L3

Tendons are vital connective tissues that anchor muscle to bone to allow the transfer of forces to the skeleton. They exhibit highly non-linear viscoelastic mechanical behaviour that arises due to their complex, hierarchical microstructure, which consists of fibrous subunits made of the protein collagen. Collagen molecules aggregate to form fibrils with diameters of tens to hundreds of nanometres, which in turn assemble into larger fibres called fascicles with diameters of tens to hundreds of microns. In this talk, I will discuss the relationship between the three-dimensional organisation of the fibrils and fascicles and the macroscale mechanical behaviour of the tendon. In particular, I will show that very simple constitutive behaviour at the microscale can give rise to highly non-linear behaviour at the macroscale when combined with geometrical effects.

Blisters form when a thin surface layer of a solid body separates/delaminates from the underlying bulk material over a finite, bounded region. It is ubiquitous in a range of industrial applications, e.g. blister test is applied to assess the strength of adhesion between thin elastic films and their solid substrates, and during natural processes, such as formation and spreading of laccoliths or retinal detachment.

We study a special case of blistering, in which a thin elastic membrane is adhered to the substrate by a thin layer of viscous fluid. In this scenario, the expansion of the newly formed blister by fluid injection occurs via a displacement flow, which peels apart the adhered surfaces through a two-way interaction between flow and deformation. If the injected fluid is less viscous than the fluid already occupying the gap, patterns of short and stubby fingers form on the propagating fluid interface in a radial geometry. This process is regulated by membrane compliance, which if increased delays the onset of fingering to higher flow rates and reduces finger amplitude. We find that the morphological features of the fingers are selected in a simple way by the local geometry of the compliant cell. In contrast, the local geometry itself is determined from a complex fluid–solid interaction, particularly in the case of rectangular blisters. Furthermore, changes to the geometry of the channel cross-section in the latter case lead to a rich variety of possible interfacial patterns. Our experiments provide a link between studies of airway reopening, Saffman-Taylor fingering and printer’s instability.   

At first glance the Interdistrict shipping problem resembles a transportation problem.  N sources with M destinations with k Stock keeping units (SKU’s); however, we want to solve for the optimal shipping frequency between each node while determining the flow of each SKU across the network.  As the replenishment quantity goes up, the shipping frequency goes down and the inventory holding cost goes up (AWI = Replenishment Qty/2 + SS).  Safety stock also increases as frequency decreases.  The relationship between replenishment quantity and shipping frequency is non-linear (frequency = annual demand/replenishment qty).  The trucks which are used to transfer the product have finite capacity and the cost to drive the truck between 2 locations is constant regardless of how many containers are actually on the truck up to the max capacity.  Each product can have a different footprint of truck capacity.  Cross docking is allowed.  (i.e. a truck may travel from Loc A to loc B carrying products X and Y.  At loc B, the truck unloads product X, picks up product Z, and continues to location C.  The key here is that product Y does not incur any handling costs at Loc B while products X and Z do.)

The objective function seeks to minimize the total costs ( distribution + handling + inventory holding costs)  for all locations, for all SKU’s, while determining how much of each product should flow across each arc such that all demand is satisfied.

In semimartingale optimal transport problem, the functional to be minimized can be considered as a “stochastic action”, which is the expectationof a “stochastic Lagrangian” in terms of differential semimartingale characteristics. Therefore it would be natural to apply variational calculus approach to characterize the minimizers. R. Lassalle and A.B. Cruzeiro have used this approach to establish a stochastic Euler-Lagrangian condition for semimartingale optimal transport by perturbing the drift terms. Motivated by their work, we want to perform the same type of calculus for martingale optimal transport problem. In particular, instead of only considering perturbations in the drift terms, we try to find a nice variational family for volatility,and then obtain the stochastic Euler-Lagrangian condition for martingale laws. In the first part of this talk we will mention some basic results regarding the existence of minimizers in semimartingale optimal transport problem. In the second part, we will introduce Lassalle and Cruzeiro’s  work, and give a simple example related to this topic, where the variational family is induced by time-changes; and then we will introduce some potential problems that are needed to be solved.

We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger,  $L^{2+\epsilon}$ (rather than $L^2$) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.  

I will talk about R^n valued random processes driven by a "noise", which is generated by a deterministic dynamical system, randomness coming from the choice of the initial condition.

Such processes were considered by D.Kelly and I.Melbourne.I will present our joint work with I.Chevyrev, P.Friz, I.Melbourne and H.Zhang, where we consider the noise with long term memory. We prove convergence to solution of a stochastic differential equation which is, depending on the noise, driven by either a Brownian motion (optimizing the assumptions of Kelly-Melbourne) or a Lévy process.Our work is made possible by recent progress in rough path theory for càdlàg paths in p-variation topology.

We consider the problem of optimally hedging a portfolio of derivatives in a scenario based discrete-time market with transaction costs. Risk-preferences are specified in terms of a convex risk-measure. Such a framework has suffered from numerical intractability up until recently, but this has changed thanks to technological advances: using hedging strategies built from neural networks and machine learning optimization techniques, optimal hedging strategies can be approximated efficiently, as shown by the numerical study and some theoretical results presented in this talk (based on joint work with Hans Bühler, Ben Wood and Josef Teichmann).

Solutions to Rough Differential Equations (RDE) may be constructed by several means. Beyond the fixed point argument, several approaches rely on using approximations of solutions over short times (Davie, Friz & Victoir, Bailleul, ...). In this talk, we present a generic, unifying framework to consider approximations of flows, called almost flows, and flows through the non-linear sewing lemma. This framework unifies the approaches mentioned above and decouples the analytical part from the algebraic part (manipulation of iterated integrals) when studying RDE. Beyond this, flows are objects with their own properties.New results, such as existence of measurable flows when several solutions of the corresponding RDE exist, will also be presented.

From a joint work with Antoine Brault (U. Toulouse III, France).

In the context of randomly fluctuating surfaces in (1+1)-dimensions two Universality Classes have generally been considered, the Kardar-Parisi-Zhang and the Edwards-Wilkinson. Models within these classes exhibit universal fluctuations under 1:2:3 and 1:2:4 scaling respectively. Starting from a modification of the classical Ballistic Deposition model we will show that this picture is not exhaustive and another Universality Class, whose scaling exponents are 1:1:2, has to be taken into account. We will describe how it arises, briefly discuss its connections to KPZ and introduce a new stochastic process, the Brownian Castle, deeply connected to the Brownian Web, which should capture the large-scale behaviour of models within this Class. 

The classical Lorentz gas model introduced by Lorentz in 1905, studied further by Sinai in the 1960s, provides a rich source of examples of chaotic dynamical systems with strong stochastic properties (despite being entirely deterministic).  Central limit theorems and convergence to Brownian motion are well understood, both with standard n^{1/2} and nonstandard (n log n)^{1/2} diffusion rates.

In joint work with Paulo Varandas, we discuss examples with diffusion rate n^{1/a}, 1<a<2, and prove convergence to an a-stable Levy process.  This includes to the best of our knowledge the first natural examples where the M_2 Skorokhod topology is the appropriate one.