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.

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.

I will discuss joint work with Gismatullin, Halupczok, and Simonetta on the following problem: given a henselian valued field of characteristic 0, possibly equipped with analytic structure (in the sense stemming originally from Denef and van den Dries), describe the possibilities for a definable group G in the valued field sort which is definably almost simple, that is, has no proper infinite definable normal subgroups. We also have results for an algebraically closed valued field K in characteristic p, but assuming also that the group is a definable subgroup of GL(n, K).

I will explain how complex varieties which have asymptotically large intersections with finite grids can be seen to correspond to projective geometries, exploiting ideas of Hrushovski. I will describe how this leads to a precise characterisation of such varieties. Time permitting, I will discuss consequences for generalised sum-product estimates and connections to diophantine problems. This is joint work with Emmanuel Breuillard.