Past

Snapshot mosaic multispectral imagery acquires an undersampled data cube by acquiring a single spectral measurement per spatial pixel. Sensors which acquire p frequencies, therefore, suffer from severe 1/p undersampling of the full data cube.  We show that the missing entries can be accurately imputed using non-convex techniques from sparse approximation and matrix completion initialised with traditional demosaicing algorithms.

Formally, the Fourier coefficients of Eisenstein series on Kac-Moody groups contain as yet mysterious automorphic L-functions relevant to open conjectures such as that of Ramanujan and Langlands functoriality. In this talk, we will consider the constant Fourier coefficient, if it even makes sense rigorously, and its relationship to the geometry and combinatorics of a Kac-Moody group. Joint work with Kyu-Hwan Lee.

This talk is concerned with the mathematical and numerical analysis of a steady phase change problem for non-isothermal incompressible viscous flow. The system is formulated in terms of pseudostress, strain rate and velocity for the Navier-Stokes-Brinkman equation, whereas temperature, normal heat flux on the boundary, and an auxiliary unknown are introduced for the energy conservation equation. In addition, and as one of the novelties of our approach, the symmetry of the pseudostress is imposed in an ultra-weak sense, thanks to which the usual introduction of the vorticity as an additional unknown is no longer needed. Then, for the mathematical analysis two variational formulations are proposed, namely mixed-primal and fully-mixed approaches, and the solvability of the resulting coupled formulations is established by combining fixed-point arguments, Sobolev embedding theorems and certain regularity assumptions. We then construct corresponding Galerkin discretizations based on adequate finite element spaces, and derive optimal a priori error estimates. Finally, numerical experiments in 2D and 3D illustrate the interest of this scheme and validate the theory.

A recent calculation of the multi-Higgs boson production in scalar theories
with spontaneous symmetry breaking has demonstrated the fast growth of the
cross section with the Higgs multiplicity at sufficiently large energies,
called “Higgsplosion”. It was argued that “Higgsplosion” solves the Higgs
hierarchy and fine-tuning problems. The phenomena looks quite exciting,
however in my talk I will present arguments that: a) the formula for
“Higgsplosion” has a limited applicability and inconsistent with unitarity
of the Standard Model; b) that the contribution from “Higgsplosion” to the
imaginary part of the Higgs boson propagator cannot be re-summed in order to
furnish a solution of the Higgs hierarchy and fine-tuning problems.

Based on our recent paper https://arxiv.org/abs/1808.05641 (A. Belyaev, F. Bezrukov, D. Ross)

Let $\Omega\subseteq\mathbb{R}^2$ be a domain and let $f\in W^{1,1}(\Omega,\mathbb{R}^2)$ be a homeomorphism (between $\Omega$ and $f(\Omega)$). Then there exists a sequence of smooth diffeomorphisms $f_k$ converging to $f$ in $W^{1,1}(\Omega,\mathbb{R}^2)$ and uniformly. This is a joint result with A. Pratelli.
 

Abstract: The edge reinforced random walk is a self-interacting process, in which the random walker prefer visited edges with a bias proportional to the number of times the edges were visited. We will gently introduce this model and talk about some of its histories and recent progresses.

 In the late seventies, Casson and Gordon developed several knot invariants that obstruct a knot from being slice, i.e. from bounding a disc in the 4-ball. In this talk, we use twisted Blanchfield pairings to define twisted generalisations of the Levine-Tristram signature function, and describe their relation to the Casson-Gordon invariants. If time permits, we will present some obstructions to algebraic knots being slice. This is joint work with Maciej Borodzik and Wojciech Politarczyk.

We will consider some problems on calculating  the average  angles of random polytopes. Some of them are open.

In this talk, we will explain how to construct embedded closed Lagrangian submanifolds in mirror quintic threefolds using tropical curves and the toric degeneration technique. As an example, we will illustrate the construction for tropical curves that contribute to the Gromov–Witten invariant of the line class of the quintic threefold. The construction will in turn provide many homologous and non-Hamiltonian isotopic Lagrangian
rational homology spheres, and a geometric interpretation of the multiplicity of a tropical curve as the weight of a Lagrangian. This is a joint work with Helge Ruddat.

