Differential equations, which express a relationship between a function and its rates of change, are used
to understand many phenomena in science and engineering. Often, such as in population modelling, we
are interested in the eventual behaviour of these functions: Do they stabilize or tend to infinity? How
fast do they grow? Valued differential fields are mathematical structures that abstractly formalize these
questions and possible answers. This project will investigate valued differential fields from the
perspective of model theory, a branch of mathematical logic which involves studying properties of
mathematical structures that can be expressed in formal languages, in turn revealing new insights into
these structures and their complexity. In recent years the ideas and tools of model theory have had
fruitful applications in other areas of mathematics, a trend this project will continue in the setting of
valued differential fields.
Tremendous results have already been established in the model theory of ordered valued differential
fields. A main example is the ordered valued differential field of real transseries, and by now decisive
results are known about its algebraic and model theoretic properties. Real transseries and similar
ordered valued differential fields do not include any oscillating behaviour. Oscillating functions, such as
sinusoidal functions, play an important role in mathematics and in describing the real world, from water
waves to wave functions in quantum physics. Therefore, the focus of this project is the valued differential
field of complex transseries and other unordered valued differential fields like it, where oscillation is
permitted. The primary goals are to study their algebraic properties and use these to establish new
results in the model theory of valued differential fields.