L3

Motion at the microscale is a fascinating field visualizing non-equilibrium behaviour of matter. Evolution has optimized the ability of microscale swimming on different length scales from tedpoles, to sperm and bacteria. A constant metabolic energy input is required to achieve active propulsion which means these systems are obeying the laws imposed by a low Reynolds number. Several strategies - including topography1 , chemotaxis or rheotaxis2 - have been used to reliably determine the path of active particles. Curiously, many of these strategies can be recognized as analogues of approaches employed by nature and are found in biological microswimmers.

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Bacteria attached to the metal caps of Janus particles3

However, natural microswimmers are not limited to being exemplary systems on the way to artificial micromotion. They certainly enable us to observe how nature overcame problems such as a lack of inertia, but natural microswimmers also offer the possibility to couple them to artificial microobjects to create biohybrid systems. Our group currently explores different coupling strategies and to create so called ‘BacteriaBots’.4

1 J. Simmchen, J. Katuri, W. E. Uspal, M. N. Popescu, M. Tasinkevych, and S. Sánchez, Nat. Commun., 2016, 7, 10598.

2 J. Katuri, W. E. Uspal, J. Simmchen, A. Miguel-López and S. Sánchez, Sci. Adv., , DOI:10.1126/sciadv.aao1755.

3 M. M. Stanton, J. Simmchen, X. Ma, A. Miguel-Lopez, S. Sánchez, Adv. Mater. Interfaces, DOI:10.1002/admi.201500505.

4 J. Bastos-Arrieta, A. Revilla-Guarinos, W. E. Uspal and J. Simmchen, Front. Robot. AI, 2018, 5, 97.

The geometry of the moduli space of 4d  N=2  moduli spaces, and in particular of their Coulomb branches (CBs), is very constrained. In this talk I will show that through its careful study, we can learn general and somewhat surprising lessons about the properties of N=2  super conformal field theories (SCFTs). Specifically I will show that we can prove that the scaling dimension of CB coordinates, and thus of the corresponding operator at the SCFT fixed point, has to be rational and it has a rank-dependent maximum value and that in general the moduli spaces of N=2 SCFTs can have metric singularities as well as complex structure singularities. 

Finally I will outline how we can explicitly perform a classification of geometries of N>=3 SCFTs and carry out the program up to rank-2. The results are surprising and exciting in many ways.

The Euler equation describing motion of ideal fluids goes back to 1755. 
The analysis of the equation is challenging since it is nonlinear and nonlocal. Its solutions are often unstable and spontaneously generate small scales. The fundamental question of global regularity vs finite time singularity formation 
remains open for the Euler equation in three spatial dimensions. In this lecture, I will review the history of this question and its connection with the arguably greatest unsolved problem of classical physics, turbulence. Recent results on small scale and singularity formation in two dimensions and for a number of related models will also be presented.

 I will describe recent progress in the study of scattering amplitudes in gauge theory and gravity at loop level, using the formalism of the scattering equations. The scattering equations relate the kinematics of the scattering of massless particles to the moduli space of the sphere. Underpinned by ambitwistor string theory, this formalism provides new insights into the relation between tree-level and loop-level contributions to scattering amplitudes. In this talk, I will describe results up to two loops on how loop integrands can be constructed as forward-limits of trees. One application is the loop-level understanding of the colour-kinematics duality, a symmetry of perturbative gauge theory which relates it to perturbative gravity.

Superstring scattering amplitudes in genus one have a low-energy expansion in terms of certain real analytic modular forms, called modular graph functions (D'Hoger, Green, Gürdogan and Vanhove). I will sketch the proof that these functions belong to a family of iterated integrals of modular forms (a generalization of Eichler integrals), recently introduced by Francis Brown, which explains many of their properties. The main tools are elliptic multiple polylogarithms (Brown and Levin), single-valued versions thereof, and elliptic multiple zeta values (Enriquez).

I will review some recent progress on D=4, N=2 superconformal field theories in what has come to be known as "Class-S". This is a huge class of (mostly non-Lagrangian) SCFTs, whose properties are encoded in the data of a punctured Riemann surface and a collection (one per puncture) of nilpotent orbits in an ADE Lie algebra.

I will talk about the Witten index of supersymmetric quantum mechanics obtained from 3d gauge theories compacted on a Riemann surface. In particular, I will show that the twisted indices of 3d N=4 theories compute enumerative invariants of the moduli space, which can be identified as a space of quasi-maps to the Higgs branch. I will also discuss 3d mirror symmetry in this context which provides a non-trivial relation between a pair of generating functions of the invariants.

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.