Finding Riemannian metrics with holonomy G_2 is a challenging problem with links in mathematics to Einstein metrics and area-minimizing submanifolds, and to M-theory in theoretical physics.  I will provide a brief survey on recent progress towards studying this problem using a geometric flow approach, including connections to calibrated fibrations.

Partially molten materials resist shearing and compaction. This resistance

is described by a fourth-rank effective viscosity tensor. When the tensor

is isotropic, two scalars determine the resistance: an effective shear and

an effective bulk viscosity. In this seminar, calculations are presented of

the effective viscosity tensor during diffusion creep for a 3D tessellation of

tetrakaidecahedrons (truncated octahedrons). The geometry of the melt is

determined by assuming textural equilibrium.  Two parameters

control the effect of melt on the viscosity tensor: the porosity and the

dihedral angle. Calculations for both Nabarro-Herring (volume diffusion)

and Coble (surface diffusion) creep are presented. For Nabarro-Herring

creep the bulk viscosity becomes singular as the porosity vanishes. This

singularity is logarithmic, a weaker singularity than typically assumed in

geodynamic models. The presence of a small amount of melt (0.1% porosity)

causes the effective shear viscosity to approximately halve. For Coble creep,

previous modelling work has argued that a very small amount of melt may

lead to a substantial, factor of 5, drop in the shear viscosity. Here, a

much smaller, factor of 1.4, drop is obtained.

How do humans process information? What are their strengths and limitations? This crash course in cognitive psychology will provide the background necessary to think realistically about how learning works.

Computational nucleic acid devices show great potential for enabling a broad range of biotechnology applications, including smart probes for molecular biology research, in vitro assembly of complex compounds, high-precision in vitro disease diagnosis and, ultimately, computational therapeutics inside living cells. This diversity of applications is supported by a range of implementation strategies, including nucleic acid strand displacement, localisation to substrates, and the use of enzymes with polymerase, nickase and exonuclease functionality. However, existing computational design tools are unable to account for these different strategies in a unified manner. This talk presents a programming language that allows a broad range of computational nucleic acid systems to be designed and analysed. We also demonstrate how similar approaches can be incorporated into a programming language for designing genetic devices that are inserted into cells to reprogram their behaviour. The language is used to characterise the genetic components for programming populations of cells that communicate and self-organise into spatial patterns. More generally, we anticipate that languages and software for programming molecular and genetic devices will accelerate the development of future biotechnology applications.

Generative adversarial networks (GANs) use neural networks as generative models, creating realistic samples that mimic real-life reference samples (for instance, images of faces, bedrooms, and more). These networks require an adaptive critic function while training, to teach the networks how to move improve their samples to better match the reference data. I will describe a kernel divergence measure, the maximum mean discrepancy, which represents one such critic function. With gradient regularisation, the MMD is used to obtain current state-of-the art performance on challenging image generation tasks, including 160 × 160 CelebA and 64 × 64 ImageNet. In addition to adversarial network training, I'll discuss issues of gradient bias for GANs based on integral probability metrics, and mechanisms for benchmarking GAN performance.

If G is a topological group, a G-flow X is a non-empty, compact, Hausdorff space on which G acts continuously; it is minimal if all G-orbits are dense. By a theorem of Ellis, there is a (unique) minimal G-flow M(G) which is universal: there is a continuous G-map to every other G-flow. 

Here, we will be interested in the case where G = Aut(K) for some structure K, usually omega-categorical. Work of Kechris, Pestov and Todorcevic and others gives conditions on K under which structural Ramsey Theory (due to Nesetril - Rodl and others) can be used to compute M(G). 

In the first part of the talk I will give a description of the above theory and when it applies (the 'tame case'). In the second part, I will describe joint work with J. Hubicka and J. Nesetril which shows that the omega-categorical structures constructed in the late 1980's by Hrushovski as counterexamples to Lachlan's conjecture are not tame and moreover, minimal flows of their automorphism groups have rather different properties to those in the tame case. 

