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Effective Methods in Tame Geometry and Applications in Arithmetic and Dynamics

Objective

Tame geometry studies structures in which every definable set has a
finite geometric complexity. The study of tame geometry spans several
interrelated mathematical fields, including semialgebraic,
subanalytic, and o-minimal geometry. The past decade has seen the
emergence of a spectacular link between tame geometry and arithmetic
following the discovery of the fundamental Pila-Wilkie counting
theorem and its applications in unlikely diophantine
intersections. The P-W theorem itself relies crucially on the
Yomdin-Gromov theorem, a classical result of tame geometry with
fundamental applications in smooth dynamics.

It is natural to ask whether the complexity of a tame set can be
estimated effectively in terms of the defining formulas. While a large
body of work is devoted to answering such questions in the
semialgebraic case, surprisingly little is known concerning more
general tame structures - specifically those needed in recent
applications to arithmetic. The nature of the link between tame
geometry and arithmetic is such that any progress toward effectivizing
the theory of tame structures will likely lead to effective results
in the domain of unlikely intersections. Similarly, a more effective
version of the Yomdin-Gromov theorem is known to imply important
consequences in smooth dynamics.

The proposed research will approach effectivity in tame geometry from
a fundamentally new direction, bringing to bear methods from the
theory of differential equations which have until recently never been
used in this context. Toward this end, our key goals will be to gain
insight into the differential algebraic and complex analytic structure
of tame sets; and to apply this insight in combination with results
from the theory of differential equations to effectivize key results
in tame geometry and its applications to arithmetic and dynamics. I
believe that my preliminary work in this direction amply demonstrates
the feasibility and potential of this approach.

Field of science

  • /natural sciences/mathematics/pure mathematics/arithmetic
  • /natural sciences/mathematics/pure mathematics/mathematical analysis/differential equations
  • /natural sciences/mathematics/pure mathematics/geometry

Call for proposal

ERC-2018-STG
See other projects for this call

Funding Scheme

ERC-STG - Starting Grant

Host institution

WEIZMANN INSTITUTE OF SCIENCE
Address
Herzl Street 234
7610001 Rehovot
Israel
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 155 027

Beneficiaries (1)

WEIZMANN INSTITUTE OF SCIENCE
Israel
EU contribution
€ 1 155 027
Address
Herzl Street 234
7610001 Rehovot
Activity type
Higher or Secondary Education Establishments