The basics of SDG will be presented axiomatically; the axioms are inconsistent on the background of classical logic, but consistent with intuitionistic logic. The basic axiom for the arithmetic of the number line is that there is a sufficient supply of nilpotent elements x (i.e. x^n =0 for some n). So the number line is not a field. Some concrete algebra for matrices with nilpotent entries is presented; it gives a geometric/combinatorial explanation for the occurrence of alternating multilinear forms in differential geometry.

A dialogue category is a monoidal category equipped with an exponentiating object. In this talk, I will introduce a notion of braided dialogue category, and explain how this notion provides a functorial bridge between proof theory and knot theory.

There is a numerical invariant of enriched categories called magnitude (defined subject to certain hypotheses). It is closely related to cardinality, Euler characteristic, entropy, and other fundamental notions of size. This talk will be a series of snapshots of the theory of magnitude: first an observation on elementary counting problems, then a bird's-eye view of the entire theory of magnitude, and finally a conjecture on the size of convex sets.

We give an updated account of the theory of strong endofunctors, in particular strong monads, on a cartesian closed category. We show that a commutative (strong) monad gives a framework for some of the reasoning with extensive quantities, in particular for distributions in a broad sense (probability distributions, mass distributions, ...)