When S is a finite set of natural numbers, a GCD-sum is a particular kind of double-sum over the elements of S, and they arise naturally in several settings. In particular, these sums play a role when one studies the local statistics of point sequences on the unit circle. There are known upper bounds for the size of a GCD-sum in terms of the size of the set S, most recently due to de la Bretèche and Tenenbaum, and these bounds are sharp. Yet the known examples of sets S for which the GCD-sum over S provides a matching lower bound all possess strong multiplicative structure, whereas in applications the set S often comes with additive structure. In this talk I will describe recent joint work with Thomas Bloom in which we apply an estimate from sum-product theory to prove a much stronger upper bound on a GCD-sum over an additively structured set. I will also describe an application of this improvement to the study of the distribution of points on the unit circle, with a further application to arbitrary infinite subsets of squares. 

We study the value of a zero-sum stopping game in which the terminal payoff function depends on the underlying process and on an additional randomness (with finitely many states) which is known to one player but unknown to the other. Such asymmetry of information arises naturally in insider trading when one of the counterparties knows an announcement before it is publicly released, e.g., central bank's interest rates decision or company earnings/business plans. In the context of game options this splits the pricing problem into the phase before announcement (asymmetric information) and after announcement (full information); the value of the latter exists and forms the terminal payoff of the asymmetric phase.

The above game does not have a value if both players use pure stopping times as the informed player's actions would reveal too much of his excess knowledge. The informed player manages the trade-off between releasing information and stopping optimally employing randomised stopping times. We reformulate the stopping game as a zero-sum game between a stopper (the uninformed player) and a singular controller (the informed player). We prove existence of the value of the latter game for a large class of underlying strong Markov processes including multi-variate diffusions and Feller processes. The main tools are approximations by smooth singular controls and by discrete-time games.

After considering motivations in symplectic geometry, I’ll give a summary of $C^\infty$-Algebraic Geometry and how to extend these concepts to manifolds with corners. 

This talk will concern the problem of inference when the posterior measure involves continuous models which require approximation before inference can be performed. Typically one cannot sample from the posterior distribution directly, but can at best only evaluate it, up to a normalizing constant. Therefore one must resort to computationally-intensive inference algorithms in order to construct estimators. These algorithms are typically of Monte Carlo type, and include for example Markov chain Monte Carlo, importance samplers, and sequential Monte Carlo samplers. The multilevel Monte Carlo method provides a way of optimally balancing discretization and sampling error on a hierarchy of approximation levels, such that cost is optimized. Recently this method has been applied to computationally intensive inference. This non-trivial task can be achieved in a variety of ways. This talk will review 3 primary strategies which have been successfully employed to achieve optimal (or canonical) convergence rates – in other words faster convergence than i.i.d. sampling at the finest discretization level. Some of the specific resulting algorithms, and applications, will also be presented.

Classical methods for X-ray computed tomography (CT) are based on the assumption that the X-ray source intensity is known. In practice, however, the intensity is measured and hence uncertain. Under normal circumstances, when the exposure time is sufficiently high, this kind of uncertainty typically has a negligible effect on the reconstruction quality. However, in time- or dose-limited applications such as dynamic CT, this uncertainty may cause severe and systematic artifacts known as ring artifacts.
By modeling the measurement process and by taking uncertainties into account, it is possible to derive a convex reconstruction model that leads to improved reconstructions when the signal-to-noise ratio is low. We discuss some computational challenges associated with the model and illustrate its merits with some numerical examples based on simulated and real data.

In this talk, we demonstrate that formation of Type I singularities of suitable weak solutions of the Navier-Stokes equations occur if there exists non-zero mild bounded ancient solutions satisfying a 'Type I' decay condition. We will also discuss some new Liouville type Theorems. Joint work with Dallas Albritton (University of Minnesota).

I will discuss Agol's proof of the Virtually Fibred Conjecture of
Thurston, focusing on the role played by the `RFRS' property. I will
then show how one can modify parts of Agol's proof by replacing some
topological considerations with a group theoretic statement about
virtual fibring of RFRS groups.
 

