Past

We review some recent progress in computing massless spectra 

and moduli in heterotic string compactifications. In particular, it was   

recently shown that the heterotic Bianchi Identity can be accounted 

for by the construction of a holomorphic operator. Mathematically,

this corresponds to a holomorphic double extension. Moduli can 

then be computed in terms of cohomologies of this operator. We 

will see how the same structure can be derived form a 

Gukov-Vafa-Witten type superpotential. We note a relation between 

the lifted complex structure and bundle moduli, and cover some 

examples, and briefly consider obstructions and Yukawa 

couplings arising from these structures.

Derived geometry and approximations - Pavel Safronov

Derived geometry has been developed to address issues arising in geometry from a consideration of spaces with intrinsic symmetry or some singular spaces arising as complicated intersections.  It has been successful both in pure mathematics and theoretical physics where derived geometric structures appear in quantum gauge field theories such as the theory of quantum electrodynamics.  Recently Lurie has developed a transparent approach to deformation theory, i.e. the theory of approximations of algebraic structures, using the language of derived algebraic geometry.  I will motivate the theory on a basic example and explain one of the theorems in the subject.

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How magnets and mathematics can help solve the current water crisis - Ian Griffiths

Although water was once considered an almost unlimited resource, population growth, drought and contamination are straining our water supplies.  Up to 70% of deaths in Bangladesh are currently attributed to arsenic contamination, highlighting the essential need to develop new and effective ways of purifying water.

Since arsenic binds to iron oxide, magnets offer one such way of removing arsenic by simply pulling it from the water.  For larger contaminants, filters with a spatially varying porosity can remove particles through selective sieving mechanisms.

Here we develop mathematical models that describe each of these scenarios, show how the resulting models give insight into the design requirements for new purification methods, and present methods for implementing these ideas with industry.

It is well known that stochasticity can play a fundamental role in 
various biochemical processes, such as cell regulatory networks and 
enzyme cascades. Isothermal, well-mixed systems can be adequately 
modeled by Markov processes and, for such systems, methods such as 
Gillespie's algorithm are typically employed. While such schemes are 
easy to implement and are exact, the computational cost of simulating 
such systems can become prohibitive as the frequency of the reaction 
events increases. This has motivated numerous coarse grained schemes, 
where the ``fast'' reactions are approximated either using Langevin 
dynamics or deterministically.  While such approaches provide a good 
approximation for systems where all reactants are present in large 
concentrations,  the approximation breaks down when the fast chemical 
species exist in small concentrations,  giving rise to significant 
errors in the simulation.  This is particularly problematic when using 
such methods to compute statistics of extinction times for chemical 
species, as well as computing observables of cell cycle models.  In this 
talk, we present a hybrid scheme for simulating well-mixed stochastic 
kinetics, using Gillepsie--type dynamics to simulate the network in 
regions of low reactant concentration, and chemical langevin dynamics 
when the concentrations of all species is large.  These two regimes are 
coupled via an intermediate region in which a ``blended'' jump-diffusion 
model is introduced.  Examples of gene regulatory networks involving 
reactions occurring at multiple scales, as well as a cell-cycle model 
are simulated, using the exact and hybrid scheme, and compared, both in 
terms weak error, as well as computational cost.

This is joint work with A. Duncan (Imperial) and R. Erban (Oxford)

We pursue robust approach to pricing and hedging in mathematical finance. We consider a continuous time setting in which some underlying assets and options, with continuous paths, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. Motivated by the notion of prediction set in Mykland [03], we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. super-replication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model--independent and model--specific settings and allows to quantify the impact of making assumptions or gaining information. We obtain a general pricing-hedging duality result: the infimum over superhedging prices is equal to supremum over calibrated martingale measures. In presence of non-trivial beliefs, the equality is between limiting values of perturbed problems. In particular, our results include the martingale optimal transport duality of Dolinsky and Soner [13] and extend it to multiple dimensions and multiple maturities.

In contrast to the Artin-Schreier Theorem, its $p$-adic analog(s) involve infinite Galois theory, e.g., the absolute Galois group of $p$-adic fields.  We plan to give a characterization of $p$-adic $p$-Henselian valuations in an essentially finite way. This relates to the $Z/p$ metabelian form of the birational $p$-adic Grothendieck section conjecture.

In contrast to the Artin-Schreier Theorem, its p-adic analog(s) involve infinite Galois theory, e.g., the absolute Galois group of p-adic fields.  We plan to give a characterization of p-adic p-Henselian valuations in an essentially finite way. This relates to the Z/p metabelian form of the birational p-adic Grothendieck section conjecture.

The talk is concerned with adapted solution of a multi-dimensional BSDE with a "diagonally" quadratic generator, the quadratic part of whose iith component only depends on the iith row of the second unknown variable. Local and global solutions are given. In our proofs, it is natural and crucial to apply both John-Nirenberg and reverse Holder inequalities for BMO martingales. 

