Mathematical Physics

Abstract: A maximum entropy method is described for estimating densities from quantum Monte Carlo simulations in high-energy physics. Standard approaches are prone to numerical overflow, limiting their reliability. We describe a dual self-scaling algorithm that is robust and efficient.

Abstract: Given a data distribution which is concentrated around a one-dimensional structure, can we infer that structure? We consider versions of this problem where the distribution resides in a metric space and the 1d structure is assumed to either be the range of an absolutely continuous curve, a connected set of finite 1d Hausdorff measure, or a general 1-rectifiable set. In each of these cases, we relate the inference task to solving a variational problem where there is a tradeoff between data fidelity and simplicity of the inferred structure; the variational problems we consider are closely related to the so-called "principal curve" problem of Hastie and Steutzle as well as the "average-distance problem" of Buttazzo, Oudet, and Stepanov. For each of the variational problems under consideration, we establish existence of minimizers, stability with respect to the data distribution, and consistency of a discretization scheme which is amenable to Lloyd-type numerical methods. Lastly, we consider applications to estimation of stochastic processes from partial observation, as well as the lineage tracing problem from mathematical biology.

Abstract: Sampling from complex probability distributions is a fundamental challenge in statistics and machine learning. When a target distribution is too complicated to sample from directly, annealing algorithms provide a solution: gradually transition from a simple, tractable distribution to the complex target through a sequence of intermediate annealing steps. This talk explores two complementary approaches to this problem: Sequential Monte Carlo and Parallel Tempering. We show how their performance depends on three key factors: the number of samples, the number of annealing steps, and the geometry of the probability distributions. Our main finding reveals a phase transition that determines when these algorithms work well and when they fail.

Abstract: Not all of us will stay in the academia. How do we prepare ourselves for the shift, especially when a lot of entry level jobs have been replaced by AI? This is a discussion of life beyond academia, and how to stay relevant as a math PhD in this competitive job market.

Abstract: Wastewater surveillance resurged during the COVID-19 pandemic as a passive, low-cost, non-invasive method for monitoring community-level disease transmission with the potential to overcome biases inherent in other surveillance methods, such as those driven by healthcare-seeking behaviour. Past efforts have primarily focused on correlating wastewater pathogen levels with clinical indicators, demonstrating its ability to forecast reported cases, Emergency Department (ED) visits, and hospitalizations. Building on this foundation, we propose TARnISHED-WW (Time-series Analysis of Random Walkers for Infections Surveillance and Hospital ED visits), a novel framework that combines multiple pathogen wastewater viral load and clinical data to model overall ED visits latent multivariate Gaussian random walks that capture shared infection dynamics across regions. This framework incorporates signals from three major respiratory viruses: Influenza A, SARS-CoV-2 and Respiratory Syncytial Virus (RSV). Due to TARnISHED-WW's hierarchical Bayesian architecture, we can infer pathogen-specific contributions to ED visits, yielding an important indicator to support hospital and emergency services planning and public health surveillance. Our work demonstrates the potential of wastewater surveillance combined with advanced quantitative modelling to provide robust public health indicators and support health care preparedness and planning. 

Abstract: In this talk, we present a framework for extracting "needles in the haystack" via agentic research workflows: decomposing a question into subqueries, routing them through multiple data collections, executing semantic search over vector embeddings, and using reflection loops to detect gaps and drive conditional repeat execution. We discuss the architecture of this system, including query routing, similarity search over chunks, reflection-based query expansion, and final synthesis. Finally, we show how the approach systematically uncovers deep patterns and hidden insights from large unstructured corpora, turning sparse signals into precise, cohesive reports.

Abstract: Quantifying the distribution of animals and understanding their movement behaviour is fundamental to their conservation. As such, ecologists increasingly collect movement data. However, characterising the distribution and behaviour of marine species is hindered by many formidable challenges. For example, marine species spend most of their life in areas difficult for us to reach (e.g., ocean depths) and many positioning systems (e.g., GPS) are not well suited to the marine environment. Using species such as Arctic terns and narwhals, I will demonstrate how advanced statistical methods can improve our understanding of their ecology and inform management and conservation.

Abstract: In the early stages of an epidemic (such as COVID-19), the number of newly reported cases typically exhibits exponential growth. Due to changes in policies or human behavior, the growth rate may shift over time. Detecting these changes in the growth rate is crucial for effective infectious disease control.

We developed an Hidden Markov Model based algorithm to monitor potential change points in the growth rate and to estimate the probability of such changes. Through both simulation studies and real data applications (using COVID-19 data from British Columbia), our results show that the proposed model can effectively detect potential change points. However, in some cases, the detection may exhibit a delay, which is an issue we plan to investigate further in future work.

Abstract: The Gottesman-Knill theorem tells us that qubit initialization in the computational basis, Clifford operations, and Pauli measurements are classically simulable in polynomial time. So where do quantum computer get their `Quantum Advantage' from? Turns out a possible answer lies in `Magic States'. My talk will introduce these magic states and also attempt to briefly go over how to prepare and do acutal computations with these magic states.

