L6

I will give a short introduction to geometric stability theory and independence relations, focussing on the tree properties. I will then introduce one of the main examples for general measureable structures, the two sorted structure of a vector space over a field with a bilinear form. I will state some results for this structure, and give some open questions. This is joint work with William Anscombe.

Cardinal characteristics of the continuum are (definitions for) cardinals that are provably uncountable and at most the cardinality c of the reals, but which (if the continuum hypothesis fails) may be strictly less than c.  Cichon's diagram is a standard diagram laying out all of the ZFC-provable inequalities between the most familiar cardinal characteristics of the continuum.  There is a natural analogy that can be drawn between these cardinal characteristics and highness properties of Turing oracles in computability theory, with implications taking the place of inequalities.  The diagram in this context is mostly the same with a few extra equivalences: many of the implications were trivial or already known, but there remained gaps, which in joint work with Brendle, Ng and Nies we have filled in.

This is a report on joint work (still in progress) with Ellen Eischen, Jian-Shu Li,
and Chris Skinner.  I will describe the general structure of our construction of p-adic L-functions
attached to families of ordinary holomorphic modular forms on Shimura varieties attached to
unitary groups.  The complex L-function is studied by means of the doubling method;
its p-adic interpolation applies adelic representation theory to Ellen Eischen's Eisenstein 
measure.

We know since almost a century that a ball can be decomposed into five pieces and these pieces rearranged so as to produce two balls of the same size as the original. This apparent paradox has led von Neumann to the notion of amenability which is now much studied in many areas of mathematics. However, the initial paradox has remained tied down to an elementary property of free groups of rotations for most of the 20th century. I will describe recent progress leading to new paradoxical groups.

Erdos, Faudree, Gould, Gyarfas, Rousseau and Schelp, conjectured that
whenever the edges of a complete graph are coloured using three colours
there always exists a set of at most three vertices which have at least
two-thirds of their neighbours in one of the colours.  We will describe a
proof of this conjecture. This is joint work with Rahil Baber

In this talk I will discuss a deformation principle for cohomology with coefficients in representations on Banach spaces. The

main idea is that small, metric perturbations of representations do not change the vanishing of cohomology in degree n, provided that

we have additional information about the cohomology in degree n+1. The perturbations considered here happen only on the generators of a

group and even for isometric representations give rise to unbounded representations. Applications include fixed point properties for

affine actions and strengthening of Kazhdan’s property (T). This is joint work with Uri Bader.

The filtration on the infinite symmetric product of spheres by number of

factors provides a sequence of spectra between the sphere spectrum and

the integral Eilenberg-Mac Lane spectrum. This filtration has received a

lot of attention and the subquotients are interesting stable homotopy

types.

In this talk I will discuss the equivariant stable homotopy types, for

finite groups, obtained from this filtration for the infinite symmetric

product of representation spheres. The filtration is more complicated

than in the non-equivariant case, and already on the zeroth homotopy

groups an interesting filtration of the augmentation ideal of the Burnside

rings arises. Our method is by `global' homotopy theory, i.e., we study

the simultaneous behaviour for all finite groups at once. In this context,

the equivariant subquotients are no longer rationally trivial, nor even

concentrated in dimension 0.

Knot Floer homology (introduced by Ozsvath-Szabo and independently by

Rasmussen) is a powerful tool for studying knots and links in the 3-sphere. In

particular, it gives rise to a numerical invariant, which provides a

nontrivial lower bound on the 4-dimensional genus of the knot. By deforming

the definition of knot Floer homology by a real number t from [0,2], we define

a family of homologies, and derive a family of numerical invariants with

similar properties. The resulting invariants provide a family of

homomorphisms on the concordance group. One of these homomorphisms can be

used to estimate the unoriented 4-dimensional genus of the knot. We will

review the basic constructions for knot Floer homology and the deformed

theories and discuss some of the applications. This is joint work with

P. Ozsvath and Z. Szabo.

We consider a large class of three dimensional continuous dynamic fluctuation models, and show that they all rescale and converge to the stochastic Allen-Cahn equation, whose solution should be interpreted after a suitable renormalization procedure. The interesting feature is that, the coefficient of the limiting equation is different from one's naive guess, and the renormalization required to get the correct limit is also different from what one would naturally expect. I will also briefly explain how the recent theory of regularity structures enables one to prove such results. Joint work with Martin Hairer.