I will discuss how the Lang–Trotter conjecture for pairs of
elliptic curves implies new cases of the Zilber–Pink conjecture for
curves in 3. Unlike previous results for curves in g, our result
does not rely on any assumption on intersections with...
Around 1921 Polya asked about the algebraicity of the indefinite
integral of an algebraic function, in terms of the integrality of
the coefficients in a power-series expansion. Polya nicely answered
the special case of rational functions, using...
Every o-minimal structure determines a collection of "tame" or
"definable" subsets of bbRn. We can then ask about the fragment of
complex geometry present in the structure: Which holomorphic
functions are definable, and which spaces are cut out by...
Ben Bakker explained to us how to construct moduli spaces of
polarized Hodge structures, and then produced period maps
associated to families of smooth projective varieties. However, in
practice one often encounters a family of smooth varieties...
The goal of these lectures is to present the fundamentals of
Simpson’s correspondence, generalizing classical Hodge theory,
between complex local systems and semistable Higgs bundles with
vanishing Chern classes on smooth projective varieties.
A Hodge structure is a certain linear algebraic datum.
Importantly, the cohomology groups of any smooth projective
algebraic variety come equipped with Hodge structures which encode
the integrals of algebraic differential forms over
topological...
Differential Galois groups are algebraic groups that describe
symmetries of some systems of differential equations. The solutions
considered can live in any differential field and thus a natural
framework to consider such symmetries is the setting...
A Hodge structure is a certain linear algebraic datum.
Importantly, the cohomology groups of any smooth projective
algebraic variety come equipped with Hodge structures which encode
the integrals of algebraic differential forms over
topological...
I'll focus specifically on point counting results in o-minimal
structures. I'll start with the classical theorem of Pila and
Wilkie and move on to improved versions that only hold in the
"sharp" variant of o-minimality.
I'll introduce o-minimality from a user's perspective assuming
zero background. I'll talk about some of the main examples of
o-minimal structures: as a user of o-minimality your first goal is
to find out whether your favorite set lives in one of...