In geometric group theory we study groups by their actions on metric spaces. Although a given group might admit many actions on different metric spaces, on a large scale these spaces will all look similar, and so the large scale properties of a space on which a group acts are intrinsic to the group. One particularly natural example of a large scale property used in this way is the Gromov boundary of a hyperbolic metric space. This is a topological space that can be thought of as compactifying the metric space at infinity. 

In this talk I will describe some constructions of spaces occurring in this way with nasty, fractal-like properties. On the other hand, there are limits to how pathological these spaces can be: theorems of Bestvina and Mess, Bowditch and Swarup imply that boundaries of hyperbolic groups are locally path connected whenever they are connected. I will discuss these results and some generalisations. 

Abstract:   Let $f$ be a continuous surjection from the compact metric space $X$ to itself. 

We say that the dynamical system $(X,f)$ has shadowing if for every $\epsilon>0$ there is a $\delta>0$ such that every $\delta$-pseudo orbit is $\epsilon$-shadowed.  Here a sequence $(x_n)$ is a $\delta$-pseudo orbit provided the distance from $f(x_n)$ to $x_{n+1}$ is less than $\delta$ and $(x_n)$ is $\epsilon$-shadowed if there is a point $z$ such that the distance from $x_n$ to $f^n(z)$ is less than $\epsilon$.  

If $f$ is a homeomorphism, $(X,f)$ is said to be expansive if there is some $c>0$, such that if the distance from $f^n(x)$ and $f^n(y)$ is less than $c$ for all $n\in \mathbb Z$, then $x=y$.

In his proof that a homeomorphism that is expansive and has shadowing is stable, Walters shows that in an expansive system with shadowing, a pseudo orbit is shadowed by exactly one point.  It turns out that the converse is also true: if the system has unique shadowing (in the above sense), then it is expansive.

In this talk, which is joint work with Joel Mitchell and Joe Thomas, we explore this notion of unique shadowing.

Suppose that you have a long strip of paper, and draw the central line through it. You then glue it together so as to make a Möbius band. Can the drawn curve be contained in a plane?

We'll answer the question in this talk, as well as introduce the concepts from the Geometry of Surfaces course required to go through it; including Gauss' one and only Theorema Egregium! (we won't prove it though).

We begin by reviewing numerical methods for problems in one variable and find that univariate polynomials are the starting point for most of them.  A similar review in several variables, however, reveals that multivariate polynomials are not so important.  Why?  On the other hand in pure mathematics, the field of algebraic geometry is precisely the study of multivariate polynomials.  Why?

Let k be a characteristic $p>0$ perfect field, V be a complete DVR whose residue field is $k$ and fraction field $K$ is of characteristic $0$. We denote by $\mathcal{E}  _K$ the Amice ring with coefficients in $K$, and by $\mathcal{E} ^\dagger _K$ the bounded Robba ring with coefficients in $K$. Berthelot's classical theory of Rigid Cohomology over varieties $X/k((t))$ gives $\mathcal{E}  _K$-valued objects.  Recently, Lazda and Pal developed a refinement of rigid cohomology,
i.e. a theory of $\mathcal{E} ^\dagger _K$-valued Rigid Cohomology over varieties $X/k((t))$. Using this refinement, they proved a semistable version of the variational Tate conjecture. 

The purpose of this talk is to introduce to a theory of arithmetic D-modules with $\mathcal{E} ^\dagger _K$-valued cohomology which satisfies a formalism of Grothendieck’s six operations. 
 

In this talk we will show how the floating point errors in the simulation of SDEs (stochastic differential equations) can be modelled as stochastic. Furthermore, we will show how these errors can be corrected within a multilevel Monte Carlo approach which performs most calculations with low precision, but a few calculations with higher precision. The same procedure can also be used to correct for errors in converting from uniform random numbers to approximate Normal random numbers. Numerical results will be generated on both CPUs (using single/double precision) and GPUs (using half/single precision).

