Past

I want to give an introduction into non-Archimedean Geometry, and show how Model Theory was used to prove the recent results of Hrushovski-Loeser on topological properties of analytic spaces. This may also be of interest with view towards Zilber's programme for syntax-semantics dualities.

The study of rational points on K3 surfaces has recently seen a lot of activity. We discuss how to compute the Picard rank of a K3 surface over a number field, and the implications for the Brauer-Manin obstruction.

We establish sharp Sobolev inequalities of order four on Euclidean $d$-balls for $d$ greater than or equal to four. When $d=4$, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean $2$-balls. Our method relies on the use of scattering theory on hyperbolic $d$-balls. As an application, we charcaterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean $4$-balls. This is joint work with Alice Chang. 

Abstract: Joint work with Oleg Zaboronsky (Warwick).

Some one dimensional nearest neighbour particle systems are examples of Pfaffian point processes - where all intensities are determined by a single kernel.In some cases these kernels have appeared in the random matrix literature (where the points are the positions of eigenvalues). We are attempting to use random matrix tools on the particle sytems, and particle tools on the random matrices.

Recent developments in 3-dimensional topological quantum field theory allow us to understand the vector spaces assigned to surfaces as spaces of string diagrams. In the Reshetikhin-Turaev model, these string diagrams live inside a handlebody bounding the surface, while in the Turaev-Viro model, they live on the surface itself. There is a "lifting map" from the former to the latter, which sheds new light on a number of constructions. Joint with Gerrit Goosen.

Abstract:   Consider the square $[0,n]^2$ with points from a Poisson point process of intensity 1 distributed within it. In a seminal work, Baik, Deift and Johansson proved that the number of points $L_n$ (length) on a maximal increasing path (an increasing path that contains the most number of points), when properly centered and scaled, converges to the Tracy-Widom distribution. Later Johansson showed that all maximal paths lie within the strip of width $n^{\frac{2}{3} +\epsilon}$ around the diagonal with probability tending to 1 as $n \to \infty$. We shall discuss recent work on the Gaussian behaviour of the length $L_n^{(\gamma)}$ of a maximal increasing path restricted to lie within a strip of width $n^{\gamma}, \gamma< \frac{2}{3}$.

 

Milnor has shown that three-dimensional Lie groups with left invariant Riemannian metrics furnish examples of 3-manifolds with principal Ricci curvatures of fixed signature --- except for the signatures (-,+,+), (0,+,-), and (0,+,+).  We examine these three cases on a Riemannian 3-manifold, and prove global obstructions in certain cases.  For example, if the manifold is closed, then the signature (-,+,+) is not globally possible if it is of the form -µ,f,f, with µ a positive constant and f a smooth function that never takes the values 0,-µ (this generalizes a result by Yamato '91).  Similar obstructions for the other cases will also be discussed.  Our methods of proof rely upon frame techniques inspired by the Newman-Penrose formalism.  Thus, we will close by turning our attention to four dimensions and Lorentzian geometry, to uncover a relation between null vector fields and exact symplectic forms, with relations to Weinstein structures. 

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.

_______________________________

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.

There is a simple way to “glue together” a coupled pair of continuum random trees to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the “interface” between the trees). We present an explicit and canonical way to embed the sphere into the Riemann sphere. In this embedding, the measure is Liouville quantum gravity with parameter gamma in (0,2), and the curve is space-filling version of SLE with kappa=16/gamma^2. Based on joint work with Bertrand Duplantier and Scott Sheffield

TBC

I will describe a “cubical flat torus theorem” for a group G acting properly and cocompactly on a CAT(0) cube complex.
This states that every “highest” free abelian subgroup of G acts properly and cocompactly on a convex subcomplex that is quasi-isometric to a Euclidean space.
I will describe some simple consequences, as well as the original motivation which was to prove the “bounded packing property” for cyclic subgroups of G.
This is joint work with Daniel Woodhouse.

We consider a controlled system, in which an input $X: [0, T] \rightarrow E:= \mathbb{R}^{d}$ is a continuous but potentially highly oscillatory path and the corresponding output $Y$ is the line integral along $X$, for some unknown function $f: E \rightarrow E$. The rough paths theory provides a general framework to answer the question on which mild condition of $X$ and $f$, the integral $I(X)$ is well defined. It is robust enough to allow to treat stochastic integrals in a deterministic way. In this paper we are interested in identification of controlled systems of this type. The difficulty comes from the high dimensionality caused by the input of a function type. We propose novel adaptive and non-parametric algorithms to learn the functional relationship between the  input and the output from the data by carefully choosing the feature set of paths based on the rough paths theory and applying linear regression techniques. The algorithms is demonstrated on a financial application where the task is to predict the P$\&$L of the unknown trading strategy.

