In this talk I will introduce the concept of the path signature and motivate its recent use in analysis of time-ordered data. I will then describe a new feature map for barcodes in persistent homology by first realizing each barcode as a path in a vector space, and then computing its signature which takes values in the tensor algebra over that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness.

In this paper we define a new object, the momentum amplituhedron, which is the long sought-after positive geometry for tree-level scattering amplitudes in N=4 super Yang-Mills theory in spinor helicity space. Inspired by the construction of the ordinary amplituhedron, we introduce bosonized spinor helicity variables to represent our external kinematical data, and restrict them to a particular positive region. The momentum amplituhedron Mn,k is then the image of the positive Grassmannian via a map determined by such kinematics. The scattering amplitudes are extracted from the canonical form with logarithmic singularities on the boundaries of this geometry.

Data analysis techniques are often highly domain specific - there are often certain patterns which should be in certain types of data but may not be apparent in data. The first part of the talk will cover a technique for finding such patterns through a tool which combines visual analytics and machine learning to provide insight into temporal multivariate data. The second half of the talk will discuss recent work on imposing high level geometric  structure into continuous optimizations including deep neural networks.
 

In existing techniques for model-based derivative-free optimisation, the computational cost of constructing local models and Lagrange polynomials can be high. As a result, these algorithms are not as suitable for large-scale problems as derivative-based methods. In this talk, I will introduce a derivative-free method based on exploration of random subspaces, suitable for nonlinear least-squares problems. This method has a substantially reduced computational cost (in terms of linear algebra), while still making progress using few objective evaluations.

Multiple scales analysis is a powerful asymptotic technique for problems where the solution depends on two scales of widely different sizes. Standard multiple scales involves the introduction of a macroscale and microscale which are assumed to be independent. A common (and usually acceptable) assumption is that when considering behaviour on the microscale, the macroscale variable can be taken as constant, however there are instances where this assumption is not valid. In this talk, I will explain one such situation, that is, when considering conductive-radiative thermal transfer within a solid matrix with spherical perforations and discuss the appropriate measures when converting the radiative boundary condition into multiple-scales form.
 

Hilbert's fifth problem asks informally what is the difference between Lie groups and topological groups. In 1950s this problem was solved by Andrew Gleason, Deane Montgomery, Leo Zippin and Hidehiko Yamabe concluding that every locally compact topological group is "essentially" a Lie group. In this talk we will show the complete proof of this theorem.