Tue, 11 May 2010

12:00 - 13:00
L3

Axions, Inflation and the Anthropic Principle

Katherine Mack (Cambridge)
Abstract

The QCD axion is the leading solution to the strong-CP problem, a

dark matter candidate, and a possible result of string theory

compactifications. However, for axions produced before inflation, high

symmetry-breaking scales (such as those favored in string-theoretic axion

models) are ruled out by cosmological constraints unless both the axion

misalignment angle and the inflationary Hubble scale are extremely

fine-tuned. I will discuss how attempting to accommodate a high-scale axion

in inflationary cosmology leads to a fine-tuning problem that is worse than

the strong-CP problem the axion was originally invented to solve, and how

this problem is exacerbated when additional axion-like fields from string

theory are taken into account. This problem remains unresolved by anthropic

selection arguments commonly applied to the high-scale axion scenario.

Thu, 28 Jan 2010
17:00
L3

Diophantine Sets of Polynomials over Number Fields

Jeroen Demeyer
(Ghent)
Abstract

 

Let R be a number field (or a recursive subring of anumber field) and consider the polynomial ring R[T].

We show that the set of polynomials with integercoefficients is diophantine (existentially definable) over R[T].

Applying a result by Denef, this implies that everyrecursively enumerable subset of R[T]^k is diophantine over R[T].

Tue, 26 Jan 2010

14:30 - 15:30
L3

Tree packing conjectures; Graceful tree labelling conjecture

Jan Hladky
(University of Warwick)
Abstract

A family of graphs $H_1,...,H_k$ packs into a graph $G$ if there exist pairwise edge-disjoint copies of $H_1,...,H_k$ in $G$. Gyarfas and Lehel conjectured that any family $T_1, ..., T_n$ of trees of respective orders $1, ..., n$ packs into $K_n$. A similar conjecture of Ringel asserts that $2n$ copies of any trees $T$ on $n+1$ vertices pack into $K_{2n+1}$. In a joint work with Boettcher, Piguet, Taraz we proved a theorem about packing trees. The theorem implies asymptotic versions of the above conjectures for families of trees of bounded maximum degree. Tree-indexed random walks controlled by the nibbling method are used in the proof.

In a joint work with Adamaszek, Adamaszek, Allen and Grosu, we used the nibbling method to prove the approximate version of the related Graceful Tree Labelling conjecture for trees of bounded degree.

In the talk we shall give proofs of both results. We shall discuss possible extensions thereof to trees of unbounded degree.

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