Topic Archive: algebraic geometry
Complexity of constructible sheaves
Constructible sheaves play an important role in several areas of mathematics, most notably in the theory of D-modules. It was proved by Kashiwara and Schapira that the category of constructible sheaves is closed under the standard six operations of Grothendieck. This result can be viewed as a far reaching generalization of the Tarski-Seidenberg principle in real algebraic geometry. In this talk I will describe some quantitative/effectivity results related to the Kashiwara-Schapira theorem – mirroring similar results for ordinary semi-algebraic sets which are now well known. For this purpose, I will introduce a measure of complexity of constructible sheaves, and bound the complexity of some of the standard sheaf operations in terms of this measure of complexity. I will also discuss quantitative bounds on the dimensions of cohomology groups of constructible sheaves in terms of their complexity, again extending similar results on bounding the the ordinary Betti numbers of semi-algebraic sets.
State of affairs in local uniformization and valuation theory in positive characteristic
This will be an informal talk summarizing roughly where we stand concerning the problem of local uniformization in positive characteristic. I will discuss the structure of valued algebraic function fields and the main open problems that are of particular interest for local uniformization. These include the dehenselization problem (an analogue of Temkin’s “decompletion”) as well as the question when the existence of a rational place implies that the ground field is existentially closed in the function field.
If time permits I will also discuss a stunning result about the badness of valuations in positive characteristic (due to Anna Blaszczok), and state a result (by Anna and myself) and an open question that are of interest for recent work by Koen Struyve et al. on “Euclidean buildings.”
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.
Why model-theorists shouldn’t think that ACF is easy
We all learned that stability theory derived many of its ideas from what happens in ACF, where everything is nice and easy. After all ACF has quantifier elimination and is strongly minimal, decidable, superstable, uncountably categorical, etc. However, my own struggles with ACF have humbled my opinion about it: it is an awfully rich theory that encodes way more than our current knowledge. I will discuss some examples showing how “difficult” ACF is: Grothendieck ring, isomorphism problem, set-theoretic intersection problem. Oddly enough, RCF seems to not have any of these problems. It is perhaps my ignorance, but I have come to think of RCF as much easier. Well, all, of course, is a matter of taste.