SR1

In this talk we present a general method using permanent estimates in order to obtain results about counting and packing Hamilton cycles in dense graphs and oriented graphs. As a warm up we prove that every Dirac graph $G$ contains at least $(reg(G)/e)^n$ many distinct Hamilton cycles, where $reg(G)$ is the maximal degree of a spanning regular subgraph of $G$. We continue with strengthening a result of Cuckler by proving that the number of oriented Hamilton cycles in an almost $cn$-regular oriented graph is $(cn/e)^n(1+o(1))^n$, provided that $c$ is greater than $3/8$. Last, we prove that every graph $G$ of minimum degree at least $n/2+\epsilon n$ contains at least $reg_{even}(G)-\epsilon n$ edge-disjoint Hamilton cycles, where $reg_{even}(G)$ is the maximal even degree of a spanning regular subgraph of $G$. This proves an approximate version of a conjecture made by Osthus and K\"uhn.  Joint work with Michael Krivelevich and Benny Sudakov.

In an Erdős--R\'enyi random graph above the phase transition, i.e.,

where there is a giant component, the size of (number of vertices in)

this giant component is asymptotically normally distributed, in that

its centred and scaled size converges to a normal distribution. This

statement does not tell us much about the probability of the giant

component having exactly a certain size. In joint work with B\'ela

Bollob\'as we prove a `local limit theorem' answering this question

for hypergraphs; the graph case was settled by Luczak and Łuczak.

The proof is based on a `smoothing' technique, deducing the local

limit result from the (much easier) `global' central limit theorem.

I'll sketch some context for future and past research around valued fields
and motivic integration, from a model theoretic viewpoint, leaving out technical details. 
The talk will be partly conjectural.

Let $X$ be a variety and $Z$ be a sub-variety. Can one glue vector bundles on $X-Z$ with vector bundles on some small neighborhood of $Z$? We survey two recent results on the process of gluing a vector bundle on the complement of a sub-variety with a vector bundle on some 'small' neighborhood of the sub-variety. This is joint work. The first with M. Temkin and is about gluing categories of coherent sheaves over the category of coherent sheaves on a Berkovich analytic space. The second with J. Block and is about gluing dg enhancements of the derived category of coherent sheaves.