We outline some concepts and issues around decidability in valued fields of positive characteristic.
For full details please visit:
http://blogs.bodleian.ox.ac.uk/adalovelace/files/2015/10/Ada-Lovelace-S…
Starting from Hilbert's 10th problem, I will explain how to characterise the set of integers by non-solubility of a set of polynomial equations and discuss related challenges. The methods needed are almost entirely elementary; ingredients from algebraic number theory will be explained as we go along. No knowledge of first-order logic is necessary.
Let $G(n,d)$ be a random $d$-regular graph on $n$ vertices. In 2004 Kim and Vu showed that if $d$ grows faster than $\log n$ as $n$ tends to infinity, then one can define a joint distribution of $G(n,d)$ and two binomial random graphs $G(n,p_1)$ and $G(n,p_2)$ -- both of which have asymptotic expected degree $d$ -- such that with high probability $G(n,d)$ is a supergraph of $G(n,p_1)$ and a subgraph of $G(n,p_2)$. The motivation for such a coupling is to deduce monotone properties (like Hamiltonicity) of $G(n,d)$ from the simpler model $G(n,p)$. We present our work with A. Dudek, A. Frieze and A. Rucinski on the Kim-Vu conjecture and its hypergraph counterpart.
Compact F-spaces play an important role in the area of compactification theory, the prototype being w*, the Stone-Cech remainder of the integers. Two curious topological characteristics of compact F-spaces are that they don’t contain convergent sequences (apart from the constant ones), and moreover, that they often contain points that don’t lie in the boundary of any countable subset (so-called weak P-points). In this talk we investigate the space of self-maps S(X) on compact zero-dimensional F-spaces X, endowed with the compact-open topology. A natural question is whether S(X) reflects properties of the ground space X. Our main result is that for zero-dimensional compact F-spaces X, also S(X) doesn’t contain convergent sequences. If time permits, I will also comment on the existence of weak P-points in S(X). This is joint work with Richard Lupton.
Manifolds have been a central object of study for over a century, and the classification of them has been a core theme for the whole of this time. This talk will give an overview of the successes and failures in this effort, with some illustrative examples.
We consider the layer potentials associated with operators $L=-\mathrm{div}A \nabla$ acting in the upper half-space $\mathbb{R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A$ is complex, elliptic, bounded, measurable, and $t$-independent. A "Calder\'{o}n--Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that solutions of the equation $Lu=0$ satisfy interior De Giorgi-Nash-Moser type estimates. In particular, we prove that $L^2$ estimates for the layer potentials imply sharp $L^p$ and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea.
Since its genesis in 1915, General Relativity has proven to be one of the most successful physical theories ever invented. Providing a description of the large scale structure of the universe it continues to be in agreement with all experimental tests to high accuracy. By merging Classical Mechanics and Electrodynamics to a consistent geometrical theory of space-time it has become one of the two pillars of modern theoretical physics alongside Quantum Mechanics. This talk aims to give an introduction to the ideas and concepts of General Relativity. After briefly reviewing Classical (Newtonian) Mechanics and experiments in contradiction with it the framework and axioms of General Relativity will be introduced. This will be followed by a survey on major implications of the (new) geometrical description of gravity. Finally an outlook on physics beyond General Relativity will be provided.