Sara Pasquetti visited the Theoretical Physics group at the Universita degli Studi de Milano-Bicocca.  During the visit she gave a seminar on Friday 22 February in the TP seminar.  The title is “Holomorphic blocks in three dimensions“.  In the talk it is shown that sphere partition functions and indices of 3 dimensional N=2 gauge theories can be decomposed into a sum of products of a universal set of holomorphic blocks. The blocks count BPS states of a theory on R2 × S1 and are in one-to-one correspondence with the theory’s massive vacua. The blocks turn out to have a wealth of surprising properties such as Stokes phenomena and have interesting dual interpretations in analytically continued Chern-Simons theory and open topological strings.

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The paper “On localised hotspots of an urban crime model” by David Lloyd and Hayley O’Farrell has been accepted for publication in Physica D.  In the paper they investigate stationary, spatially localised crime hotspots on the real line and the plane of an urban crime model of Short et al (2008).  Extending the weakly nonlinear analysis of Short et al, they show in one dimension that localised hotspots should bifurcate off the background spatially homogenious state at a Turing instability provided the bifurcation is subcritical.  Using path-following techniques, they continue these hotspots and show that the bifurcating pulses can undergo the process of homoclinic snaking near the singular limit.  We analyse the singular limit to explain the existence of spike solutions and compare the analyical results with the numerical computations.  In two dimensions, we show that localised radial spots should also bifurcate off the spatially homogeneous background state.  Localised planar hexagon fronts and hexagon patches are found and depending on the proximity to the singular limit these solutions either undergo homoclinic snaking or act like “multi-spot” solutions.  Finally, we discuss applications of these localised patterns in the urban crime context and the full agent-based model.  A copy of the final form preprint can be downloaded here.

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The paper “A regularization of the carbon cycle data fusion problem” has been accepted for oral presentation at the 2013 general assembly of the European Geophysical Union.  It will appear in session NP5.1.  The presenter is Sylvain Delahaies and his co-authors are Ian Roulstone and Nancy Nicholls (Reading).  AGU2013 will take place the week of 07 April in Vienna.  The website for AGU2013 can be found here.

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Tom Bridges gave a talk in the Industrial and Applied Mathematics Seminar series at the University of Nottingham on 20th February and a talk in the Applied Mathematics Series at the University of Reading on 26th February.  The title of the talk in both cases was “Reappraisal of two myths in the theory of nonlinear waves“.  The first myth is that the shallow water equations (SWEs) are a model for shallow water hydrodynamics.  It is shown that in fact the shallow water equations can not control the growth of horizontal vorticity and so the equations will in general break down.  Although the SWEs are a perfectly valid mathematical model, they may be invalid when applied to realistic shallow water oceanographic flows.  The second myth is how the KdV equation arises as a model PDE.  The conventional view is that the KdV equation arises as a model when the dispersion relation of the linearization of some system of PDEs has the appropriate form, and the nonlinearity is quadratic.  A new mechanism is presented which shows that the KdV equation always takes a universal form where the coefficients in the KdV equation are completely determined from the modulation properties of the background state — even when the background state is the trivial solution!   One outcome is that “shallow water” is neither necessary nor sufficient for the emergence of the KdV equation.

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The paper “On the massless modes of the AdS3/CFT2 integrable systems” has been accepted for publication in the Journal of High Energy Physics (JHEP).  The 83 page paper is co-authored by Olof Ohlsson Sax (University of Utrecht), Bogdan Steganski Jr (City University London) and Alessandro Torrielli.  In the paper they make a proposal for incorporating massless modes into the spin-chain of the AdS3/CFT2 integrable system.  They do this by considering the alpha –> 0 limit of the alternating delta(2,1;alpha)^2 spin-chain constructed in arXiv:1106.2558.  In the process they encounter integrable spin-chains with non-irreducible representations at some of their sites.  They investigate their properties and construct their R-matrices in terms of Yangians.  A final-form preprint of the paper can be found on the arXiv.

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Gianne Derks visited the University of Keele on Wednesday 20 February to give a talk in the Applied Mathematics Seminar series.   The title of the talk is “Viscosity-induced instability for Euler and averaged Euler equations in a circular domain“.

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Sara Pasquetti gave a talk on Wednesday 13 February at King’s College London.  It was an exceptional seminar associated with the inaugural meeting of the “London Polygon”.  Her talk was on “Holomorphic blocks in three dimensions“.  The London Polygon is an outgrowth of the “London Triangle”.  The homepage is here.

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Pablo Shmerkin visited the University of St Andrews from 26 February to 1 March to work with the fractal and dynamical systems group.  He also gave an analysis seminar on “Multifractal structure of Bernoulli convolutions” and a pure mathematics colloquium on “On sets containing circles/squares centered at every point“.  The abstract for the colloqium follows:
Let A be a subset of the plane with the property that, for each point of the plane, A contains a circle centered at that point. A deep classical result due independently to Marstrand and to Bourgain says that A must have positive Lebesgue measure.
I will review the history of the circle problem and tell about some recent developments on the similar problem where circles are replaced by (boundaries of) squares. Unlike the circle case, it was known that a set containing a square centered at every point may have zero Lebesgue measure. The question we study is: how small can A be? As we will see, the answer surprisingly depends on the notion of  “smallness” used. The talk is based on joint work with T. Keleti and D. Nagy.

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Gianne Derks and Anne Skeldon have been awarded MILES funding to work on “Mathematical modelling of circadian processes“.  The project is joint with Dr Simon Archer from the Department of Biochemistry and Physiology and Prof Derk-Jan Dijk from the Surrey Sleep Research Centre.  A short description of the project follows.

The primary components that form the clock in mammals are relatively well understood but internal resonance and how the core clock mechanism relates to circadian-regulated processes is a challenging and important question.  It has been established that the circadian clock is composed of an auto-regulatory transcriptional network with interlocked feedback loops involving the transcriptional activators such as BMAL1, CLOCK and NPAS2 that activate the Per1, Per2, Cry1 and Cry2 genes.  While the core clock is believed to consist of just a few components, many thousands of transcripts have been shown to express circadian oscillations.  Very little is known about the overall architecture of the interactions between the core circadian process and the biorhythms they drive. For example, how is it that different transcripts peak at a wide range of different points in the day when they all driven by the same core circadian clock?

In this project gene transcription data that has been collected under different conditions will be fitted to a model of the circadian clock.  Understanding the dynamics of the circadian model and the way that parameters in the model need to change in order to mimic the different experimental conditions will lead to insight into what underlying biological processes are affected in the different scenarios.

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James Grant attended the Central European Relativity Seminar III in Golm, Germany for 7-9 February.  His talk was titled “On differentiability of volume time functions“.   In the talk he showed differentiability of a class of Geroch’s volume functions on globally hyperbolic manifolds. Furthermore, it was proved that every volume function satisfies a local anti-Lipschitz condition over causal curves, and that locally Lipschitz time functions which are locally anti-Lipschitz can be uniformly approximated by smooth time functions with timelike gradient.  Finally, it was proved that in stably causal spacetimes Hawking’s time function can be uniformly approximated by smooth time functions with timelike gradient.  The presentation was based on joint work with P.T. Chrusciel and E. Minguzzi.  The website for the conference is here.

 

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