Logic, Set Theory and Topology Ical Atom

Zilber showed that a compact complex manifold $M$, equipped with the structure generated by the collection of all complex analytic subsets of each $M^n$, is well-behaved from a logical perspective, forming a Zariski geometry in the sense of Hrushovski and Zilber. This has led to fruitful model-theoretic developments, including a classification of definable groups, the isolation of the canonical base property, and a theory of generic automorphisms.

Motivated by this example, we will examine the possibility of emulating this theory in the setting of an almost complex manifold, a real manifold equipped with a smoothly-varying complex vector space structure on each tangent space. In particular, we will define the notion of a pseudoanalytic subset of an almost complex manifold. We develop some rudimentary almost complex analytic geometry, including an identity principle for almost complex maps, and an analysis of the singular part of a pseudoanalytic subset under some algebraic conditions. The lack of a true algebraic theory means that geometric methods, including pseudoholomorphic curves and almost complex connections, have to pick up the slack. These results hint at routes towards an almost complex analogue of Zilber’s theorem.

In the first part of the talk I briefly outline Hindman’s finite sums theorem, a famous Ramsey-theoretic result in algebra, its precursors and some generalizations, and the algebra of ultrafilters, a powerful technique providing a tool for getting similar results in combinatorics, algebra, and dynamics, see [1]. In the second (main) part I apply a multidimensional generalization of Hindman’s theorem (proved by Hindman and Bergelson) to show topologizability of certain algebras.

The topologization problem for groups and rings was first posed by Markov Jr. and then studied by various authors. I consider universal algebras consisting of an Abelian group and a family of additional operation (of arbitrary arity) distributive w.r.t. the group addition. Such algebras are called here polyrings; their instances include rings, modules, algebras over a field, differential rings, etc. Given a polyring $K$, a closed subbasis of the Zariski topology on the Cartesian product $K^n$ consists of finite unions of sets of roots of equations $t(x_1,\ldots,x_n)=0$ for all terms $t$ in $n$ variables.

The main theorem (a proof of which I plan to sketch) states that, for every infinite polyring $K$ and every $n>0$, sets definable by terms in $<n$ variables are nowhere dense in the space $K^n$. In particular, $K^n$ is nowhere dense in $K^{n+1}$. A fortiori, all the spaces $K^n$ are non-discrete (this fact was earlier stated by Arnautov for $K$ a ring and $n=1$). For more details, see [2].

References [1] N. Hindman, D. Strauss, Algebra in the Stone-Cech compactification: Theory and applications, W. de Gruyter, 2nd ed., 2012. [2] D. I. Saveliev, On Zariski topologies on polyrings, Russian Math. Surveys, 72:4 (2017), in press.

Zilber’s trichotomy conjecture, in modern formulation, distinguishes three flavours of geometries of strongly minimal sets — disintegrated/trivial, modular, and the geometry of an ACF. Each of these three flavours has a classic ``template’’ — a set with no structure, a projective space over a prime field, and an algebraically closed field, respectively. The class of ab initio constructions with which Hrushovski refuted the conjecture features a new flavour of geometries — non-modular, yet prohibiting any algebraic structure.

In this talk we take a step towards defining ``template’’ structures for the class of (CM-trivial) ab initio Hrushovski constructions. After presenting intuitively the standard ab initio Hrushovski construction, we generalize Hrushovski’s predimension function, showing that the geometries associated to certain Hrushovski constructions are, essentially, ab initio constructions themselves. If time permits, we elaborate on how these \emph{geometric} structures may generate the class of geometries of ab initio constructions under the Hrushovski fusion operation.