Applied Mathematics Seminar

Abstract: Deep learning has achieved remarkable success in computer vision, natural language processing, and scientific data analysis, primarily due to its ability to extract hierarchical representations from data. At the heart of training deep neural networks lies gradient descent, whose effectiveness depends crucially on how model parameters are initialized. Classical initialization strategies such as Xavier, He, and orthogonal initialization aim to preserve variance or approximate isometry, and they have enabled significant progress in stabilizing training. However, as network depth increases, these schemes often fail to prevent neuron inactivation ("dying ReLU") and suffer from instability of activations and gradients.

In this talk, we will first introduce the key ideas behind deep learning and gradient descent, then provide an overview of standard initialization methods and their limitations. I will then present recent joint work on optimized weight initialization on the Stiefel manifold for deep ReLU networks. By formulating an optimization problem on the Stiefel manifold, we derive an orthogonal initialization that not only preserves scale but also calibrates pre-activation statistics at the outset. A family of closed-form solutions and an efficient sampling scheme are established. Theoretical analysis demonstrates prevention of the dying ReLU problem, slower variance decay, and mitigation of gradient vanishing, ensuring more stable signal propagation. Empirical studies on image benchmarks, tabular data, and few-shot settings show that the proposed method consistently outperforms existing initializations and enables reliable training in very deep architectures.

Abstract: Glioblastoma multiforme (GBM) is the most aggressive form of brain cancer with the very poor survival and high recurrence rate. Tumor-associated neutrophils (TANs) play a pivotal role in regulation of the tumor microenvironment. In this study, we developed a new multi-scale model of the critical GBM-TAN interaction in the heterogeneous brain tissue. The model reveals that the dual and complex role of TANs (either anti-tumorigenic N1 and the pro-tumorigenic N2 type) regulates the phenotypic trajectory of the evolution of tumor growth and the invasive patterns in white and gray matter via mediators such as IFN-beta and TGF-beta. We investigated the effect of normalizing the immune environment on glioma growth by applying a therapeutic antibody and developed several strategies for eradication of tumor cells by neutrophil-mediated transport of nanoparticles. We also developed a strategy of combination therapy (surgery + Trojan neutrophils) for effective control of the infiltration of the glioma cells in one hemisphere before crossing the corpus callosum (CC) in order to prevent recurrence in the other hemisphere. This alternative approach compared to the extended resection of the glioma including CC or butterfly GBM may provide the greater anti-tumor efficacy and reduce side effects such as cognitive impairment. We also studied the asthma-mediated control of optic glioma growth. Our results indicate that asthma-induced T cell reprogramming inhibits tumor growth by promoting the release of decorin and a subsequent suppression of CCR8 and the intercellular binding kinetics in microglia followed by blocking of CCL5 production in TME via suppression of NFκB. By using the mathematical model, we tested several hypothesis in prevention of optic glioma in athma patients.

Abstract: Consider some population evolving stochastically in time. Conditional on the population surviving until some large time T, take a sample of individuals from those alive. What does the ancestral tree drawn out by this sample look like? Some special cases were known, e.g. Durrett (1978), O’Connell (1995), but we will discuss some more recent advances for Bienyamé-Galton-Watson (BGW) branching processes conditioned to survive.

In near-critical or in critical varying environment BGW settings, the same universal limiting sample genealogy always appears up to some deterministic time change (which only depends on the mean and variance of the offspring distributions). This genealogical tree has the same binary tree topology as the classical Kingman coalescent, but where the coalescent (or split) times are quite different due to stochastic population size effects, with a representation as a mixture of independent identically distributed times. In contrast, in critical infinite variance offspring settings, we find that more complex universal limiting sample genealogies emerge that exhibit multiple-mergers, these being driven by rare but massive birth events within the underlying population (eg. `superspreaders’ in an epidemic). Some ongoing work, open problems, and potential downstream applications will be discussed.

This talk is based on collaborative works in Annals of Applied Probability (2020, 2024) and Annals of Probability (2024) with collaborators S.Palau (UNAM), J.C.Pardo (CIMAT), S.Johnston (Kings College London), and M.Roberts (Bath).

I would also like to acknowledge the support of the New Zealand Royal Society Te Apārangi Marsden fund.