measure,
then...
This talk is based on a joint work with Steve Lester.
We review the Gauss circle problem, and Hardy's conjecture
regarding the order of magnitude of the remainder term. It is
attempted to rigorously formulate the folklore heuristics behind
Hardy's...
Ruzsa asked whether there exist Fourier-uniform subsets of ℤ/Nℤ
with very few 4-term arithmetic progressions (4-AP). The standard
pedagogical example of a Fourier uniform set with a "wrong" density
of 4-APs actually has 4-AP density much higher than...
I will discuss pointwise ergodic theory as it developed out of
Bourgain's work in the 80s, leading up to my work with Mirek and
Tao on bilinear ergodic averages.
In 1996 Manjul Barghava introduced a notion of P-orderings for
arbitrary sets S of a Dedekind domain, with respect to a prime
ideal P, which defined associated invariants called P-sequences. He
combined these invariants to define generalized...
In its dynamical formulation, the Furstenberg—Sárközy theorem
states that for any invertible measure-preserving system (X,μ,T),
any set A⊆X with μ(A) greater than 0, and any integer polynomial P
with P(0)=0,
The goal of this talk is to present new results dealing with the
asymptotic joint independence properties of commuting strongly
mixing transformations along polynomials. These results form
natural strongly mixing counterparts to various weakly and...