With growing population of humans being clustered in large cities and connected by fast routes more suitable environments for epidemics are being created. Topped by rapid mutation rate of viral and bacterial strains, epidemiological studies stay a relevant topic at all times. From the beginning of 2019, the World Health Organization publishes at least five disease outbreak news including Ebola virus disease, dengue fever and drug resistant gonococcal infection, the latter is registered in the United Kingdom.

To control the outbreaks it is necessary to gain information on mechanisms of appearance and evolution of pathogens. Close to all disease-causing virus and bacteria undergo a specialization towards a human host from the closest livestock or wild fauna of a shared habitat. Every strain (or subtype) of a pathogen has a set of characteristics (e.g. infection rate and burst size) responsible for its success in a new environment, a host cell in case of a virus, and with the right amount of skepticism that set can be framed as fitness of the pathogen. In our model, we consider a population of a mutating strain of a virus. The strain specialized towards a new host usually remains in the environment and does not switch until conditions get volatile. Two subtypes, wild and mutant, of the virus share a host. This talk will illustrate findings on an explicitly independent cycling coexistence of the two subtypes of the parasite population. A rare transcritical bifurcation of limit cycles is discussed. Moreover, we will find conditions when one of the strains can outnumber and eventually eliminate the other strain focusing on an infection rate as fitness of strains.

Mysteries of isolated horizons: the Near Horizon Geometry equation, geometric characterizations of the non-extremal Kerr horizon, spacetimes foliated by non-expanding horizons.

3-dimensional null surfaces  that are  Killing horizons to the second order  are  considered. They are embedded in 4-dimensional spacetimes that satisfy the vacuum Einstein equations with arbitrary cosmological constant. Internal geometry of 2-dimensional cross sections of  the horizons  consists of induced metric tensor and a rotation 1-form potential. It is subject to the type D equation. The equation is interesting from the both, mathematical and physical points of view. Mathematically it  involves  geometry, holomorphic structures and algebraic topology.  Physically, the equation knows the secrete of black holes: the only  axisymmetric solutions on topological sphere  correspond  to the the Kerr / Kerr-de Sitter / Kerr-anti-de-Sitter non-extremal black holes or to the near horizon limit  of the extremal ones.  In the case of bifurcated  horizons the type D equation implies another spacial  symmetry. In this way the axial symmetry may be ensured without the rigidity theorem. The type D equation does not allow rotating horizons of topology different then that of the  sphere (or its quotient). That completes a new local non-her theorem. The type D equation is also  an integrability condition for the  Near Horizon Geometry equation and leads to new results on the solution existence issue.
 

Seeking income during World War II, Piet Hein created the game now called Hex, marketing it through the Danish newspaper Politiken.  The game was popular but disappeared in 1943 when Hein fled Denmark.

The game re-appeared in 1948 when John Nash introduced it to Princeton's game theory group, and became popular again in 1957 after Martin Gardner's column --- "Concerning the game of Hex, which may be played on the tiles of the bathroom floor" --- appeared in Scientific American.

I will survey the early history of Hex, highlighting the war's influence on Hein's design and marketing, Hein's mysterious puzzle-maker, and Nash's fascination with Hex's theoretical properties.

In a joint work with Matt Tointon, we study the fine structure of approximate groups. We deduce various applications on growth, isoperimetry and quantitative estimates for the the simple random walk on finite vertex transitive graphs.

The hypoelliptic Laplacian is a family of operators indexed by $b \in \mathbf{R}^*_+$, acting on the total space of the tangent bundle of a Riemannian manifold, that interpolates between the ordinary Laplacian as $b \to 0$ and the generator of the geodesic flow as $b \to +\infty$. These operators are not elliptic, they are not self-adjoint, they are hypoelliptic. One can think of the total space of the tangent bundle as the phase space of classical mechanics; so that the hypoelliptic Laplacian produces an interpolation between the geodesic flow and its quantisation. There is a dynamical counterpart, which is a natural interpolation between classical Brownian motion and the geodesic flow.

The hypoelliptic deformation preserves subtle invariants of the Laplacian. In the case of locally symmetric spaces (which are defined via Lie groups), the deformation is essentially isospectral, and leads to geometric formulas for orbital integrals, a key ingredient in Selberg's trace formula.

