We give a construction of a boundary (the Morse boundary) which can be assigned to any proper geodesic metric space and which is rigid, in the sense that a quasi-isometry of spaces induces a homeomorphism of boundaries. To obtain a more workable invariant than the homeomorphism type, I will introduce the metric Morse boundary and discuss notions of capacity and conformal dimensions of the metric Morse boundary. I will then demonstrate that these dimensions give useful invariants of relatively hyperbolic and mapping class groups. This is joint work with Matthew Cordes (Technion).

Inspired by the theory of JSJ decomposition for 3-manifolds, one can define the JSJ decomposition of a group as a maximal canonical way of cutting it up into simpler pieces using amalgamated products and HNN extensions. If the group in question has some sort of non-positive curvature property then one can define a boundary at infinity for the group, which captures its large scale geometry. The JSJ decomposition of the group is then reflected in the treelike structure of the boundary. In this talk I will discuss this connection in the case of hyperbolic groups and explain some of the ideas used in its proof by Brian Bowditch.

Asymptotic dimension is a large-scale analogue of Lebesgue covering dimension. I will give a gentle introduction to asymptotic dimension, prove some basic propeties and give some applications to group theory. I will then define coarse homology and explain how when defined, virtual cohomological dimension gives a lower bound on asymptotic dimension.

For a smooth projective scheme Y over W(k) we consider an element in the motivic Chow group of the reduction Y_m over the truncated Witt ring W_m(k) and give a "Hodge" criterion - using the crystalline cycle class in relative crystalline cohomology - for the element to the lift to the continuous Chow group of Y. The result extends previous work of Bloch-Esnault-Kerz on the p-adic variational Hodge conjecture to a relative setting. In the course of the proof we derive two new results on the relative de Rham-Witt complex and its Nygaard filtration, and work with relative syntomic complexes to define relative motivic complexes for a smooth, formal lifting of Y_m over W(W_m(k)).

to come

Oxford Mathematics Public Lectures

The Law of the Few - Sanjeev Goyal

The study of networks offers a fruitful approach to understanding human behaviour. Sanjeev Goyal is one of its pioneers. In this lecture Sanjeev presents a puzzle:

In social communities, the vast majority of individuals get their information from a very small subset of the group – the influencers, connectors, and opinion leaders. But empirical research suggests that there are only minor differences between the influencers and the others. Using mathematical modelling of individual activity and networking and experiments with human subjects, Sanjeev helps explain the puzzle and the economic trade-offs it contains.

Professor Sanjeev Goyal FBA is the Chair of the Economics Faculty at the University of Cambridge and was the founding Director of the Cambridge-INET Institute.

28 June 2017, 5.00-6.00pm, Lecture Theatre 1, Mathematical Institute Oxford.

Please email @email to register

Given two $k$-graphs $H$ and $F$, a perfect $F$-packing in $H$ is a collection of vertex-disjoint copies of $F$ in $H$ which together cover all the vertices in $H$. In the case when $F$ is a single edge, a perfect $F$-packing is simply a perfect matching. For a given fixed $F$, it is generally the case that the decision problem whether an $n$-vertex $k$-graph $H$ contains a perfect $F$-packing is NP-complete.

In this talk we describe a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect $F$-packings is polynomial time solvable. We then give applications of this tool. For example, we give a minimum $\ell$-degree condition for which it is polynomial time solvable to determine whether a $k$-graph satisfying this condition has a perfect matching (partially resolving a conjecture of Keevash, Knox and Mycroft). We also answer a question of Yuster concerning perfect $F$-packings in graphs.

This is joint work with Jie Han (Sao Paulo).
 

Groups which are "attached" to theories of fields, appearing in models of the theory  
as the automorphism groups of intermediate fields fixing an elementary submodel are called geometrically represented. 
We will discuss the concept ``geometric representation" in the case of pseudo finite fields.  Then will show that any group which is geometrically represented in a complete theory of a pseudo-finite field must be abelian. 
This result also generalizes to bounded PAC fields. This is joint work with Zoe Chatzidakis.