# Blog Archives

These are the items posted in this seminar, currently ordered by their post-date, rather than by the event date. We will create improved views in the future. In the meantime, please click on the Seminar menu item above to find the page associated with this seminar, which does have a more useful view order.# Mordell-Lang and Manin-Mumford in positive characteristic, revisited

We give a reduction of function field Mordell-Lang to function field Manin-Mumford, in positive characteristic. The upshot is another account of or proof of function field Mordell-Lang in positive characteristic, avoiding the recourse to difficult results on Zariski geometries.

(This work is joint with Benoist and Bouscaren.)

# Picard-Vessiot structures

# Logarithmic-Exponential Series

I will survey some old work of van den Dries, Macintyre and myself. We construct an algebraic nonstandard model of the theory of the real exponential field. There is a natural derivation on the LE-series which is compatible with the exponential and the archimedean valuation.

# D-Fields as a Common Formalism for Difference and Differential Algebra

In a series of papers with Rahim Moosa, I have developed a theory of D-rings unifying and generalizing difference and differential algebra. Here we are given a ring functor D whose underlying additive group scheme is isomorphic to some power of the additive group. A D-ring is a ring R given together with a homomorphism f : R → D(R). A first motivating example is when D(R) = R[ε]/(ε2), so that the data of D-ring is that of an endomorphism σ:R → R and a σ-derivation ∂:R → R (that is, ∂(rs) = ∂(r)σ(s)+σ(r)∂(s)). Another example is when D(R) = R, where a D-ring structure is given by an endomorphism of R.

We develop a theory of prolongation spaces, jet spaces, and of D-algebraic geometry. With our most recent paper, we draw out the model theoretic consequences of this work showing that in characteristic zero, the theory of D-fields has a model companion, which we call the theory of D-closed fields, and that many of the refined model theoretic theorems (eg the Zilber trichotomy) hold at this level of generality. As a complement, we show that no such model companion exists in characteristic p under a mild hypothesis on D.

# Stability revisited

I will discuss an observation Ivo Herzog and I made in the last millennium that yields a purely topological definition of stability of a complete first-order theory in terms of their Stone spaces.

# A Set-Theoretic Approach to Model Theory

Although one always employs logic in proofs, the foundations of many branches of mathematics appear to be predominantly set-theoretic: one defines a topological space to be a pair (X, τ) consisting of a set X and a collection τ of subsets satisfying certain well-known properties; one defines a group to be a pair (G, μ) consisting of a set G and a subset μ ⊂ G × G × G as the binary operation satisfying certain well-known properties (of course, for a group one needs a bit more to handle the identity); etc. There are advantages to this commonality, particularly if one is well-versed in category theory: one can move from one area to the other and still have a fairly good idea of what the major problems are and the sort of techniques one might expect to see. In contrast, in Model Theory, the foundation appears to be heavily based on logic, and as a result the language and terminology can seem foreign to those who work in more widely publicized areas of mathematics. Rather than “sets of groups”, one hears about “sets of formulas”; rather than products (Cartesian, fibered, direct, or semi-direct), one hears of “ultraproducts”; rather than “reducing to a simpler case”, one is told about “eliminating quantifiers”.

In this talk I will indicate how some of the basic ideas of Model Theory can be formulated set-theoretically, that is, in the topological and algebraic spirit indicated above.