L3

An externally definable set of a first order structure $M$ is a set of the form $X\cap M^n$ for a set $X$ that is parametrically definable in some elementary extension of $M$. By a theorem of Shelah, these sets form again a first order structure if $M$ is NIP. If $M$ is a real closed field, externally definable sets can be described as some sort of limit sets (to be explained in the talk), in the best case as Hausdorff limits of definable families. It is conjectured that the Shelah structure on a real closed field is generated by expanding the field with convex subsets of the line. This is known to be true in the archimedean case by van den Dries (generalised by Marker and Steinhorn). I will report on recent progress around this question, mainly its confirmation on real closed fields that are close to being maximally valued with archimedean residue field. The main tool is an algebraic characterisation of definable types in real closed valued fields. I also intend to give counterexamples to a localized version of the conjecture. This is joint work with Francoise Delon.

The Schelling segregation model has been extensively studied, by researchers in fields as diverse as economics, physics and computer science. While the explicit concern when the model was first introduced back in 1969, was to model the kind for racial segregation observed in large American cities, the model is sufficiently abstract to apply to almost situation in which agents or nodes arrange themselves geographically according to a preference not to be of a minority type within their own neighbourhhood. Kirman and Vinkovik have established, for example, that Schelling's model is a finite difference version of a differential equation describing interparticle forces (and applied in the modelling of cluster formation). Despite the large literature relating to the model, however, it has largely resisted rigorous analysis -- it has not been possible to prove the segregation behaviour easily observed when running simulations. For the first time we have now been able to rigorously analyse the model, and have also established some rather surprising threshold behaviour.

This talk will require no specialist background knowledge.

It has long been a challenge to synthesize the complementary insights offered by model theory and category theory. A small fragment of that challenge is to understand ultraproducts categorically. I will show that, granted some general categorical machinery, the notions of ultrafilter and ultraproduct follow inexorably from the notion of finiteness of a set. The machine in question, known as the codensity monad, has existed in an underexploited state for nearly fifty years. To emphasize that it was not constructed specifically for this purpose, I will mention some of its other applications. This talk represents joint work with an anonymous referee. Little knowledge of category theory will be assumed.

              Conventional decoherence (usually called 'Environmental

Decoherence') is supposed to be a result of correlations

established between some quantum system and the environment.

'Intrinsic decoherence' is hypothesized as being an essential

feature of Nature - its existence would entail a breakdown of

quantum mechanics. A specific mechanism of some interest is

'gravitational decoherence', whereby gravity causes intrinsic

decoherence.

I will begin by discussing what is now known about the mechanisms of

environmental decoherence, noting in particular that they can and do

involve decoherence without dissipation (ie., pure phase decoherence).

I will then briefly review the fundamental conflict between Quantum

Mechanics and General Relativity, and several arguments that suggest

how this might be resolved by the existence of some sort of 'gravitational

decoherence'.  I then outline a theory of gravitational decoherence

(the 'GR-Psi' theory) which attempts to give a quantitative discussion of

gravitational decoherence, and which makes predictions for

experiments.

The weak field regime of this theory (relevant to experimental

predictions) is discussed in detail, along with a more speculative

discussion of the strong field regime.

Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.

A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.

The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.

Cutting-edge experiments in quantum communications are reaching regimes

where relativistic effects can no longer be neglected. For example, there

are advanced plans to use satellites to implement teleportation and quantum

cryptographic protocols. Relativistic effects can be expected at these

regimes: the Global Positioning System (GPS), which is a system of

satellites that is used for time dissemination and navigation, requires

relativistic corrections to determine time and positions accurately.

Therefore, it is timely to understand what are the effects of gravity and

motion on entanglement and other quantum properties exploited in quantum

information.

In this talk I will show that entanglement can be created or degraded by

gravity and non-uniform motion. While relativistic effects can degrade the

efficiency of teleportation between moving observers, the effects can also

be exploited in quantum information. I will show that the relativistic

motion of a quantum system can be used to perform quantum gates. Our

results, which will inform future space-based experiments, can be

demonstrated in table-top experiments using superconducting circuits.

We prove that quasi-trees of spaces satisfying the axiomatisation given by Bestvina, Bromberg and Fujiwara are quasi-isometric to tree-graded spaces in the sense of Dru\c{t}u and Sapir. We then present a technique for obtaining `good' embeddings of such spaces into $\ell^p$ spaces, and show how results of Bestvina-Bromberg-Fujiwara and Mackay-Sisto allow us to better understand the metric geometry of such groups.

A-infinity algebras arise whenever one has a multiplication which is "associative up to homotopy". There is an important theory of minimal models which involves studying differential graded algebras via A-infinity structures on their homology algebras. However, this only works well over a ground field. Recently Sagave introduced the more general notion of a derived A-infinity algebra in order to extend the theory of minimal models to a general commutative ground ring.

Operads provide a very nice way of saying what A-infinity algebras are - they are described by a kind of free resolution of a strictly associative structure. I will explain the analogous result for derived A_infinity algebras - these are obtained in the same manner from a strictly associative structure with an extra differential.

This is joint work with Muriel Livernet and Constanze Roitzheim.