To each additive definable category there is attached its category of pp-imaginaries. This is abelian and every small abelian category arises in this way. The connection may be expressed as an equivalence of 2-categories. We describe two associated spectra (Ziegler and Zariski) which have arisen in the model theory of modules.
Brittle failure through multiple cracks occurs in a wide variety of contexts, from microscopic failures in dental enamel and cleaved silicon to geological faults and planetary ice crusts. In each of these situations, with complicated stress geometries and different microscopic mechanisms, pairwise interactions between approaching cracks nonetheless produce characteristically curved fracture paths. We investigate the origins of this widely observed "en passant" crack pattern by fracturing a rectangular slab which is notched on each long side and then subjected to quasistatic uniaxial strain from the short side. The two cracks propagate along approximately straight paths until they pass each other, after which they curve and release a lens-shaped fragment. We find that, for materials with diverse mechanical properties, each curve has an approximately square-root shape, and that the length of each fragment is twice its width. We are able to explain the origins of this universal shape with a simple geometrical model.
I will review the basic properties of the DFT and describe how these can be exploited to efficiently compute degree 1 L-functions.
Computing the eigenvalue decomposition of a symmetric matrix and the singular value decomposition of a general matrix are two of the central tasks in numerical linear algebra. There has been much recent work in the development of linear algebra algorithms that minimize communication cost. However, the reduction in communication cost sometimes comes at the expense of significantly more arithmetic and potential instability.
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In this talk I will describe algorithms for the two decompositions that have optimal communication cost and arithmetic cost within a small factor of those for the best known algorithms. The key idea is to use the best rational approximation of the sign function, which lets the algorithm converge in just two steps. The algorithms are backward stable and easily parallelizable. Preliminary numerical experiments demonstrate their efficiency.
I will give a brief introduction into how Elliptic curves can be used to define complex oriented
cohomology theories. I will start by introducing complex oriented cohomology theories, and then move onto
formal group laws and a theorem of Quillen. I will then end by showing how the formal group law associated
to an elliptic curve can, in many cases, allow one to define a complex oriented cohomology theory.
Consider the solution of a scalar Helmholtz equation where the potential (or index) takes two positive values, one inside a disk or a ball (when d=2 or 3) of radius epsilon and another one outside. For this classical problem, it is possible to derive sharp explicit estimates of the size of the scattered field caused by this inhomogeneity, for any frequencies and any contrast. We will see that uniform estimates with respect to frequency and contrast do not tend to zero with epsilon, because of a quasi-resonance phenomenon. However, broadband estimates can be derived: uniform bounds for the scattered field for any contrast, and any frequencies outside of a set which tends to zero with epsilon.
If $A$ is a set of $n$ positive integers, how small can the set
$\{ x/(x,y) : x,y \in A \}$ be? Here, as usual, $(x,y)$ denotes the highest common factor of
$x$ and $y$. This elegant question was raised by Granville and Roesler, who
also reformulated it in the following way: given a set $A$ of $n$ points in
the integer grid ${\bf Z}^d$, how small can $(A-A)^+$, the projection of the difference
set of $A$ onto the positive orthant, be?
Freiman and Lev gave an example to show that (in any dimension) the size can
be as small as $n^{2/3}$ (up to a constant factor). Granville and Roesler
proved that in two dimensions this bound is correct, i.e. that the size is
always at least $n^{2/3}$, and they asked if this holds in any dimension.
After some background material, the talk will focus on recent developments.
Joint work with B\'ela Bollob\'as.
When modelling the motion of a sperm cell in the female reproductive tract, the Reynolds number is found to be very small, thus allowing the nonlinear Navier-Stokes equations to simplify to the linear Stokes equations stating that pressure, viscous and body forces balance each other at any instant in time. A wide range of analytical techniques can be applied to investigate the Stokes flow past a moving body. In this talk, we introduce various Stokes flow singularities and illustrate how they can provide a handy starting point (ansatz) when trying to determine the form of the flow field around certain bodies, from simple translating spheres to beating sperm tails.
The infamous inverse Galois problem asks whether or not every finite group can be realised as a Galois group over Q. We will see some techniques that have been developed to attack it, and will soon end up in the realms of class field theory, étale fundamental groups and modular representations. We will give some concrete examples and outline the so called 'rigidity method'.
Morgan and Culler proved in the 1980’s that a minimal action of a free group on a tree is
completely determined by its length function. This theorem has been of fundamental importance in the
study of automorphisms of free groups. In particular, it gives rise to a compactification of Culler-Vogtmann's
Outer Space. We prove a 2-dimensional analogue of this theorem for right-angled Artin groups acting on
CAT(0) rectangle complexes. (Joint work with M. Margolis)
I will describe the recent construction of new supersymmetric compactifications of the heterotic string which yield just the spectrum of the MSSM at low energies. The starting point is the standard embedding solution on a Calabi-Yau manifold with Euler number -6 with various choices of Wilson lines, i.e., various choices of discrete holonomy for the gauge bundle. Although they yield three net generations of standard model matter, such models necessarily have a larger gauge group than the standard model, as well as exotic matter content. Families of stable bundles can be obtained by deformation of these simple solutions, the deformation playing the dual role of partially breaking the gauge group and reducing the matter content, and in this way we construct more realistic models. The moduli space breaks up into various branches depending on the initial choice of Wilson lines, and on eight of these branches we find models with exactly the standard model gauge group, three generations of quarks and leptons, two Higgs doublets, and no other massless charged states. I will also comment on why these are possibly the unique models of this type.
The Eilenberg-Ganea conjecture is the statement that every group of cohomological dimension two admits a two-dimensional classifying space. This problem is unsolved after 50 years. I shall discuss the background to this question and negative answers to some other related questions. This includes recent joint work with Martin Fluch.
In a market with one safe and one risky asset, an investor with a long
horizon and constant relative risk aversion trades with constant
investment opportunities and proportional transaction costs. We derive
the optimal investment policy, its welfare, and the resulting trading
volume, explicitly as functions of the market and preference parameters,
and of the implied liquidity premium, which is identified as the
solution of a scalar equation. For small transaction costs, all these
quantities admit asymptotic expansions of arbitrary order. The results
exploit the equivalence of the transaction cost market to another
frictionless market, with a shadow risky asset, in which investment
opportunities are stochastic. The shadow price is also derived
explicitly. (Joint work with Paolo Guasoni, Johannes Muhle-Karbe, and
Walter Schachermayer)