This talk features a self-contained introduction to persistent homology, which is the main ingredient of topological data analysis. 

Injection of a gas into a gas/liquid foam is known to give rise to instability phenomena on a variety of time and length scales. Macroscopically, one observes a propagating gas-filled structure that can display properties of liquid finger propagation as well as of fracture in solids. Using a discrete model, which incorporates the underlying film instability as well as viscous resistance from the moving liquid structures, we describe brittle cleavage phenomena in line with experimental observations. We find that  the dimensions of the foam sample significantly affect the speed of the  cracks as well as the pressure necessary to sustain them: cracks in wider samples travel faster at a given driving stress, but are able to avoid arrest and maintain propagation at a lower pressure (the  velocity gap becomes smaller). The system thus becomes a study case for stress concentration and the transition between discrete and continuum systems in dynamical fracture; taking into account the finite dimensions of the system improves agreement with experiment.

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.

[[{"fid":"53543","view_mode":"media_square","fields":{"format":"media_square","field_file_image_alt_text[und][0][value]":false,"field_file_image_title_text[und][0][value]":false},"type":"media","field_deltas":{"1":{"format":"media_square","field_file_image_alt_text[und][0][value]":false,"field_file_image_title_text[und][0][value]":false}},"attributes":{"class":"media-element file-media-square","data-delta":"1"}}]]

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.

Graph products are a class of groups that 'interpolate' between direct and free products, and generalise the notion of right-angled Artin groups. Given a property that free products (and maybe direct products) are known to satisfy, a natural question arises: do graph products satisfy this property? For instance, it is known that graph products act on tree-like spaces (quasi-trees) in a nice way (acylindrically), just like free products. In the talk we will discuss a construction of such an action and, if time permits, its relation to solving systems of equations over graph products.

One occasionally encounters computational problems that work just fine on ill-posed inputs, even though they should not. One example is polynomial eigenvalue problems, where standard algorithms such as QZ can find a desired solution to instances with infinite condition number to machine precision, while being completely oblivious to the ill-conditioning of the problem. One explanation is that, intuitively, adversarial perturbations are extremely unlikely, and "for all practical purposes'' the problem might not be ill-conditioned at all. We analyse perturbations of singular polynomial eigenvalue problems and derive methods to bound the likelihood of adversarial perturbations for any given input in different stochastic models.

Joint work with Vanni Noferini
 

In this talk, we shall introduce various identities among partitions of integers, and how these can be expressed via formal power series. In particular, we shall look at the Rogers Ramanujan identities of power series, and discuss possible combinatorial proofs using partitions and Durfree squares.

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.

Recovery of reaction pathways under the DCSC framework.