Oxford University

TQFTs lie at the intersection of maths and theoretical physics. Topologically, they are a recipe for calculating an invariant of manifolds by cutting them into elementary pieces; physically, they describe the evolution of the state of a particle. These two viewpoints allow physical intuition to be harnessed to shed light on topological problems, including understanding the topology of 4-manifolds and calculating geometric invariants using topology.

Recent results have provided classifications of certain types of TQFTs as algebraic structures. After reviewing the definition of TQFTs and giving some diagrammatic examples, I will give informal arguments as to how these classifications arise. Finally, I will show that in many cases these algebras are in fact free, and give an explicit classification of them in this case.
 

Morse theory explores the topology of a smooth manifold $M$ by looking at the local behaviour of a fixed smooth function $f : M \to \mathbb{R}$. In this talk, I will explain how we can construct ordinary homology by looking at the flow of $\nabla f$ on the manifold. The talk should serve as an introduction to Morse theory for those new to the subject. At the end, I will state a new(ish) proof of the functoriality of Morse homology.

Geometric Invariant Theory is a central tool in the construction of moduli spaces, and shares the property ubiquitous among such tools that certain so-called 'unstable' objects must be excluded if the moduli space is to be well behaved. However, instability in GIT is a structured phenomenon: after making a choice of a certain invariant inner product, one has the HKKN stratification of the parameter space which, morally, sorts the objects according to how unstable they are. I will explain how one can use recent results of Berczi-Doran-Hawes-Kirwan in Non-Reductive GIT to perform quotients of these unstable strata as well, extending the classifications given by classical moduli spaces. This can be carried out, at least in principle, for any moduli problem that can be posed using GIT, and I will discuss two examples in particular: unstable (i.e. singular) curves, and coherent sheaves of fixed Harder-Narasimhan type. The latter of these is joint work with Gergely Berczi, Victoria Hoskins and Frances Kirwan.
 

(Joint work with Coralia Cartis) The problem of finding the most extreme value of a function, also known as global optimization, is a challenging task. The difficulty is associated with the exponential increase in the computational time for a linear increase in the dimension. This is known as the ``curse of dimensionality''. In this talk, we demonstrate that such challenges can be overcome for functions with low effective dimensionality --- functions which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations.
We propose the use of random subspace embeddings within a(ny) global minimisation algorithm, extending the approach in Wang et al. (2013). We introduce a new framework, called REGO (Random Embeddings for GO), which transforms the high-dimensional optimization problem into a low-dimensional one. In REGO, a new low-dimensional problem is formulated with bound constraints in the reduced space and solved with any GO solver. Using random matrix theory, we provide probabilistic bounds for the success of REGO, which indicate that this is dependent upon the dimension of the embedded subspace and the intrinsic dimension of the function, but independent of the ambient dimension. Numerical results demonstrate that high success rates can be achieved with only one embedding and that rates are for the most part invariant with respect to the ambient dimension of the problem.
 

This talk will review the main Tikhonov and Bayesian smoothing formulations of inverse problems for dynamical systems with partially observed variables and parameters. The main contenders: strong-constraint, weak-constraint and penalty function formulations will be described. The relationship between these formulations and associated optimisation problems will be revealed.  To close we will indicate techniques for maintaining sparsity and for quantifying uncertainty.

In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. Holes die when a subset of their concepts appear in the same article, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the dimension of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We also show that authors' conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.

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Single parameter persistent homology has proven to be a useful data analytic tool and single parameter persistence modules enjoy a concise description as a barcode, a complete invariant. [Bubenik, 2012] derived a topological summary closely related to the barcode called the persistence landscape which is amenable to statistical analysis and machine learning techniques.

The theory of multidimensional persistence modules is presented in [Carlsson and Zomorodian, 2009] and unlike the single parameter case where one may associate a barcode to a module, there is not an analogous complete discrete invariant in the multiparameter setting. We propose an incomplete invariant derived from the rank invariant associated to a multiparameter persistence module, which generalises the single parameter persistence landscape in [Bubenik, 2012] and satisfies similar stability properties with respect to the interleaving distance. Our invariant naturally lies in a Banach Space and so is naturally endowed with a distance function, it is also well suited to statistical analysis since there is a uniquely defined mean associated to multiple landscapes. We shall present computational examples in the 2-parameter case using the RIVET software presented in [Lesnick and Wright, 2015].