Past

I will report on $p$-adic Beilinson's formulas, relating the values of certain Rankin $p$-adic L-functions outside their range of classical interpolation, to $p$-adic syntomic regulators of Beilinson-Kato and Beilinson-Flach elements. Applications to the theory of Euler systems and to the Birch and Swinnerton-Dyer conjecture will also be discussed. This is joint work with Henri Darmon and Victor Rotger.

In this lecture I shall introduce certain generalised Gross-Kudla-Schoen diagonal cycles in the product of three Kuga-Sato varieties and a $p$-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the $p$-adic Abel-Jacobi map to special values of the $p$-adic L-function attached to the Garrett triple convolution of three  Hida families of  modular forms. This formula has applications to the Birch--Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. (Joint work with Henri Darmon.)

The main result of the talk is that two curves over a finite field are isomorphic, up to automorphisms of the ground field, if and only if there is an isomorphism of groups of Dirichlet characters such that the corresponding L-series are all equal. This can be shown by combining Uchida's proof of the anabelian theorem for global function fields with methods from (noncommutative) dynamical systems. I will also discuss how to turn this theorem into a theoretical algorithm that, given a listing of L-functions, determines an equation for the corresponding curve(s).

I will present an overview of a series of joint works with Akio Tamagawa about l-adic representations of etale fundamental group of curves (to simplify, over finitely generated fields of characteristic 0).
More precisely, when the generic representation is GLP (geometrically Lie perfect) i.e. the Lie algebra of the geometric etale fundamental group is perfect, we show that the associated local $\ell$-adic Galois representations satisfies a strong uniform open image theorem (ouside a `small' exceptional locus). Representations on l-adic cohohomology provide an important example of GLP representations. In that case, one can even provethat the exceptional loci that appear in the statement of our stronguniform open image theorem are independent of $\ell$, which was predicted by motivic conjectures.
Without the GLP assumption, we prove that the  associated local l-adic Galois representations still satisfy remarkable rigidity properties: the codimension of the image at the special fibre in the image at the generic fibre is at most 2 (outside a 'small' exceptional locus) and its Lie algebra is controlled by the first terms of the derived series of the Lie algebra of the image at the generic fibre.
I will state the results precisely, mention a few applications/open questions and draw a general picture of the proof in the GLP case (which,in particular, intertwins via the formalism of Galois categories, arithmetico-geometric properties of curves and $\ell$-adic geometry). If time allows, I will also give a few hints about the $\ell$-independency of the exceptional loci or the non GLP case.

We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large. What about when p-deficiency is exactly one? We also generalise a result of Grigorchuk on Coxeter groups to odd primes.

In this talk, I will describe a construction of a $p$-adic L-function attached to a triple of $p$-adic Coleman families of cusp forms.  This function interpolates algebraic parts of special values of Garrett triple product L-functions at balanced triples of weights.  Our construction is complementary to that of Harris and Tilouine which treats the case of unbalanced weights.

Given a $p$-adic weight and a finite slope we describe a Hecke and Galois equivariant geometric map relating elliptic overconvergent modular symbols and overconvergent modular forms of that slope, appropriate weights and $\mathbf{C}_p$-coefficients. We show that for a fixed slope, with the possible exception of a discrete family of weights, this map is an isomorphism.

Using Faltings' theory of the Hodge-Tate sequence of an abelian scheme we construct certain sheaves $\Omega^\kappa$, where $\kappa$ is a not-necessarily integral weight, over formal subschemes of modular varieties over which the canonical subgroup exists.   These sheaves generalize the integral powers, $\omega^k$, of the sheaf $\omega$ of relative differentials on a modular curve.   Global sections of $\Omega^\kappa$ provide geometric realizations of overconvergent automorphic forms of non-integral weight.  Applications of this approach to the theory of $p$-adic Hilbert modular forms will be given.   This is joint work with Fabrizio Andreotti and Adrian Iovita.

