Time Delay Embeddings

• Speaker: Thomas Whitely

Time-delay embeddings are a way to reconstruct the attractor of a dynamical system by using partial information of the system collected over time. This approach allows us to distinguish between deterministic and random (noisy) behaviour without requiring access to the full state of the system. However, in order to do this conversion in practice, we require a choice of parameters: lag and dimension. Traditionally, these parameters are chosen by first minimizing the mutual information between lags, then applying the false nearest neighbours to determine dimension. The limitation of this method is that mutual information requires choosing a grid size. There is currently no universal way to determine the best grid size given the data, and an incorrect choice of grid size can result in undesirable embeddings. We aim to resolve this issue by investigating a new method of choosing the parameters based on distance correlation.

Subgraphs of Random Graphs with Specified Degrees

• Speaker: John Larkin

Given a degree sequence, let G be a uniformly chosen simple random graph with that degree sequence. What is the probability G contains some edges and forbids others? We obtain improved bounds on this probability via the method of switchings and build on previous results, presenting theorems with weaker conditions and stronger conclusions. We investigate the same problem restricting to a uniformly chosen bipartite random graph with a given degree sequence.

Rigid Frameworks in 2 Dimensions

• Speaker: Luke Doherty

We will define and characterize rigid frameworks in 2 dimensions, giving the intuition behind Laman’s condition and discussing how one can produce a combinatorial property for a geometric problem. Our aim is to give the audience a basic understanding of rigidity as well as a the main results proved on the subject so far.  We will then move onto a special case of rigidity we call symmetric rigidity, defining this and giving necessary but not sufficient conditions for rigidity in this case.

Determinantal representations of singular cubic surfaces

• Speaker: Charlotte Sherratt

BRST Quantization of Gauge field Theories

• Speaker: Noah Holicky

It is an innately human question to wonder about our place in the universe and why things happen the way they do. The goal of physics is to find theories that answer such questions. Our best theory, to date, is quantum field theory for it has lead to the most accurate predictions of physical phenomena. However, it is riddled with issues, like the divergence of integrals and infinite dimensionality of measures, and hence not mathematically rigorous. A current goal in mathematical physics is thus to find a mathematically rigorous quantum field theory. For a certain class of quantum field theories, called gauge quantum field theories, there have been tremendous advancements in the rigour and solvability of them. In a revolutionary 1975 paper by Becchi, Rouet, Stora and Tyutin, the theory of BRST quantization was introduced. The purpose of this talk is to outline this construction, highlighting how it helps to resolve the issues of the path integral formalism and thus, make quantum gauge field theories more mathematically coherent.