Week 10.03.2024 – 16.03.2024

Diverse equilibrium systems with heterogeneous interactions lie at the edge of stability. Such marginally stable states are dynamically selected as the most abundant ones or as those with the largest basins of attraction. On the other hand, systems with non-reciprocal (or asymmetric) interactions are inherently out of equilibrium, and exhibit a rich variety of steady states, including fixed points, limit cycles and chaotic trajectories. How are steady states dynamically selected away from equilibrium? We address this question in a simple neural network model, with a tunable level of non-reciprocity. Our study reveals different types of ordered phases and it shows how non-equilibrium steady states are selected in each phase. In the spin-glass region, the system exhibits marginally stable behaviour for reciprocal (or symmetric) interactions and it smoothly transitions to chaotic dynamics, as the non-reciprocity (or asymmetry) in the couplings increases. Such region, on the other hand, shrinks and eventually disappears when couplings become anti-symmetric.

I will present a simple but curious observation on the zeros of centered stationary Gaussian processes (SGP) on $\mathbb{R}$. The object of interest is
$N_T$ which is the number of zeros in the interval $[0,T]$. We restrict our attention to SGP with compactly supported spectral measure $\mu$. Let $A > 0$ be the smallest number such that supp$(\mu) \subseteq [-A, A]$.

Our primary interest is in the probability of overcrowding (resp. under crowding) event, which is the event that there is an excess (resp. deficit) of zeros in $[0,T]$ compared to the expected number, which is proportional to $T$. Comparing a couple of known results, we observe that there is a change in the behaviour of the probability $\p(N_T \geq \eta T)$, as $\eta$ varies. We show that there is indeed a \textit{sharp transition}. That is, this probability is at least of the order of $\exp(-C_{\eta}T)$ for small $\eta$, and at most of order $\exp(-c_{\eta}T^2)$ for large $\eta$. We identify the critical $\eta$ where this transition happens to be $\eta_c = A/\pi$. We also prove a similar result for under crowding probability when supp$(\mu)$ has a gap at the origin.

This talk is based on a joint work with Naomi Feldheim $\&$ Ohad Feldheim.

In this talk, I will explain how to apply the Fock-Goncharov construction to the representation theory of a class of algebras introduced by Etingof, Oblomkov and Rains.

Although STEM programs adequately equip students with the disciplinary knowledge required for the workplace, research suggests that STEM graduates have insufficient professional competencies. Further, employers expect STEM graduates to be able to link their areas of expertise to other disciplines (Sarkar et al., 2016) so that “a subject is not divided by watertight bulkheads from all others.

In this talk, I will reflect on the journey thus far towards fostering professional competencies in Mathematical Sciences.

A colouring rule is a way to colour the points x of a probability space according to the colours of finitely many measure preserving transformations
of x. The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none whose inverse images are measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We show that there is a paradoxical colouring rule when the rule is continuous and the measure preserving transformations generate a group.

Routinely collected healthcare data is becoming more commonly used for healthcare research. The increasing availability of such data promises advantages in the shape of largescale, representative data, but also brings many challenges which require statistical innovation. I will highlight some of these promises and challenges using four examples illustrating the use of routinely collected data, including modelling lung function trajectories of cystic fibrosis patients, dynamic prediction of cardiovascular disease, multi-state modelling of multimorbidity and predicting outcomes for intensive care patients.

Ngoc Khanh Nguyen:
Zero-knowledge proofs allow a party to convey that a given statement is true without leaking any secret information. These proofs form the foundations of many complex privacy-oriented protocols, such as electronic voting, verifiable computation, and blockchain. In this talk, we will discuss how the theory of cyclotomic fields helps with designing efficient zero-knowledge proofs from lattice-based assumptions.

Eamonn Postlethwaite:
In this talk I will briefly introduce the lattice isomorphism problem – LIP – and structured variants. I will describe approaches to solving generic LIP instances, and some recent progress in solving particular structured variants. For some brief cryptographic context, LIP has recently been used to build signature schemes. These are asymmetric cryptographic primitives that allow one party to authenticate data by appending a "signature", which takes the form of a short bitstring. Only an entity in possession of the secret key should be able to create valid signatures, but the validity of signatures can be checked publicly using the public key. Key recovery in recent proposals of such schemes from LIP – that is, computing the secret key from the public key – is exactly a LIP instance sampled from some distribution.