We apply the theory of the radius of convergence of a $p$-adic connection to the special case of the direct image of the constant connection via a finite morphism of compact $p$-adic curves, smooth in the sense of rigid geometry. We show that a trivial lower bound for that radius implies a global form of Robert's $p$-adic Rolle theorem. The proof is based on a widely believed, although unpublished, result of simultaneous semistable reduction for finite morphisms of smooth $p$-adic curves. We also clarify the relation between the notion of radius of convergence used in our previous work and the more intrinsic one used by Kedlaya. (The paper is available athttp://arxiv.org/abs/1209.0081)

For a proper semistable curve over a DVR of mixed characteristics we re prove the ``invariant cycles theorem'' with trivial coefficients  by Chiarellotto i.e. that the group of elements annihilated by the monodromy operator on the first de Rham cohomology group of the generic fiber coincides with the first rigid cohomology group of the special fiber, without the hypothesis that the residue field is finite. This is done using the explicit description of the monodromy operator on the de Rham cohomology of the generic fiber with coefficients convergent F-isocrystals given in a work of Coleman and Iovita. We apply these ideas to the case where the coefficients are unipotent convergent F-isocrystals defined on the special fiber: we show that the invariant cycles theorem does not hold in general in this setting. Moreover we give a sufficient condition for the non exactness. It is a joint work with B. Chiarellotto, R. Coleman and A. Iovita.

In 1986 Mazur, Tate and Teitelbaum came up with a $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for elliptic curves over the rationals. In this talk I will report on joint work with Jennifer Balakrishnan and William Stein on a generalization of this conjecture to the case of modular abelian varieties and primes $p$ of good ordinary reduction. I will discuss the theoretical background that led us to the formulation of the conjecture, as well as numerical evidence supporting it in the case of modular abelian surfaces and the algorithms that we used to gather this evidence.

We extend the following theorem of H. Imai in several ways: If  $A$  is an abelian variety with potentially good reduction over a finite extension  $K$  of  $\mathbf{Q}_p$, then it has only finitely many rational torsion points over the maximal $p$-cyclotomic extension of  $K$. In particular, we prove the finiteness over $K(K^{1/p^\infty})$.

The overconvergence of the canonical subgroup of the universal abelian variety is one of the key ingredients of the theory of overconvergent modular forms. In this talk, I will show the overconvergence of the canonical subgroup with expected properties via the Breuil--Kisin classification, including the case of $p=2$.

I will explain recent joint work with Matthew Emerton and David Savitt, in which we relate the geometry of various tamely potentially Barsotti--Tate deformation rings for two-dimensional Galois representations to the integral structure of the cohomology of Shimura curves. As a consequence, we establish some conjectures of Breuil regarding this integral structure.
The key technique is the Taylor--–Wiles--–Kisin patching argument, which,when combined with a new, geometric perspective on the Breuil–--Mezard conjecture, forges a tight link between the structure of cohomology (a global automorphic invariant) and local deformation rings (local Galois-theoretic invariants).

Composite membranes comprised of an ultra-thin coating film formed over a porous support membrane are the basis for state-of-the-art reverse osmosis (RO) and nanofiltration (NF) membranes, offering the possibility to independently optimize the support membrane and the coating film. However, limited information exists on transport through such composite membrane structures. Numerical calculations have been carried out in order to probe the impacts of the support membrane skin-layer pore size and porosity, support membrane bulk micro-porosity, and coating film thickness and morphology (i.e. surface roughness) on solvent and solute transport through composite membranes. Results suggest that the flux and rejection of a composite membrane may be fine-tuned, by adjusting support membrane skin layer porosity and pore size, independent of the properties of the coating film. Further, the water flux over the membrane surface is unevenly distributed, creating local ‘hot spots’ of high flux that may govern initial stages of membrane fouling and scaling. The analysis provides important insight on how the non-trivial interaction of support properties and film roughness may result in widely varying transport properties of the composite structure. In particular, the simulations reveal inherent trade-offs between flux, rejection and fouling propensity (the latter due to ‘hot spots’), which are purely consequences of geometrical factors, irrespective of materials chemistry.

I will be concerned with initial-boundary value problems for systems of conservation laws in one space variable. First, I will go over some of the most relevant features of these problems. In particular, I will stress that different viscous approximation lead, in general, to different limits.

Next, I will discuss possible ways of characterizing the limit of a given viscous approximation. Also, I will establish a uniqueness criterion that allows to conclude that the limit of a self-similar approximation introduced by Dafermos et al. actually coincide with the limit of the physical viscous approximation. Finally, if time allows I will mention consequences on the design of numerical schemes. The talk will be based on joint works with S. Bianchini, C. Christoforou and S. Mishra.

• Alfonso Bueno - Recent advances in mathematical modelling of cardiac tissue: A fractional step forward
• Matt Moore - Oblique water entry
• Matt Hennessy - Mathematical problems relating to organic solar cell production

Russian mathematician V.I.Arnold conjectured that convex, homogeneous bodies with less than four equilibria (also called mono-monostatic) may exist. Not only did his conjecture turn out to be true, the newly discovered objects show various interesting features. Our goal is to give an overview of these findings as well as to present some new results. We will point out that mono-monostatic bodies are neither flat, nor thin, they are not similar to typical objects with more equilibria and they are hard to approximate by polyhedra. Despite these "negative" traits, there seems to be strong indication that these forms appear in Nature due to their special mechanical properties.

