L3

I will present some new results on classifying 123 TQFTs,
using a 2-categorical approach. The invariants defined by a TQFT are
described using a new graphical calculus, which makes them easier to
define and to work with. Some new and interesting physical phenomena
are brought out by this perspective, which we investigate. I will
finish by banishing some TQFT myths! This talk is based on joint work
with Bruce Bartlett, Chris Schommer-Pries and Chris Douglas.

Recent papers by Butler-Campbell-Kovàcs, Rump, Prihoda-Puninski and others introduce over an order O over a Dedekind domain D a notion of "generalized lattice", meaning a D-projective O-module.

We define a similar notion over Dedekind-like rings -- a class of rings intensively studied by Klingler and Levy. We examine in which cases every generalized lattices is a direct sum of ordinary -- i.e., finitely generated -- lattices. We also consider other algebraic and model theoretic questions about generalized lattices.

"We will talk on stability, simplicity, nip, etc of partial types. We will review some known results and we will discuss some open problems."

The points in question can be found on  any semi-abelian surface over an elliptic curve with complex multiplication. We will show that they provide counter-examples to natural expectations in a variety of fields :  Galois representations (following K. Ribet's initial study from the 80's), Lehmer's problem on heights, and more recently, the relative  analogue of the Manin-Mumford conjecture. However, they do support Pink's general conjecture on special subvarieties of mixed Shimura varieties.