Oxford

Auctioneers may wish to sell related but different indivisible goods in

a single process. To develop such techniques, we study the geometry of

how an agent's demanded bundle changes as prices change. This object

is the convex-geometric object known as a `tropical hypersurface'.

Moreover, simple geometric properties translate directly to economic

properties, providing a new taxonomy for economic valuations. When

considering multiple agents, we study the unions and intersections of

the corresponding tropical hypersurfaces; in particular, properties of

the intersection are deeply related to whether competitive equilibrium

exists or fails. This leads us to new results and generalisations of

existing results on equilibrium existence. The talk will provide an

introductory tour to relevant economics to show the context of these

applications of tropical geometry. This is joint work with Paul

Klemperer.

We will introduce a few class of generalised metrisable

properties; that is, properties that hold of all metrisable spaces that

can be used to generalise results and are in some sense 'close' to

metrisability. In particular, we will discuss Moore spaces and the

independence of the normal Moore space conjecture - Is every normal

Moore space metrisable?

In some of their recent work Derbez and Wang studied volumes of representations of 3-manifold groups into the Lie groups $$Iso_e \widetilde{SL_2(\mathbb{R})} \mbox{ and }PSL(2,\mathbb{C}).$$ They computed the set of all volumes of representations for a fixed prime closed oriented 3-manifold with $$\widetilde{SL_2(\mathbb{R})}\mbox{-geometry}$$ and used this result to compute some volumes of Graph manifolds after passing to finite coverings.

In the talk I will give a brief introduction to the theory of volumes of representations and state some of Derbez' and Wang's results. Then I will prove an additivity formula for volumes of representations into $$Iso_e \widetilde{SL_2(\mathbb{R})}$$ which enables us to improve some of the results of Derbez and Wang.

We explore the technique of elementary submodels to prove 
results in topology and set theory. We will in particular prove the 
delta system lemma, and Arhangelskii's result that a first countable 
Lindelof space has cardinality not exceeding continuum.

I will explain why one can symplectically embed closed symplectic manifolds (with integral symplectic form) into CPⁿ and compute the weak homotopy type of the space of all symplectic embeddings of such a symplectic manifold into CP^∞.