Topological defects (TDs) are unavoidable consequence of continuous symmetry breaking phase transitions. They exhibit several universal features and often span apparently completely different systems. Particularly convenient testing ground to study basic physics of TDs are liquid crystals (LCs) due to their softness, liquid character and optical anisotropy. In the lecture I will present our recent theoretical studies of TDs in nematic LCs, which are of interest also to other branches of physics.

I will first focus on coarsening dynamics of TDs following the isotropic-nematic phase transition. Among others we have tested the validity of the Kibble-Zurek [1,2] prediction on the size of the so called protodomains, which was originally derived to estimate density of TDs as a function of inflation time in the early universe. Next I will consider nematic LC shells [3]. These systems are of interest because they could pave path to mm sized scaled crystals exhibiting different symmetries. Particular attention will be paid to curvature induced unbinding of pairs of topological defects. This process might play important role in membrane fission processes.  

[1] W.H. Zurek, Nature 317, 505 (1985).

[2] Z. Bradac et al.,  J.Chem.Phys 135, 024506 (2011)

[3] S. Kralj et al.,  Soft Matter 7, 670 (2011); 8, 2460  (2012).

Telomeres, non-coding terminal structures of DNA strands, consist of repetitive long tandem repeats of a specific length. An absence of an enzyme, telomerase, in certain cellular structures requires an alternative telomerase-independent pathway for telomeric sequence length regulation. Besides linear telomeres other configurations such as telomeric circles and telomeric loops were experimentally observed. They are suspected to play an important role in a universal mechanism for stabilization of the ends of linear DNA that possibly dates back to pre-telomerase ages. We propose a mathematical model that captures biophysical interactions of various telomeric structures on a short time scale and that is able to reproduce experimental measurements in mtDNA of yeast. Moreover, the model opens up a couple of interesting mathematical problems such as validity of a quasi-steady state approximation and dynamic properties of discrete coagulation-fragmentation systems. We also identify and estimate key factors influencing the length distribution of telomeric circles, loops and strand invasions using numerical simulations.

For a given positive measure on a fixed domain, Gaussian quadrature routines can be defined via their property of integrating an arbitrary polynomial of degree $2n+1$ exactly using only $n+1$ quadrature nodes. In the special case of Gauss--Jacobi quadrature, this means that $$\int_{-1}^1 (1+x)^\alpha(1-x)^\beta f(x) dx = \sum_{j=0}^{n} w_j f(x_j), \quad \alpha, \beta > -1, $$ whenever $f(x)$ is a polynomial of degree at most $2n+1$. When $f$ is not a polynomial, but a function analytic in a neighbourhood of $[-1,1]$, the above is not an equality but an approximation that converges exponentially fast as $n$ is increased.

An undergraduate mathematician taking a numerical analysis course could tell you that the nodes $x_j$ are roots of the Jacobi polynomial $P^{\alpha,\beta}_{n+1}(x)$, the degree $n+1$ polynomial orthogonal with respect to the given weight, and that the quadrature weight at each node is related to the derivative $P'^{\alpha,\beta}_{n+1}(x_j)$. However, these values are not generally known in closed form, and we must compute them numerically... but how?

Traditional approaches involve applying the recurrence relation satisfied by the orthogonal polynomials, or solving the Jacobi matrix eigenvalue problem in the algorithm of Golub and Welsch, but these methods are inherently limited by a minimal complexity of $O(n^2)$. The current state-of-the-art is the $O(n)$ algorithm developed by Glasier, Liu, and Rokhlin, which hops from root to root using a Newton iteration evaluated with a Taylor series defined by the ODE satisfied by $P^{\alpha,\beta}_{n+1}$.

We propose an alternative approach, whereby the function and derivative evaluations required in the Newton iteration are computed independently in $O(1)$ operations per point using certain well-known asymptotic expansions. We shall review these expansions, outline the new algorithm, and demonstrate improvements in both accuracy and efficiency. 

Organisers: Hilary Priestley, Drew Moshier and Leo Cabrer.

This will be devoted to the applications of dualities to logic and algebra, focusing on general techniques. Thus it will seek to complement the specialised coverage in meetings devoted to, for example, modal logic, residuated structures and many-valued logics, or coalgebras. The featured topics for the Workshop will be drawn from completions of ordered structures, and applications; admissible rules, unification theory, interpolation and amalgamation; aspects of many-valued and substructural logics and ordered algebraic structures. Keynote speakers will be Leo Cabrer and Mai Gehrke.