In a first part of the talk, I will describe the geometric construction of the hypoelliptic Laplacian in the context of de Rham theory. In a second part, I will explain applications to the trace formula.

In a joint-work with Andrea Collevecchio and Vladas Sidoravicius,  we study  phase transitions in the recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define the branching-ruin number of a tree, which is  a natural way to measure trees with polynomial growth and therefore provides a polynomial version of the branching number defined by Furstenberg (1970) and studied by R. Lyons (1990). We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk on this tree. We will also mention two other results where the branching-ruin number arises as critical parameter: first, in the context of random walks on heavy-tailed random conductances on trees and, second, in the case of Volkov's M-digging random walk.

A conjecture from the '80s claims that the isometry type of a closed, negatively curved Riemannian manifold should be uniquely determined by the lengths of its closed geodesics. By work of Otal, this is essentially equivalent to the problem of extending cross-ratio preserving maps between Gromov boundaries of simply connected, negatively curved manifolds. Progress on the conjecture has been remarkably slow, with only the 2-dimensional and locally symmetric cases having been solved so far (Otal '90 and Hamenstädt '99).
Still, it is natural to try leaving the world of manifolds and address the conjecture in the general context of non-positively curved metric spaces. We restrict to the class of CAT(0) cube complexes, as their geometry is both rich and well-understood. We introduce a new notion of cross ratio on their horoboundary and use it to provide a full answer to the conjecture in this setting. More precisely, we show that essential, hyperplane-essential cubulations of Gromov-hyperbolic groups are completely determined by their combinatorial length functions. One can also consider non-proper non-cocompact actions of non-hyperbolic groups, as long as the cube complexes are irreducible and have no free faces.
Joint work with J. Beyrer and M. Incerti-Medici.

Stable commutator length (scl) is a well established invariant of elements g in the commutator subgroup (write scl(g)) and has both geometric and algebraic meaning.  A group has a \emph{gap} in stable commutator length if for every non-trivial element g, scl(g) > C for some C > 0.
SCL may be interpreted as an 'algebraic translation length' and such a gap may be thus interpreted an 'algebraic injectivity radius'.
Many classes of groups have such a gap, like hyperbolic groups, mapping class groups, Baumslag-Solitar groups and graph of groups.
In this talk I will show that Right-Angled Artin Groups have the optimal scl-gap of 1/2. This yields a new invariant for the vast class of subgroups of Right-Angled Artin Groups.

Consider the asymmetric simple exclusion process with k particles on a linear lattice of N sites. I will present results on the asymptotic of the time needed for the system to reach its equilibrium distribution starting from the worst initial configuration (also called mixing time). Two main regimes appear according to the strength of the asymmetry (in terms of k and N), and in both regimes, the system displays a cutoff phenomenon: the distance to equilibrium falls abruptly from 1 to 0. This is a joint work with Hubert Lacoin (IMPA).

In this talk, we are going to talk about the Type I singularity on 4-dimensional manifolds foliated by homogeneous S3 evolving under the Ricci
flow. We review the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as by Isenberg, Knopf and Sesum. In the latter, a global frame for the tangent bundle, called the Milnor frame, was used to set up the problem. We shall look at the symmetries of the manifold, derived from Lie groups and its ansatz metrics, and this global tangent bundle frame developed by Milnor and Bianchi. Numerical simulations of the Ricci flow on these manifolds are done, following the work by Garfinkle and Isenberg, providing insight and conjectures for the main problem. Some analytic results will be proven for the manifolds S1×S3 and S4 using maximum principles from parabolic PDE theory and some sufficiency conditions for a neckpinch singularity will be provided. Finally, a problem from general relativity with similar metric symmetries but endowed on a manifold with differenttopology, the Taub-Bolt and Taub-NUT metrics, will be discussed.

Symmetric spaces and lattices are important objects to model spaces in geometry and topology. They have been studied from many different viewpoints. We will concentrate on their coarse geometry view point in this talk. I will first quickly go over some well-known results about quasi-isometry of those spaces. Then I will move to the study about quasi-isometric embeddings. While results in this direction are far less complete and well-studied, there are some rigidity phenomenons still happening here.

