Past

Dynatomic curves parametrize n-periodic orbits of a one-parameter family of polynomial dynamical systems. These curves lack the structure of their arithmetic-geometric analogues (modular curves of level n) but can be studied dynamically.  Morton and Silverman conjectured a dynamical analogue of the uniform boundedness conjecture (theorems of Mazur, Merel), asserting uniform bounds for the number of rational periodic points for such a family.  I will discuss recent work towards the function field version of their conjecture, including results on the reduction mod p of dynatomic curves for the quadratic polynomial family z^2+c.

For many applications in science and engineering, the ability to efficiently and accurately approximate solutions to elliptic PDEs dictates what physical phenomena can be simulated numerically.  In this seminar, we present a high-order accurate discretization technique for variable coefficient PDEs with smooth coefficients.  The technique comes with a nested dissection inspired direct solver that scales linearly or nearly linearly with respect to the number of unknowns.  Unlike the application of nested dissection methods to classic discretization techniques, the constant prefactors do not grow with the order of the discretization.  The discretization is robust even for problems with highly oscillatory solutions.  For example, a problem 100 wavelengths in size can be solved to 9 digits of accuracy with 3.7 million unknowns on a desktop computer.  The precomputation of the direct solver takes 6 minutes on a desktop computer.  Then applying the computed solver takes 3 seconds.  The recent application of the algorithm to inverse media scattering also will be presented.

I will  address the study of decay rates of solutions to dissipative equations. The characterization of these rates will first be given for a wide class of linear systems by the decay character, which is a number associated to the initial datum that describes the behavior of the datum near the origin in frequency space. The understanding of the behavior of the linear  combined with the decay character and the Fourier Splitting method is then used to obtain some  upper and lower bounds for decay of solutions to appropriate dissipative nonlinear equations, both in the incompressible and compressible case. 

I will talk about a result on meromorphic continuation of Euler products over primes p of definable p-adic or motivic integrals, and applications to zeta functions of groups. If time permitting, I'll state an analogue for counting rational points of bounded height in some adelic homogeneous spac

The goal of this talk will be to give an overview of recent work, joint with Kim Moore, on a short time existence problem in Lagrangian mean curvature flow. More specifically, we consider a compact initial Lagrangian submanifold with a finite number of singularities, each asymptotic to a pair of transversely intersecting planes. We show it is possible to construct a smooth Lagrangian mean curvature flow, existing for positive times, that attains the singular Lagrangian as its initial condition in a suitable weak sense.  The construction uses a family of smooth solutions whose initial conditions approximate the singular Lagrangian. In order to appeal to compactness theorems and produce the desired solution, it is necessary to first establish uniform curvature estimates on the approximating family. As time allows I hope to focus in particular on the proof of these estimates, and their role in the proof of the main theorem.

Each year Prof. Trefethen gives the Problem Solving Squad a sequence of problems with no hints, one a week, where the solution of each problem is a single real number to be computed by any method available.  We will present this year's three problems, involving (1) an S-shaped bifurcation curve, (2) shortest path around a random web, and (3) switching a time-varying system to maximize a matrix norm.

The 14 students this year are Simon Vary plus InFoMM cohort 2: Matteo Croci, Davin Lunz, Michael McPhail, Tori Pereira, Lindon Roberts, Caoimhe Rooney, Ian Roper, Thomas Roy, Tino Sulzer, Bogdan Toader, Florian Wechsung, Jess Williams, and Fabian Ying.  The presentations will be by (1) Lindon Roberts, (2) Florian Wechsung, and (3) Thomas Roy.

Delsarte-Goethals frames are a popular choice for deterministic measurement matrices in compressive sensing. I will show that it is possible to construct extremely sparse matrices which share precisely the same row space as Delsarte-Goethals frames. I will also describe the combinatorial block design underlying the construction and make a connection to Steiner equiangular tight frames.
 

Part of the series "What do historians of mathematics do?"

The talk will set out the key debate in England at the Restoration, the need for a new orientation in mathematics towards algebra and the new "analysis". It will focus on efforts by three central players in England's mathematical community, John Pell, John Collins, and the Oxford mathematician John Wallis to produce an English language algebra text which would play a pioneering role in promoting this change. What was the background to the work we now call Pell's Algebra and why was it so significant?

We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations. We also numerically compute a special class of such solutions, namely those possessing icosahedral symmetry. We overcome several difficulties along the way. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly under-determined mid-plane deformation. We then use well known group-theoretic selection techniques combined with global bifurcation methods to obtain our results.

We will discuss a construction of cobordism maps on the full link complex for decorated link cobordisms. We will focus on some formal properties, such as grading change formulas and local relations. We will see how several expressions for mapping class group actions can be interpreted in terms of pictorial relations on decorated surfaces. Similarly, we will see how these pictorial relations give a "connected sum formula" for the involutive concordance invariants of Hendricks and Manolescu.

Lyons’ theory of rough paths allows us to solve stochastic differential equations driven by a Gaussian processes X of finite p-variation. The rough integral of the solutions against X again exists. We show that the solution also belong to the domain of the divergence operator of the Malliavin derivative, so that the 'Skorohod integral' of the solution with respect to X can also be defined. The latter operation has some properties in common with the Ito integral, and a natural question is to find a closed-form conversion formula between this rough integral and its Malliavin divergence. This is particularly useful in applications, where often one wants to compute the (conditional) expectation of the rough integral. In the case of Brownian motion our formula reduces to the classical Stratonovich-to-Ito conversion formula. There is an interesting difference between the formulae obtained in the cases 2<=p<3 and 3<=p<4, and we consider the reasons for this difference. We elaborate on the connection with previous work in which the integrand is generally assumed to be the gradient of a smooth function of X_{t}; we show that our formula can recover these results as special cases. This is joint work with Nengli Lim.