The energy of a deformed crystal is calculated in the context of a central force lattice model in two dimensions. When the crystal shape is a lattice polygon, it is shown that the energy equals the bulk elastic energy, plus the boundary integral of a surface energy density, plus the sum over the vertices of a corner energy function. This is an exact result when the interatomic potential has finite range; for an infinite-range potential it is asymptotically valid as the lattice parameter tends to zero. The surface energy density is obtained explicitly as a function of the deformation gradient and boundary normal. The corner energy is found as an explicit function of the deformation gradient and the normals of the two facets meeting at the corner. A new bond counting approach is used, which reduces the problem to certain lattice point problems of number theory. The approach is then extended to more general convex regions with possibly curved boundary. The resulting surface energy density depends on the unit normal in a striking way. It is continuous at irrational directions, discontinuous at rational ones and nowhere differentiable. The method also yields an explicit interfacial energy for twin and phase boundaries.

\[

%\large

We study nonnegative radial solutions to the problem

\begin{equation*}

\left\{

\begin{split}

-\Delta u = \lambda K(\left|x \right|) f(u), \quad x \in \Omega

\\u = 0 \quad \qquad \quad \qquad \mbox{if } \left|x \right| = r_0

\\u \rightarrow0 \quad \qquad \quad \qquad \mbox{as } \left|x \right|\rightarrow\infty,

\end{split} \right.

\end{equation*}

where $\lambda$ is a positive parameter, $\Delta u=\mbox{div} \big(\nabla u\big)$ is the Laplacian of $u$,

$\Omega=\{x\in\ \mathbb{R}^{n}; n \textgreater 2, \left|x \right| \textgreater r_0\}$ and $K$ belongs to a class of functions such that $\lim_{r\rightarrow \infty}K(r)=0$. For classes of nonlinearities $f$ that are negative at the origin and sublinear at $\infty$ we discuss existence and uniqueness results when $\lambda$ is large.

\]

• James Herterich         -           Mathematical modelling of water purification
• Paul Roberts               -           Mathematical models of retinal oxygen distribution
• Stephen O'Keeffe       -           Mathematical modelling of growth and stability in biological structures
• Andrey Melnik             -           Dynamics of anisotropic remodelling in elastic tissues

We consider two independent random sequences of length n.
We consider optimal alignments according to a scoring function S.
We show that when the scoring function S is chosen at random
then with probability 1, the frequency of the aligned letter pairs
converges to a unique distribution as n goes to infinity. We also show
some concentration of measure phenomena.

• Yujia Chen - Solving Surface Poisson's Equation via the Closest Point Method
• Alex Lewis - Modelling liquid crystal devices
• Georgina Lang - Modelling of Brain Tissue Swelling

Chief Organiser: Shari Levine.  Main speakers: Alexander Arhangel'skii, Alan Dow, Aisling McCluskey, Jan van Mill, Frank Tall, Vladimir Tkachuk

Contact for further information: @email

Respiration is a redox reaction in which oxidation of a substrate (often organic) is coupled to the reduction of a terminal electron acceptor (TEA) such as oxygen. Iron oxides in various mineral forms are abundant in sediments and sedimentary rocks, and many subsurface microbes have the ability to respire using Fe(III) as the TEA in anoxic conditions. This process is environmentally important in the degradation of organic substrates and in the redox-cycling of iron. But low mineral solubility limits the bioavailability of Fe(III), which microbes access primarily through reductive dissolution. For aqueous nutrients, expressions for microbial growth and nutrient uptake rates are standardly based on Monod kinetics. We address the question of what equivalent description is appropriate when solid phase Fe(III) is the electron acceptor.

Given a diffusion in R^n, we prove a small-noise expansion for its density. Our proof relies on the Laplace method on Wiener space and stochastic Taylor expansions in the spirit of Benarous-Bismut. Our result applies (i) to small-time asymptotics and (ii) to the tails of the distribution and (iii) to small volatility of volatility.

We shall study applications of this result to stochastic volatility models, recovering the Berestycki- Busca-Florent formula (using (i)), the Gulisashvili-Stein expansion (from (ii)) and Lewis' expansions (using (iii)).

This is a joint work with J.D. Deuschel (TU Berlin), P. Friz (TU Berlin) and S. Violante (Imperial College London).