The InFoMM CDT presents The Reddick Lecture Dr. Nira Chamberlain (Holland & Barrett) Modelling the Competition Friday, 15 February 2019 17:00- 18:00 Mathematical Institute, L1 Followed by a drinks reception

It can be argued that any market would not survive without competition. It is everywhere; you can't run away from it. Competition can cause a business to either thrive, survive or die. So one might ask, why is there a need to mathematically model the competition? Two quotes may help to answer this: "Business is a game played for fantastic stakes, and you're in competition with experts. If you want to win, you have to learn to be a master of the game" Anon. “You can't look at the competition and say you're going to do it better. You have to look at the competition and say you're going to do it differently." Steve Jobs In this talk, I wish to demonstrate how mathematical modelling can be used to "master the game" and "do things differently". I will be focusing on three real life examples: Bidding to provide service support for a complex communication asset - dynamic travelling repairman Increasing market share in the Energy Sector - Markov Chain Retail's shop Location Location Location Location - Agent Based Simulation

Mathematicians need to talk and writeabout their mathematics.  This includes undergraduates and MSc students, who may be writing a dissertation or project report, preparing a presentation on a summer research project, or preparing for a job interview.  We think that it can be helpful to think of this as a form of story telling, as this can lead to more effective communication.  For a story to be engaging you also need to know your audience.In this session, we'll discuss what we mean by telling a mathematical story, give you some top tips from our experience, and give you a chance to think about how you might put this into practice.  The session will be of relevance to all undergraduates and MSc students, not only those currently writing a dissertation or preparing an oral presentation.

There is great interest in the molecular characterisation of intestinal metaplasia, such as Barrett’s esophagus (BE), to understand the basic biology of metaplastic development from a tissue of origin. BE is asymptomatic, so it is not generally known how long a patient has lived with this precursor of esophageal adenocarcinoma (EAC) when initially diagnosed in the clinic. We previously constructed a BE clock model using patient-specific methylation data to estimate BE onset times using Bayesian inference techniques, and thus obtain the biological age of BE tissue (Curtius et al. 2016). We find such epigenetic drift to be widely evident in BE tissue (Luebeck et al. 2017) and the corresponding tissue ages show large inter-individual heterogeneity in two patient populations.               

From a basic biological mechanism standpoint, it is not fully understood how the Barrett’s tissue first forms in the human esophagus because this process is never observed in vivo, yet such information is critical to inform biomarkers of risk based on temporal features (e.g., growth rates, tissue age) reflecting the evolution toward cancer. We analysed multi-region samples from 17 BE patients to

1) measure the spatial heterogeneity in biological tissue ages, and 2) use these ages to calibrate mathematical models (agent-based and continuum) of the mechanisms for formation of the segment itself. Most importantly, we found that tissue must be regenerated nearer to the stomach, perhaps driven by wound healing caused by exposure to reflux, implying a gastric tissue of origin for the lesions observed in BE. Combining bioinformatics and mechanistic modelling allowed us to infer evolutionary processes that cannot be clinically observed and we believe there is great translational promise to develop such hybrid methods to better understand multiscale cancer data.

References:

Curtius K, Wong C, Hazelton WD, Kaz AM, Chak A, et al. (2016) A Molecular Clock Infers Heterogeneous Tissue Age Among Patients with Barrett's Esophagus. PLoS Comput Biol 12(5): e1004919

Luebeck EG, Curtius K, Hazelton WD, Made S, Yu M, et al. (2017) Identification of a key role of epigenetic drift in Barrett’s esophagus and esophageal adenocarcinoma. J Clin Epigenet 9:113

Data science has become a topic of great interest lately and has triggered new widescale research activities around efficientl first order methods for optimisation and Bayesian sampling. The National Physical Laboratory is addressing some of these challenges with particular focus on  robustness and confidence in the solution.  In this talk, I will present some problems and recent results concerning i. robust learning in the presence of outliers based on the Median of Means (MoM) principle and ii. stability of the solution in super-resolution (joint work with A. Thompson and B. Toader).