Week 17.11.2024 – 23.11.2024

Active matter systems are attracting great interest in physics, as non-equilibrium systems with rich emergent behaviour. Some of their features can be captured by multi-species exclusion processes, whose hydrodynamic limits can be derived exactly. I will discuss the behaviour of several such models, including their hydrodynamic limits [1,2], and large (rare) fluctuations [3].

[1] J. Mason, C. Erignoux, R. L. Jack, and M. Bruna, Proc. Roy. Soc. A 479, 2279 (2023).
[2] J. Mason, R. L. Jack, and M. Bruna, arXiv:2408.03932.
[3] T. Agranov, M. E. Cates and R. L. Jack, J. Stat. Mech. (2022), 123201.

Scattering amplitudes provide crucial theoretical input in collider and gravitational wave physics, and at the same time exhibit a remarkable mathematical structure. These lectures will introduce essential concepts and modern techniques exploiting this structure so as to efficiently compute amplitudes and their building blocks, Feynman integrals, in perturbation theory. We will start by decomposing gauge theory amplitudes into simpler pieces based on colour and helicity information. Focusing on tree level, we will then show how these may be determined from their analytic properties with the help of Britto-Cachazo-Feng-Witten recursion. Moving on to loop level, we will define the the class of polylogarithmic functions amplitudes and integrals often evaluate to, and explain their properties as well as relate them to the universal framework for predicting their singularities, known as the Landau equations. Time permitting, we will also summarise the state of the art in the calculation of the aforementioned singularities, and their intriguing relation to mathematical objects known as cluster algebras.

Ultracold atoms have emerged as powerful and versatile platforms for simulating quantum many-body systems. Yet, there remain several obstacles preventing them from unleashing their full potential. One major bottleneck is the limited available methods for reading out information from the simulators. In this talk, I will focus on a pair of low-dimensional (1D and 2D) quantum gases, whereby spatial correlation of the phase fields can be extracted from matter-wave interference. I will start by examining the reliability of the standard method to extract relative phase profiles from density interference patterns [1]. Then, I will present a new method to extract the total phase, i.e. the sum rather than the difference of phase fields, from density ripples after time of flight [2]. Lastly, I will discuss applications of this new measurement towards pushing quantum simulators to new regimes previously unexplored, such as probing anharmonic correction to Luttinger liquid and non-equilibrium dynamics of sine-Gordon field theory.

[1] Murtadho, T., Gluza, M., Arifa, K. Z., Erne, S., Schmiedmayer, J., & Ng, N.H. (2024). Systematic analysis of relative phase extraction in one-dimensional Bose gases interferometry. https://arxiv.org/abs/2403.05528v2
[2] Murtadho, T., Cataldini, F., Erne, S., Gluza, M., Schmiedmayer, J., & Ng, N. H. (2024). Measurement of total phase fluctuation in cold-atomic quantum simulator. https://arxiv.org/abs/2408.03736

Scattering amplitudes provide crucial theoretical input in collider and gravitational wave physics, and at the same time exhibit a remarkable mathematical structure. These lectures will introduce essential concepts and modern techniques exploiting this structure so as to efficiently compute amplitudes and their building blocks, Feynman integrals, in perturbation theory. We will start by decomposing gauge theory amplitudes into simpler pieces based on colour and helicity information. Focusing on tree level, we will then show how these may be determined from their analytic properties with the help of Britto-Cachazo-Feng-Witten recursion. Moving on to loop level, we will define the the class of polylogarithmic functions amplitudes and integrals often evaluate to, and explain their properties as well as relate them to the universal framework for predicting their singularities, known as the Landau equations. Time permitting, we will also summarise the state of the art in the calculation of the aforementioned singularities, and their intriguing relation to mathematical objects known as cluster algebras.

Biological neural network models of associative memory (such as the seminal Hopfield Network, which earned
its creator the Physics Nobel prize last month) suffer from an artificial separation of storage and recall phases: the network's natural dynamics, which enable pattern completion at recall, are suspended during the storage phase.
I will present a cortical circuit mechanism which was previously implicated in a range of nonlinear computations in
sensory cortex, and show how it can solve the above problem. The mechanism involves a network of supralinear neurons, with strong recurrent excitatory connections,
which is stabilized against runaway excitation by recurrent inhibition. We show that such networks can naturally cross over from a regime of
memory storage to a regime of recall by simply scaling the external inputs. Finally, we propose that these regimes could correspond to
different phases of theta oscillations in the hippocampus, which have been previously linked to memory recall and storage.

STACK is a KEATS quiz question type which uses a computer algebra system (maxima) to support the creation of online quiz questions. It allows for randomised questions to be written with dynamic feedback which adapts to the students answer. It also allows students to input mathematical answer such as polynomials or other functions, matrices, of even complete calculations and check that they are correct.

In this talk, I will introduce STACK with examples of the types of questions which can be written using it. I will discuss the types of mathematical skills STACK can assess, including proof, (and what it can’t). I will then talk about how STACK is already being used in the department, and how the funding will be used to develop this further.

Join using this link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZWY2M2M4ODMtNjJmYi00NGZjLWE2M2MtMDMzMzk3MDBjZjY0%40thread.v2/0?context=%7b%22Tid%22%3a%228370cf14-16f3-4c16-b83c-724071654356%22%2c%22Oid%22%3a%22babbe2bf-c194-4f10-9c9c-d05f13435b1c%22%7d

The spectrum of elliptic operators on a smooth domain encodes geometric information. One way of seeing this effect is via the asymptotics of the counting function, whose form is known as the Weyl law. Less is known about what happens in the case of geometric singularities, but it is an area of active research. This talk will cover the Weyl law of the Dirichlet to Nuemann map, both historical results for smooth domains, and recent results for piecewise smooth domains.

There has been a surge of interest in incorporating neural networks into particle filters, e.g. differentiable particle filters, to perform joint sequential state estimation and model learning for non-linear non-Gaussian state-space models in complex environments. Existing approaches primarily use vanilla neural networks which do not allow density estimation. As a result, they are often restricted to a bootstrap particle filtering framework or employ predefined distribution families (e.g. Gaussian distributions), limiting their performance in more complex real-world scenarios. In this talk I will introduce a differentiable particle filtering framework that uses conditional normalising flows to build its dynamic model, proposal distribution, and measurement model. This approach not only enables valid probability densities but also allows the proposed method to adaptively learn these modules in a flexible way, without being restricted to predefined distribution families. I will discuss the theoretical properties of the proposed filters and present their performance through a series of numerical experiments.

TBA