University of Oxford

We will answer the following question: given a finite simplicial complex X acted on by a finite group G, which object stores the minimal amount of information about the symmetries of X in such a way that we can reconstruct both X and the group action? The natural first guess would be the quotient X/G, which remembers one representative from each orbit. However, it does not tell us the size of each orbit or how to glue together simplices to recover X. Our desired object is, in fact, a complex of groups. We will understand two processes: compression and reconstruction and see primarily through an example how to answer our initial question.

Every Cayley graph of a finitely generated group has some basic properties: they are locally finite, connected, and vertex-transitive. These are not sufficient conditions, there are some well known examples of graphs that have all these properties but are non-Cayley. These examples do however "look like" Cayley graphs, which leads to the natural question of if there exist any vertex-transitive graphs that are completely unlike any Cayley graph. I plan to give some of the history of this question, as well as the construction of the example that finally answered it.

The dynamics of many physical systems often evolve to asymptotic states that exhibit periodic spatial and temporal variations in their properties such as density, temperature, etc. Such regular patterns look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. In contrast, an aperiodic crystal displays long range spatial order but no translational symmetry. 

Recently, quasicrystals which are related to aperiodic crystals have been observed to form in diverse physical systems such as metallic alloys (atomic scale) and dendritic-, star-, and block co-polymers (molecular scale). Such quasicrystals lack the lattice symmetries of regular crystals, yet have discrete Fourier spectra. We look to understand the minimal mechanism which promotes the formation of such quasicrystalline structures using a phase field crystal model. Direct numerical simulations combined with weakly nonlinear analysis highlight the parameter values where the quasicrystals are the global minimum energy state and help determine the phase diagram. 

By locating parameter values where multiple patterned states possess the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a quasicrystal) is present in the background of another (for example, the homogeneous liquid state) in the form of spatially localized dodecagonal (in 2D) and icosahedral (in 3D) quasicrystals. In two dimensions, we compute several families of spatially localized quasicrystals with dodecagonal structure and investigate their properties as a function of the system parameters. The presence of such meta-stable localized quasicrystals is significant as they may affect the dynamics of the crystallisation in soft matter.

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.

The time bottleneck in the manufacturing process of Besi (company involved in ESGI 149 Innsbruck) is the extraction of undamaged dies from a component wafer. The easiest way for them to speed up this process is to reduce the number of 'selections' made by the robotic arm.  Each 'selection' made by this robotic arm can be thought of as choosing a 2x2 submatix of a large binary matrix, and editing the 1's in this submatrix to be 0's.  The quesiton is: what is the fewest number of 2x2 submatrices required to cover the full matrix, and how can we find this number. This problem can be solved exactly using integer programming methods, although this approach proves to be prohibitively expensive for realistic sizes. In this talk I will describe the approach taken by my team at EGSI 149, as well as directions for further improvement.

This talk is a brief introduction to the analysis of Donaldson theory, a branch of gauge theory. Roughly, this is an area of differential topology that aims to extract smooth structure invariants from the geometry of the space of solutions (moduli space) to a system of partial differential equations: the Yang-Mills equations.

I will start by discussing the differential geometric background required to talk about Yang-Mills connections. This will involve introducing the concepts of principal fibre bundles, connections and curvature. In the second half of the talk I will attempt to convey the flavour of the mathematics used to address technical issues in gauge theory. I plan to do this by presenting a sketch of the proof of Uhlenbeck's compactness theorem, the main technical tool involved in the compactification of the moduli space.