I will explain the formulation and proof of Arthur's multiplicity formula for automorphic representations of special orthogonal groups and certain inner forms of symplectic groups $G$ over a number field $F$. I work under an assumption that substantially simplifies the use of the stabilisation of the trace formula, namely that there exists a non-empty set $S$ of real places of $F$ such that $G$ has discrete series at places in $S$ and is quasi-split at places outside $S$, and restricting to automorphic representations of $G(A_{F})$ which have algebraic regular infinitesimal character at the places in $S$. In particular, this proves the general multiplicity formula for groups $G$ such that $F$ is totally real, $G$ is compact at all real places of $F$ and quasi-split at all finite places of $F$. Crucially, the formulation of Arthur's multiplicity formula is made possible by Kaletha's recent work on local and global Galois
gerbes and their application to the normalisation of Kottwitz-Langlands-Shelstad transfer factors.

Michael Gomez:

Title: The role of ghosts in elastic snap-through
Abstract: Elastic `snap-through' buckling is a striking instability of many elastic systems with natural curvature and bistable states. The conditions under which bistability exists have been reasonably well studied, not least because a number of engineering applications make use of the rapid transitions between states. However, the dynamics of the transition itself remains much less well understood. Several examples have been studied that show slower dynamics than would be expected based on purely elastic timescales of motion, with the natural conclusion drawn that some other effect, such as viscoelasticity, must play a role. I will present analysis (and hopefully experiments) of a purely elastic system that shows similar `anomalous dynamics'; however, we show that here this dynamics is a consequence of the ‘ghost’ of the snap-through bifurcation.

Andrew Krause:

Title: Fluid-Growth Interactions in Bioactive Porous Media   
Abstract: Recent models in Tissue Engineering have considered pore blocking by cells in a porous tissue scaffold, as well as fluid shear effects on cell growth. We implement a suite of models to better understand these interactions between cell growth and fluid flow in an active porous medium. We modify some existing models in the literature that are spatially continuous (e.g. Darcy's law with a cell density dependent porosity). However, this type of model is based on assumptions that we argue are not good at describing geometric and topological properties of a heterogeneous pore network, and show how such a network can emerge in this system. Therefore we propose a different modelling paradigm to directly describe the mesoscopic pore networks of a tissue scaffold. We investigate a deterministic network model that can reproduce behaviour of the continuum models found in the literature, but can also exhibit finite-scale effects of the pore network. We also consider simpler stochastic models which compare well with near-critical Percolation behaviour, and show how this kind of behaviour can arise from our deterministic network model.

We will briefly review the notions of Dirac cohomology and of $A_{\mathfrak{q}}(\lambda)$ modules of real reductive groups, and recall a formula for the Dirac cohomology of an $A_{\mathfrak{q}}(\lambda)$ module. Then we will discuss to what extent an $A_{\mathfrak{q}}(\lambda)$ module is determined by its Dirac cohomology. This is joint work with Jing-Song Huang and David Vogan.

In numerical atmosphere models, values of relevant physical parameters are often uncertain by more than 100% and weather forecast skill is significantly reduced after a couple of days. Still, numerical operations are typically calculated in double precision with 15 significant decimal digits. If we reduce numerical precision, we can reduce power consumption and increase computational performance significantly. If savings are reinvested to build larger supercomputers, this would allow an increase in resolution in weather and climate models and might lead to better predictions of future weather and climate. 
I will discuss approaches to reduce numerical precision beyond single precision in high performance computing and in particular in weather and climate modelling. I will present results that show that precision can be reduced significantly in atmosphere models and that potential savings can be huge. I will also discuss how rounding errors will impact model dynamics and interact with model uncertainty and predictability.

In 1943 Fisher, together with Corbet and Williams, published a study on the relation between the number of species and the number of individuals, which has since been recognized as one of the most influential papers in 20th century ecology. It was a combination of empirical work backed up by a simple theoretical argument, which describes a sort of universal law governing random partitions, such as the celebrated Ewens partition whose original derivation flows from the Fisher-Wright model. This talk will discuss several empirical studies of a similar sort, including Taylor's law and recent work related to Fairfield-Smith's work on the variance of spatial averages.

Over a decade ago Welschinger defined invariants of real symplectic manifolds of complex dimension 2 and 3, which count $J$-holomorphic disks with boundary and interior point constraints. Since then, the problem of extending the definition to higher dimensions has attracted much attention.
  We generalize Welschinger's invariants with boundary and interior constraints to higher odd dimensions using the language of $A_\infty$-algebras and bounding chains. The bounding chains play the role of boundary point constraints. The geometric structure of our invariants is expressed algebraically in a version of the open WDVV equations. These equations give rise to recursive formulae which allow the computation of all invariants for $\mathbb{CP}^n$.
  This is joint work with Jake Solomon.

A system of linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic solution. The finite partition regular systems were completely characterised by Rado in terms of a simple property of their matrix of coefficients. As a result, finite partition regular systems are very well understood.

Much less is known about infinite systems. In fact, only a very few families of infinite partition regular systems are known. I'll explain a relatively new method of constructing infinite partition regular systems, and describe how it has been applied to settle some basic questions in the area.

Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution.  This result is based on work of Crawley-Bouvey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.
 

I will introduce classical results on finiteness theorem with a way of connecting them to idea of covering spaces. I will talk about the proof of FLT under this connection.