Valuations are a classical instrument which allows the study of the geometric properties of an object.
Suppose the object in question is convex. That is, it contains every segment joining every couple of
points in the object. In this case, such geometric descriptors can be classified under suitable topological
and geometric assumptions. A notable example is Hadwiger`s theorem, which characterizes all
continuous valuations on convex bodies invariant under translations and rotations. The theorem states
that intrinsic volumes span this space of valuations. These are functionals which are the staples of the so-
called Brunn-Minkowski theory. Hadwiger`s theorem has applications, for example, in integral geometry,
leading to kinematic and Kubota-type formulas. More recently, Alesker, Bernig, and Fu have initiated the
study of algebraic structures on spaces of valuations, which has led to substantial advancements and
new interpretations of the aforementioned integral-geometric formulas. Later, Colesanti, Ludwig, and
Mussnig introduced the theory of valuations on convex functions, which extended the theory of
valuations on convex bodies. In this framework, the study of algebraic structures on spaces of valuations
on convex functions is an untapped line of research that this project aims to pursue. We plan to
introduce algebraic operations on these functional spaces and study the structure that arises from them.
The final aim is to connect these structures with integral geometric formulas, as it happens in the
geometric case. This project interacts with many different fields of mathematics, such as geometric
measure theory, algebraic geometry, and calculus of variations, and we expect them to have substantial
interplay with our results.