This year’s Trinity Mathematical Society Symposium is running from 11:00 to 18:00 on Sunday 26nd February. We have talks by fellows and PhD students, ranging across all areas of mathematical research. The event is free and open to all; no particular specialist knowledge is assumed. There is no need to stay for the whole day – just drop in on talks you find interesting. Click on the speakers/titles in order to view abstracts.
The program is:
Symplectic geometry provides the framework for classical mechanics, but it is also a rapidly evolving major area of modern pure mathematics. I will give a brief introduction to symplectic structures, and then discuss a fundamental property of symplectomorphisms, maps which preserve the symplectic structure. Gromov’s non-squeezing theorem is one of the first major results about such maps, answering the question why a symplectic camel cannot pass through the eye of a needle: Its ribcage cannot be symplectically compressed!
Finally, I will touch on quantum mechanics and how the non-squeezing property manifests itself there in the form of the Heisenberg uncertainty relation.
In this talk, we will go on a journey through the realm of algebraic number theory. We will talk a little bit about Galois theory before moving on to class field theory (CFT).
The goal of class field theory is to describe all abelian extensions of a given number field. We will first do this for the field of rational numbers, using roots of unity.
Then we will give an introduction to elliptic curves (EC) and the theory of complex multiplication (CM). This allows us in the end to describe all abelian extensions of a given imaginary quadratic number field (IQNF).
BRUNCH
The Devil Facial Tumour Disease (DFTD), a unique case of a transmissible cancer, had a devastating effect on the Tasmanian Devils, leading to an overall population decline of over 80%. A number of single-population epidemiological models have predicted the likely extinction of the Tasmanian Devils. However, despite extensive surveys across Tasmania providing data on the spatial and temporal spread of DFTD, the metapopulation dynamics of this disease has yet to be modelled. Here we fit a stochastic spatial metapopulation model of the origin and spread of the DFTD to empirical observations of local disease trajectories and time-stamped observations of diseased animals across Tasmania, using the Approximate Bayesian Computation statistical method to account for the randomness of the population dynamics and spread of the disease. We confirm a most likely origin of the disease in the north-east corner of the island, and highlight the importance of interpopulation contacts in the fast spread of the tumour. We then use the inferred metapopulation dynamics to predict the fate of this host-pathogen system. Surprisingly, we find that the devils are predicted to coexist with the tumour, in contrast with predictions from single population models. The key process allowing long-term persistence of the species is the repeated reinvasion of extinct patches from neighbouring areas where the disease has flared up and died out, resulting in a dynamic equilibrium with different levels of spatial heterogeneity. However, this dynamic equilibrium is predicted to keep the population of this apex predator at roughy 10% of its original density, with possible dramatic effects on the Tasmanian ecosystem.
How well can you imagine high-dimensional objects? What can you say about the behaviour of convex bodies in n-dimensional space as n grows to infinity? During the talk I will try to convince you that our intuition based on low dimensional geometry can be far off and that unexpected phenomena occur. For example, we will see that the mass of a high dimensional ball is concentrated on a thin band around any equator. This result plays an important role in many theorems in asymptotic geometric analysis, such as Dvoretzky’s theorem, which suggests that all convex bodies admit “almost ellipsoidal” sections. We will make this statement precise.
A multitude of important real-world datasets come together with some form of graph structure: social networks, citation networks, protein-protein interactions, brain connectome data, etc. Extending neural networks to be able to properly deal with this kind of data is, therefore, a very important direction for machine learning research, but one that has received comparatively rather low levels of attention until very recently.
Unlike the structure that one might typically find in an image or text, the graph structure is often highly irregular, and producing a general operator (such as a convolution for images or a recurrent cell for text) that satisfies all the criteria we would like for an operator like this is known to be very difficult. Several approaches that have been recently proposed (a lot of them only in 2017!) attempt to handle the problem from a variety of perspectives: structural, stochastic and/or spectral. In this talk I will give a comprehensive overview of these approaches, outlining their relative strengths and limitations, and exposing some cool application areas where they’ve been applied.
Special emphasis will be given to the work I’ve done in collaboration with Adriana Romero and Yoshua Bengio (Montréal Institute for Learning Algorithms), on Graph Attention Networks (GATs: https://arxiv.org/abs/1710.10903), which might offer the first neural network layer to satisfy all of the desirable criteria simultaneously.
TEA/BREAK
Non-linear (Ordinary) Differential Equations (ODEs) are typically impossible to solve analytically – instead one has to refer to one of the many available *approximate* methods (accompanied by an enormous amount of research on the characterisation of their convergence properties) to numerically represent the ODE and develop an understanding of its behavior.
A similar scenario arises in the context of *Stochastic* Differential Equations (SDEs). However, it appears that the presence of randomness allows for much more flexibility compared to the deterministic case. We describe a recent, surprisingly simple algorithm, that solves non-linear SDEs without any approximation error. We stress that exact solution of SDEs is a notorious problem within the applied probability community, and the talk will present a direction toward the solution of this problem
Newton’s differential calculus provides a framework for studying smooth transformations of smooth functions. Many quantities of important in physics, engineering and economics follow an irregular time evolution, leading to irregular and non-differentiable signals whose (ir)regularity may be described by the notion of p-th order variation along a refining sequence of time partitions. We will show that smooth transformations of such irregular functions obey a higher-order version of Newton’s calculus. In the case p=2, one retrieves the Ito formula from stochastic calculus, but without any probabilistic assumptions.
19:45 – Annual Dinner