Let Q be a uniformly random quadrangulation with simple boundary decorated by a critical (p=3/4) face percolation configuration.  We prove that the chordal percolation exploration path on Q between two marked boundary edges converges in the scaling limit to SLE(6) on the Brownian disk (equivalently, a Liouville quantum gravity surface).  The topology of convergence is the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces.  Our method of proof is robust and, up to certain technical steps, extends to any percolation model on a random planar map which can be explored via peeling.  Joint work with E. Gwynne.

I shall discuss Zauner's conjecture about the existence of n^2 mutually equidistant points in complex projective space CP^{n-1} with its standard Fubini-Study metric. This is the so-called SIC-POVM problem, and is related to properties of the moment mapping that embeds CP^{n-1} into the Lie algebra su(n). In the case n=3, there is an obvious 1-parameter family of such sets of 9 points under the action of SU(3) and we shall sketch a proof that there are no others. This is joint work with Lane Hughston.

Understanding the structure of hadrons in terms of their fundamental constituents requires an understanding of QCD at large distances, a vastly complex and unsolved dynamical problem. I will discuss in this talk a new approach to hadron structure based on superconformal quantum mechanics in the light-front and its holographic embedding in a higher dimensional gravity theory. This approach captures essential aspects of the confinement dynamics which are not apparent from the QCD Lagrangian, such as the emergence of a mass scale and confinement, the occurrence of a zero mode: the pion, universal Regge trajectories for mesons and baryons and precise connections between the light meson and nucleon spectra. This effective semiclassical approach to relativistic bound-state equations in QCD can be extended to heavy-light hadrons where heavy quark masses break the conformal invariance but the underlying dynamical supersymmetry holds.
 

Professor Uta Frith FRS is a distinguished developmental psychologist who is well known for her pioneering research on autism spectrum disorders. She also has a long-standing interest in matters relating to diversity in science, and is the Chair of the Royal Society's Diversity Committee. Oxford Mathematician Dr Maria Bruna is a Junior Research Fellow in Mathematics at St John's College, and has won prizes such as the L'Oréal-UNESCO UK and Ireland For Women in Science Fellowship and the Olga Taussky Pauli Fellowship, Wolfgang Pauli Institute. This informal discussion will no doubt include a range of topics -- but it is hard to say in advance where the conversation might go!

In my research I model three components of the Earth system: the ice sheets, the ocean, and the solid Earth. In the first half of this talk I will describe the traditional approach that is used to model the impact of ice sheet growth and decay on global sea-level change and solid Earth deformation. I will then go on to explain how collaboration across the fields of glaciology, geodynamics and seismology is providing exciting new insight into feedbacks between ice dynamics and solid Earth deformation.

Biomedical research and clinical practice rely on complex and multimodality

datasets for the characterisation of human organs in health and disease. In

computational biomedicine, we often argue that multiscale computational

models are and will be increasingly required as tools for data integration,

for probing the established knowledge of physiological systems, and for

predictions of the effects of therapies and disease. But what has

computational biomedicine delivered so far? This presentation will describe

successes, failures and future directions of computational models in

cardiac research from basic to translational science.

I spent a number of years trading government bonds and interest-rate derivatives for Barclays Capital. This included the period of the financial crisis, and I was a colleague of some of the Barclays traders charged with fraud related to LIBOR rate manipulation. I will present a some examples of common trading scenarios, and some of the ethical issues these might raise for quants.
 

Let A be an abelian variety over the function field K of a curve over a finite field of characteristic p>0. We shall show that the group A(K^{p^{-\infty}}) is finitely generated, unless severe restrictions are put on the geometry of A. In particular, we shall show that if A is ordinary and has a point of bad reduction then A(K^{p^{-\infty}}) is finitely generated. This result can be used to give partial answers to questions of Scanlon, Ziegler, Esnault, Voloch and Poonen.

Conventional neutron reflection is a very powerful tool to characterise surfactants, polymers and other materials at the solid/liquid and air/liquid interfaces. Usually the analysis considers molecular layers with coherent addition of reflected waves that give the resultant reflected intensity. In this short workshop talk I will illustrate recent developments in this approach to address a wide variety of challenges of academic and commercial interest. Specifically I will introduce the challenges of using substrates that are thick on the coherence lengthscale of the radiation and the issues that brings in the structural analysis. I also invite the audience to consider if there may be some mathematical analysis that might lead us to exploit this incoherence to optimise our analysis. In particular, facilitating the removal of the 'background substrate contribution' to help us focus on the adsorbed layers of most interest.

I will review the notion of classifying topos of a first-order (geometric) theory and explain the central role enjoyed by theories of presheaf type (i.e. classified by a presheaf topos) in the context of the topos-theoretic investigation of the model theory of geometric theories. After presenting a few main results and characterizations for theories of presheaf type, I will illustrate the generality of the point of view provided by this class of theories by discussing a topos-theoretic framework unifying and generalizing Fraissé’s construction in model theory and topological Galois theory and leading to an approach to the problem of the independence from l of l-adic cohomology.

In 2016, Habegger and Pila published a proof of the Zilber-Pink conjecture for curves in abelian varieties (all defined over $\mathbb{Q}^{\rm alg}$). Their article also contained a proof of the same conjecture for a product of modular curves that was conditional on a strong arithmetic hypothesis. Both proofs were extensions of the Pila-Zannier strategy based in o-minimality that has yielded many results in this area. In this talk, we will explain our generalisation of the strategy to the Zilber-Pink conjecture for any Shimura variety. This is joint work with J. Ren.