L3

(Joint work with Artem Chernikov.) In the talk, we will first recall some basic results on valued difference fields, both from an algebraic and from a model-theoretic point of view. In particular, we will give a description, due to Hrushovski, of the theory VFA of the non-standard Frobenius acting on an algebraically closed valued field of residue characteristic 0, as well as an Ax-Kochen-Ershov type result for certain valued difference fields which was proved by Durhan. We will then present a recent work where it is shown that VFA does not have the tree property of the second kind (i.e., is NTP2); more generally, in the context of the Ax-Kochen-Ershov principle mentioned above, the valued difference field is NTP2 iff both the residue difference field and the value difference group are NTP2. The property NTP2 had already been introduced by Shelah in 1980, but only recently it has been shown to provide a fruitful ‘tameness’ assumption, e.g. when dealing with independence notions in unstable NIP theories (work of Chernikov-Kaplan).

Newelski suggested that topological dynamics could be used to extend "stable group theory" results outside the stable context. Given a group G, it acts on the left on its type space S_G(M), i.e. (G,S_G(M)) is a G-flow. If every type is definable, S_G(M) can be equipped with a semigroup structure *, and it is isomorphic to the enveloping Ellis semigroup of the flow. The topological dynamics of (G,S_G(M)) is coded in the Ellis semigroup and in its minimal G-invariant subflows, which coincide with the left ideals I of S_G(M). Such ideals contain at least an idempotent r, and r*I forms a group, called "ideal group". Newelski proved that in stable theories and in o-minimal theories r*I is abstractly isomorphic to G/G^{00} as a group. He then asked if this happens for any NIP theory. Pillay recently extended the result to fsg groups; we found instead a counterexample to Newelski`s conjecture in SL(2,R), for which G/G^{00} is trivial but we show r*I has two elements. This is joint work with Jakub Gismatullin and Anand Pillay.

We discuss the question of the existence of the smallest size of a family of Banach spaces of a given density which embeds all Banach spaces of that same density. We shall consider two kinds of embeddings, isometric and isomorphic. This type of question is well studied in the context of separable spaces, for example a classical result by Banach states that C([0,1]) embeds all separable Banach spaces. However, the nonseparable case involves a lot of set theory and the answer is independent of ZFC.

I shall present a very general class of virtual elements in a structure, ultraimaginaries, and analyse their model-theoretic properties.

The lecture will discuss some applications of topology to a number of interesting physical systems:

1. Classifications of Phases, 2. Classifications of one-dimensional textures in Nematics and Superfluid HE-3,

3. Classification of defects, 4. Phase transition in Liquid membranes.

The solution of these problems leads to interesting mathematics but the talk will also include some historical remarks.

Motived by the desire to study geometry over the 'field with one element', in the past decade several authors have constructed extensions of scheme theory to geometries locally modelled on algebraic objects more general than rings. Semi-ring schemes exist in all of these theories, and it has been suggested that schemes over the semi-ring T of tropical numbers should describe the polyhedral objects of tropical geometry. We show that this is indeed the case by lifting Payne's tropicalization functor for subvarieties of toric varieties to the category of T-schemes. There are many applications such as tropical Hilbert schemes, tropical sheaf theory, and group actions and quotients in tropical geometry. This project is joint work with N. Giansiracusa (Berkeley).

Given a flat torus, we consider certain discrete graph approximations of

it and determine the asymptotics of the number of spanning trees

("complexity") of these graphs as the mesh gets finer. The constants in the

asymptotics involve various notions of determinants such as the

determinant of the Laplacian ("height") of the torus. The analogy between

the complexity of graphs and the height of manifolds was previously

commented on by Sarnak and Kenyon. In dimension two, similar asymptotics

were established earlier by Barber and Duplantier-David in the context of

statistical physics.

Our proofs rely on heat kernel analysis involving Bessel functions, which

in the torus case leads into modular forms and Epstein zeta functions. In

view of a folklore conjecture it also suggests that tori corresponding to

densest regular sphere packings should have approximating graphs with the

largest number of spanning trees, a desirable property in network theory.

Joint work with G. Chinta and J. Jorgenson.