Abstract: The Flag variety is a basic object of geometry and linear algebra. It is a space that parameterizes flag: sequences of nested linear spaces. In the process of studying the topology of the space of polynomial curves on this space, we find a surprising connection to linear control systems. This led us to formulate conjectures which were proved with assistance from Google Deep Mind. 

Abstract: The application of machine learning to study physical systems is an extremely active area of research. We are specifically interested in data-driven approximations of Hamiltonians and study schemes that preserve the qualitative behaviour of the physical system, that is they need at the very least to preserve the conserved quantities. We know from Noether’s theorem that the conserved quantities are equivalent to the presence of continuous symmetries, for instance conservation of angular momentum arises from rotational invariance. Standard data driven models only lead to approximate symmetries, and in general, approximate symmetries will not lead to approximate conservation of conserved quantities. This motivates the need to understand the symmetrisation error in terms of the data provided.

We then present and study 2 different ways of augmenting data in order to improve the symmetry properties of an approximation scheme. These are augmenting the data through a quadrature rule for the Haar measure on the compact Lie group whose representation is the continuous symmetry of interest and a Monte-Carlo sampling of that Haar measure. Intuitively, these 2 methods can be understood as comparing an optimal way of augmenting
and random augmentations. We show that the quadrature augmentation leads to exact symmetry preservation while the MC augmentation has a standard square root decay at best.

Abstract: Take two points and pick a path between them `uniformly at random'. Of course there are infinitely many paths, and the question does not really literally make sense. However, there is a way to make sense of it: Brownian motion. Although this theory is rather classical in 1d, in two dimensions or more, things are much less understood. I will give a gentle overview of the type of problems we tackle in this area, and maybe even some theorems.

Abstract: Understanding what lies beneath the Earth’s surface is essential for addressing challenges such as locating critical minerals, managing groundwater, monitoring CO₂ storage, and tracking permafrost change. Much like how medical imaging techniques such as MRI or X-ray reveal internal structure non-invasively, geophysical surveys measure signals—electric, magnetic, seismic, or gravitational—that are influenced by the physical properties of the subsurface.

Recovering those properties from surface data leads to a PDE-constrained inverse problem, where we seek models of the Earth that reproduce the observed data while incorporating prior knowledge from geology, petrophysics, and complementary geophysical methods. In this talk, I will introduce the mathematical formulation of the geophysical inverse problem and discuss the strategies we use to tackle its inherent non-uniqueness and ill-posedness. I will also highlight current research directions, including how machine learning can be integrated with physics-based approaches, and show examples from applications in mineral exploration and environmental monitoring.

Abstract: Accurate, robust and fast image reconstruction is a critical task in many scientific, industrial and medical applications. Deep learning is currently causing a revolution in modern image reconstruction, leading to powerful new techniques with seemingly breakthrough performance. In this talk, I will first briefly survey several different deep learning methodologies. Then I will highlight some of their pitfalls, in particular, their tendency to hallucinate – a serious problem for safety-critical applications such as healthcare. Finally, I will end with insights into how mathematics can help guarantee the robustness of deep learning-based methods.

Abstract: Turbulent fluid flow is characterized by chaotic swirling motions across various length scales. Current state-of-the-art (SOTA) approaches utilize computational fluid dynamics (CFD) solvers. Direct Numerical Simulations (DNS) explicitly resolve all length scales of a turbulent flow, which yields accurate but computationally expensive results. Other methods (such as Reynolds averaging and Large Eddy Simulations) use a coarser grid to decrease expense, but require closure equations, which often have yet to be derived from first principles and remain poorly understood. Further, closure equations are semi-empirical and require parameter fitting to match experiments, which limits the efficiency, accuracy, and predictive capability of CFD results. The limitations of current CFD simulations often lead to the need for physical prototyping in industrial design and optimization. We employ mathematical transformations of the Navier-Stokes equations to a Hilbert space to allow for analytical solutions to our eigenvalue problem. This makes it easier to numerically solve the Navier-Stokes equations with faster computation through the use of Runge-Kutta rather than finite element methods, and increased interpretability due to the strong physical connections between the mathematics and physics.

Abstract: The presence and classification of generic stagnation points in fluid flows are essential to understanding the behaviour of various phenomena such as mixing, transition to turbulence, stability and modelling of a variety of physical processes in fluid dynamics. Stagnation points can be approximated, to first approximation, as being either elliptical or hyperbolic. Understanding the dynamics of perturbations around these stagnation points yields insight into the stability of the flow. In this talk, I will overview work done in this direction for the case of hyperbolic stagnation points and present some recent findings regarding the evolution of perturbations around a hyperbolic stagnation point in a fluid flow experiencing global rotation. We show the existence of a class of initial conditions leading to transient, bounded and unbounded energy growth. For the case of unbounded growth, this shows that there exist hyperbolic instabilities in fluid flows undergoing strong rotation.