From a complex analytic perspective, both Teichmüller spaces and
symmetric spaces can be realised as contractible bounded domains, that
have several features in common but also exhibit many differences. In
this talk we will study isometric maps between these two important
classes of bounded domains equipped with their intrinsic Kobayashi metric.

Calabi-Yau manifolds without flux are perhaps the best-known
supergravity backgrounds that leave some supersymmetry unbroken. The
supersymmetry conditions on such spaces can be rephrased as the
existence and integrability of a particular geometric structure. When
fluxes are allowed, the conditions are more complicated and the
analogue of the geometric structure is not well understood.

In this talk, I will define the analogue of Calabi-Yau geometry for
generic D=4, N=2 backgrounds with flux in both type II and
eleven-dimensional supergravity. The geometry is characterised by a
pair of G-structures in 'exceptional generalised geometry' that
interpolate between complex, symplectic and hyper-Kahler geometry.
Supersymmetry is then equivalent to integrability of the structures,
which appears as moment maps for diffeomorphisms and gauge
transformations. Similar structures also appear in D=5 and D=6
backgrounds with eight supercharges.

As a simple application, I will discuss the case of AdS5 backgrounds
in type IIB, where deformations of these geometric structures give
exactly marginal deformations of the dual field theories.

In the course of work with Jamshid Derakhshan on definability in adele rings, we came upon various problems about definability and model completeness for possibly infinite dimensional algebraic extensions of p-adic fields (sometimes involving uniformity across p). In some cases these problems had been closely approached in the literature but never  explicitly considered.I will explain what we have proved, and try to bring out many big gaps in our understanding of these matters. This  seems appropriate just over 50 years after the breakthroughs of Ax-Kochen and Ershov.

Abstract: Four manifolds are some of the most intriguing objects in topology. So far, they have eluded any attempt of classification and their behaviour is very different from what one encounters in other dimensions. On the other hand, Einstein metrics are among the canonical types of metrics one can find on a manifold. In this talk I will discuss many of the peculiarities that make dimension four so special and see how Einstein metrics could potentially help us understand more about four manifolds.

Solving systems of linear equations $Ax=b$ is easy, but how can we solve such a system when given a "noisy" version of $b$? Over the reals one can use the least squares method, but the problem is harder when working over a finite field. Recently this subject has become very important in cryptography, due to the introduction of new cryptosystems with interesting properties.

The talk will survey work in this area. I will discuss connections with coding theory and cryptography. I will also explain how Fourier analysis in finite groups can be used to solve variants of this problem, and will briefly describe some other applications of Fourier analysis in cryptography. The talk will be accessible to a general mathematical audience.

Feedforward layers are integral step in processing and transmitting sensory information across different regions the brain. Yet experiments reveal the difficulty of stable propagation through layers without causing neurons to synchronize their activity. We study the limits of stable propagation in a discrete feedforward model of binary neurons. By analyzing the spectral properties of a mean-field Markov chain model, we show when such information transmission persists. Addition of inhibitory neurons and synaptic noise increases the robustness of asynchronous rate transmission. We close with an example of feedforward processing in the input layer to cerebellum. 

We propose the use of a constraint preconditioner for the iterative solution of the linear system arising from the finite element discretization of the coupled Stokes-Darcy system. The Stokes-Darcy system is a set of coupled PDEs that can be used to model a freely flowing fluid over porous media flow. The fully coupled system matrix is large, sparse, non-symmetric, and of saddle point form. We provide for exact versions of the constraint preconditioner spectral and field-of-values bounds that are independent of the underlying mesh width. We present several numerical experiments, using the deal.II finite element library, that illustrate our results in both two and three dimensions. We compare exact and inexact versions of the constraint preconditioner against standard block diagonal and block lower triangular preconditioners to illustrate its favorable properties.

I will explain how algebraic spaces can be presented as Zariski geometries and prove some classical facts about algebraic spaces using the theory of Zariski geometries.