AMATH Graduate Mini-Conference

CAR (chimeric antigen receptor) T cell therapy is a cancer treatment that utilizes immune cells known as T cells. In this treatment, a patient’s T cells are engineered with a receptor that allows them to target and destroy tumor cells; these T cells are then replicated for re-injection into the patient. While CAR-T therapy can be highly effective, individual patient responses vary significantly and are difficult to predict. As a result, mathematical modelling is a valuable tool in better understanding and predicting treatment responses. We present our extension of an ordinary differential equation (ODE) model of CAR-T cell therapy proposed by Kirouac et al. (2023), such that it describes the dynamics of two T cell subtypes: CD8+ cells (which directly kill cancer cells) and CD4+ cells (which serve more of an activating role). These T cells transition between different functional states (memory, active, and exhausted) during their interactions with tumor cells, and influence each other's dynamics directly and indirectly. This model can be used to investigate how the ratio of CD4+ to CD8+ T cells in CAR-T therapy influences treatment success. Through sensitivity analysis and simulations with virtual patients, we explore the effects of varying CD4+:CD8+ ratios. Our work provides an important step towards understanding how patient-specific immune differences impact CAR-T therapy outcomes, and how treatment can be optimized through patient stratification or personalized approaches.

In network communication, ensuring that messages remain confidential among multiple receivers is essential. In the case of quantum networks, classical-quantum and fully quantum broadcast channels provide important fundamental mathematical models for multi-receiver scenarios. Classical communication rates for several fully classical broadcast channel communication scenarios, like degraded messages (one receiver "stronger" than the other), simultaneous transmission of common and individual messages, and mutually confidential communication, are known. The quantum generalization of these results is not straightforward; for example, non-cloning and non-commutativity of operators in quantum theory impose strict constraints on simultaneous decoding schemes. In this talk, I will present an overview of known quantum generalizations of achievable rate regions for classical-quantum broadcast channels, along with the coding strategies and mathematical tools used to construct decoding POVMs. However, the rate region for a (classical-)quantum broadcast channel with mutually confidential messages is not known, even for the two-receiver case. We aim to determine upper and lower bounds for the secret key distillation rates, and the private capacity for the two-receiver quantum broadcast channel, using the simultaneous pinching method for "packing" messages and the obfuscation error as our criterion for secrecy. The next goal is to use the private codes to determine the quantum capacity region for such a channel.

In this accessible talk I will discuss how physicists use the idea of "symmetries" as a powerful tool for investigation. Whether its how particles scatter off each other, or how an apple falls from a tree, or even the equations we use to describe these phenomenon, symmetries underpin all of physics. I will in particular focus on the relationship between symmetries and "constraints", meaning how symmetries help us narrow down possible physical phenomena even in a complex or not well understood system.

Solving inverse problems with Physics-Informed Neural Networks (PINNs) is computationally expensive for multi-query scenarios, as each new set of observed data requires a new, expensive training procedure. We present Inverse-Parameter Basis PINNs (IP-Basis PINNs), a meta-learning framework that extends the foundational work of Desai et al. (2022) to enable rapid and efficient inference for inverse problems. Our method employs an offline-online decomposition: a deep network is first trained offline to produce a rich set of basis functions that span the solution space of a parametric differential equation. For each new inverse problem online, this network is frozen, and solutions and parameters are inferred by training only a lightweight linear output layer against observed data. Key innovations that make our approach effective for inverse problems include: (1) a novel online loss formulation for simultaneous solution reconstruction and parameter identification, (2) a significant reduction in computational overhead via forward-mode automatic differentiation for PDE loss evaluation, and (3) a non-trivial validation and early-stopping mechanism for robust offline training. We demonstrate the efficacy of IP-Basis PINNs on three diverse benchmarks, including an extension to universal PINNs for unknown functional terms—showing consistent performance across constant and functional parameter estimation, a significant speedup per query over standard PINNs, and robust operation with scarce and noisy data.

Switched dynamical systems arise when we are allowed to transition in time between different subsystems, typically modelled by autonomous ODEs. A fundamental example involves switched systems where all subsystems are linear. Studying their stability under arbitrary switching is important for practical applications. However, even for switched linear systems, there is no set of necessary and sufficient conditions to guarantee stability under arbitrary switching for arbitrary dimension of the phase space. I will present some new necessary and sufficient conditions for uniform asymptotic stability of the origin under arbitrary switching in two-dimensional switched linear systems.

Stabilization results for abstract linear and nonlinear partial differential equations have significant applications in the analysis and control of various mathematical models. In this talk, I will introduce the notion of stability for semigroups and their characterizations, together with tools such as LaSalle’s invariance principle and unique continuation results. I will then present our preliminary work on the linear stabilization problem in abstract spaces, demonstrating the robustness of exponential stability under perturbations, followed by an application to a uniformly parabolic partial differential equation.

Accurate and fast calibration and benchmarking of quantum gates is crucial for understanding and enhancing the performance of quantum hardware. We compare and contrast some of the most commonly used benchmarking protocols for individual quantum gate operations and show how the twirling group affects the systematic uncertainties on the estimates of process infidelities. We also show how local vs global fitting of fidelity decay curves are responsible for accurately capturing crosstalk errors present across parallel gate operations.

We introduce a new family of multi-mode, rotationally symmetric bosonic codes inspired by the group-theoretic framework of [Phys. Rev. Lett. 133, 240603 (2024)]. Such a construction inverts the traditional paradigm of code design by identifying codes from the requirement that a group of chosen logical gates should be implemented by means of physically simple logical operations, such as linear optics. Leveraging previously unexplored degrees of freedom within this framework, our construction preserves rotational symmetry across multiple modes, enabling linear-optics implementation of the full Pauli group. These codes exhibit improved protection against dephasing noise, outperforming both single-mode analogues and earlier multi-mode constructions. Notably, they allow exact correction of correlated dephasing and support qudit encoding in arbitrary dimensions. We analytically construct and numerically benchmark two-mode binomial codes instances, and demonstrate that, unlike single-mode rotationally symmetric bosonic codes, these exhibit no trade-off between protection against dephasing and photon loss.

Blockchains and decentralized finance have introduced new mathematical challenges spanning cryptography, statistics, optimization, and algorithm design. Despite their imperfections, these systems offer low-cost transactions and verifiable programs, drawing growing attention.

This talk focuses on the routing problem: finding the best price for an asset across all markets and intermediate forms. Traditional finance has only a few hundred equity markets, but crypto supports hundreds of thousands of venues competing for liquidity.

As a consequence, finding optimal prices in sub-second time is a difficult problem. In this talk, we will outline the history of this problem, survey some existing approaches, and show where they fall short.