A pseudofinite field is a perfect pseudo-algebraically closed (PAC) field which

has $\hat{\mathbb{Z}}$ as absolute Galois group. Pseudofinite fields exists and they can

be realised as ultraproducts of finite fields. A group $G$ is geometrically

represented in a theory $T$ if there are modles $M_0\prec M$ of $T$,

substructures $A,B$ of $M$, $B\subset acl(A)$, such that $M_0\le A\le B\le M$

and $Aut(B/A)$ is isomorphic to $G$. Let $T$ be a complete theory of

pseudofinite fields. We show that, geometric representation of a group whose order

is divisibly by $p$ in $T$ heavily depends on the presence of $p^n$'th roots of unity

in models of $T$. As a consequence of this, we show that, for almost all

completions of the theory of pseudofinite fields, over a substructure $A$, algebraic

closure agrees with definable closure, if $A$ contains the relative algebraic closure

of the prime field. This is joint work with Ehud Hrushovski.

The computational analysis of a mathematical model describing a complex system is often based on the following roadmap: first, an experiment is conceived, in which the measured data are (either directly or indirectly) related to the input data of the model equations; second, such equations are computationally solved to provide iconographic reconstructions of the unknown physical or physiological parameters of the system; third, the reconstructed images are utilized to validate the model or to inspire appropriate improvements. This talk will adopt such framework to investigate three applied problems, respectively in solar physics, neuroscience and physiology. The solar physics problem is concerned with the exploitation of hard X-ray data for the comprehension of energy transport mechanisms in solar flares. The neuroscientific problem is the one to model visual recognition in humans with the help of a magnetocencephalography experiment. Finally, the physiological problem investigates the kinetics of the kidney-bladder system by means of nuclear data.

While a general framework of approximating the solution to Hamilton-Jacobi-Bellman (HJB) equations by difference methods is well established, and efficient numerical algorithms are available for one-dimensional problems, much less is known in the multi-dimensional case. One difficulty is the monotone approximation of cross-derivatives, which guarantees convergence to the viscosity solution. We propose a scheme combining piecewise freezing of the policies in time with a suitable spatial discretisation to establish convergence for a wide class of equations, and give numerical illustrations for a diffusion equation with uncertain parameters. These equations arise, for instance, in the valuation of financial derivatives under model uncertainty.

This is joint work with Peter Forsyth.

A method of defining non-equilibrium entropy for a chaotic dynamical system is proposed which, unlike the usual method based on Boltzmann's principle $S = k\log W$, does not involve the concept of a macroscopic state. The idea is illustrated using an example based on Arnold's `cat' map. The example also demonstrates that it is possible to have irreversible behaviour, involving a large increase of entropy, in a chaotic system with only two degrees of freedom.

This is intended as an introductory talk about one of the most

important (and most geometric) aspects of Geometric Group Theory. No

prior knowledge of any maths will be assumed.

We present some recent results concerning domain wall motion in one-dimensional nanowires, including the existence, velocity and stability of travelling-wave and precessing solutions.  We consider the case of unixial anisotropy, characteristic of wires with symmetrical (e.g., circular) cross-section, as opposed to strongly anisotropic geometries (films and strips) that have received greater attention.  This is joint work with Arseni Goussev and Valeriy Slastikov.

Organisers: Hilary Priestley, Drew Moshier and Leo Cabrer.

This will be dedicated principally to extensions of duality theory beyond zero-dimensional structures and to its application in novel settings. Topics that are likely to feature include duality for bilattice-based structures and associated semantics; extensions to compact Hausdorff spaces, bitopological duality, and duality for continuous data; applications to coalgebraic logic. We shall be seeking two-way interaction between those focused on a particular application and those who are seeking to extend the theory. Keynote speakers will be Mike Mislove and Drew Moshier. Samson Abramsky will be away from Oxford fromJune 12, but we are grateful for his offer to give a talk on June 11. We are also pleased to announce that, through the good offices of Georg Gottlob (Oxford Department of Computer Science), we are able to include within W1 a tutorial lecture on the applications of bilattice semantics to computer science; this will be given by Ofer Arieli.

 Nuclear fusion offers the prospect of abundant clean energy production, but the physical and engineering challenges are very great. In nuclear fusion reactors, the fuel is in the form of a plasma (charged gas) which is confined at high temperature and density using a toroidal magnetic field. This configuration is susceptible to turbulence, which transports heat out of the plasma and prevents fusion. It is believed that rotating the plasma suppresses turbulence, but experiments are expensive and even modest numerical simulation requires hundreds of thousands of CPU hours. We present a numerical technique for one of the five phase-space dimensions that both improves the accuracy of the calculation and greatly reduces the resolution required.