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The Post Correspondence Problem (PCP) is a classical problem in computer science that can be stated as: is it decidable whether given two morphisms g and h between two free semigroups $A$ and $B$, there is any nontrivial $x$ in $A$ such that $g(x)=h(x)$? This question can be phrased in terms of equalisers, asked in the context of free groups, and expanded: if the `equaliser' of $g$ and $h$ is defined to be the subgroup consisting of all $x$ where $g(x)=h(x)$, it is natural to wonder not only whether the equaliser is trivial, but what its rank or basis might be. 

While the PCP for semigroups is famously insoluble and acts as a source of undecidability in many areas of computer science, the PCP for free groups is open, as are the related questions about rank, basis, or further generalisations. In this talk I will give an overview of what is known about the PCP in hyperbolic groups, nilpotent groups and beyond (joint work with Alex Levine and Alan Logan).

One of the more common ways to study a residually finite group (or its profinite completion) is via breaking it down into a graph of groups in some way. The descriptions of this theory generally found in the literature are highly algebraic and difficult to digest. I will present alternative, more geometric, definitions and perspectives on these theories based on properties of virtually free groups and their profinite completions.

The celebrated Kirchberg-Phillips classification theorem classifies so-called Kirchberg algebras by K-theory. Many examples of Kirchberg algebras can be constructed via the crossed product construction starting from a group action on a compact space. One might ask: When exactly does the crossed product construction produce a Kirchberg algebra? In joint work with Gardella, Geffen, and Naryshkin, we obtained a dynamical answer to this question for a large class of nonamenable groups which we call "groups with paradoxical towers". Our class includes many non-positively curved groups such as acylindrically hyperbolic groups and lattices in Lie groups. I will try to advertise our notion of paradoxical towers, outline how we use it, and pose some open questions.

In a recent work with Sam Mellick and Amanda Wilkens, we proved that higher rank semisimple Lie groups satisfy a generalization of Gaboriau fixed price property (originally defined for countable groups) to the setting of locally compact second countable groups. As one of the corollaries, under mild conditions, we can prove that the rank (minimal number of generators) or the first mod-p Betti number of a higher rank lattice grow sublinearly in the covolume.  The proof relies on surprising geometric properties of Poisson-Voronoi tessellations in higher-rank symmetric spaces, which could be of independent interest. 

We shall consider a crossing equation of the Euclidean conformal group in terms of conformal partial waves and in particular, a position independent representation of this equation. We shall briefly discuss the relevance of this equation to the problem of cosmological bootstrap. Thereafter, we shall sketch the derivation of the Biedenharn-Eliiot identity (a pentagon identity) for the 6j symbols of the conformal group and show how this provides us with an exact solution to said crossing equation. For the conformal group (which is non-compact), this involves some careful bookkeeping of the spinning representations. Finally, we shall discuss some consistency checks on the result obtained, and some open questions. 

Recently, a relation was introduced connecting codes of various types with the space of abelian (Narain) 2d CFTs. We extend this relation to provide holographic description of code CFTs in terms of abelian Chern-Simons theory in the bulk. For codes over the alphabet Z_p corresponding bulk theory is, schematically, U(1)_p times U(1)_{-p} where p stands for the level. Furthermore, CFT partition function averaged over all code theories for the codes of a given type is holographically given by the Chern-Simons partition function summed over all possible 3d geometries. This provides an explicit and controllable example of holographic correspondence where a finite ensemble of CFTs is dual to "topological/CS gravity" in the bulk. The parameter p controls the size of the ensemble and "how topological" the bulk theory is. Say, for p=1 any given Narain CFT is described holographically in terms of U(1)_1^n times U(1)_{-1}^n Chern-Simons, which does not distinguish between different 3d geometries (and hence can be evaluated on any of them). When p approaches infinity, the ensemble of code theories covers the whole Narain moduli space with the bulk theory becoming "U(1)-gravity" proposed by Maloney-Witten and Afkhami-Jeddi et al.

In recent years, methods and concepts of algebraic geometry, particularly those of real and computational algebraic geometry, have been used in many applied domains. In this talk, aimed at a broad audience, I will review applications to molecular biology. The goal is to analyze standard models in systems biology to predict dynamic behavior in regions of parameter space without the need for simulations. I will also mention some challenges in the field of real algebraic geometry that arise from these applications.

Predictive scoring modelling is a common approach to measure financial health and credit worthiness in banking. Whilst the latter is a key factor in making decisions for lending, evaluating financial health helps to identify vulnerable customers trending towards financial hardship, who need support. The current macroeconomic uncertainties amplify the importance of extensive flexibility in modelling data solutions so that they can remain effective and adaptive to a volatile economic environment. This workshop is focused on discussing relevant techniques and mathematical methodologies which can help modernise traditional scoring models and accelerate innovation. In summary, the problem definition in the banking context is how automation and optimality can be achieved in a multiobjective problem where a subset of existing data features should be selected by relevance and uniqueness, assigned scoring weights by importance and how a pool of customers can be categorised accordingly using their individual scores and auto-adjusting thresholds of risk classification scales. The key challenge is imposed by the mutual dependency of the three sub-problems and their objectives. Introducing or removing constraints in any of them can change the feasibility and optimality of the others and the overall solution. It is common for traditional scoring models to be mainly focused on the predictive accuracy and their setup is often defined and revised manually, following ad-hoc exploratory data analysis and business-led decision making. An automated optimisation of the data features’ selection, scoring weights and classification thresholds definition can achieve respectively: ▪ Precise financial health evaluation and book classification under changing economic climate; ▪ Development of innovative data-driven solutions to enhance prevention from financial hardship and bankruptcies.

I will discuss how to place conformal boundary conditions on free higher derivative theories of scalars and spinors as well as the zoo of boundary renormalization group flows that connect the different boundary conditions.  Historically, there are connections to Poisson’s ratio and classical equations governing the bending of thin steel plates.  As these higher derivative theories are often invoked in the context of cosmology, there may be cosmological applications for the boundary conditions discussed here.