陽明交通大學應用數學系

Abstract. The heat equation with a random potential driven by space-time white noise serves as a basic model for continuum directed polymers in random media, a topic of longstanding interest in the physics literature. While this model generates systems exhibiting scale invariance and universality, establishing their fundamental properties has posed deep mathematical challenges. In particular, although classical stochastic analysis can construct the corresponding Gibbs measures for polymers as two-dimensional curves, extending these results to three dimensions introduces substantial complexity due to the critical, non-Gaussian behaviour.

This talk will present the basic ideas of continuum directed polymers in random media, focusing on their behaviour as three-dimensional curves.

Abstract. Newton-type methods are among the most effective tools for solving large-scale nonlinear systems arising in scientific computing. Despite their fast local convergence, their global behavior can be unpredictable, with common issues such as overshooting, stagnation, and sensitivity to problem scaling—especially in stiff or highly unbalanced systems.

In this talk, we present a dynamical systems perspective for understanding these behaviors by interpreting Newton iterations as discrete approximations of an underlying continuous-time flow. This viewpoint provides an intuitive characterization of nonlinear imbalance in terms of stiffness, offering insight into why classical globalization strategies, particularly line search, may become ineffective or overly restrictive.

Motivated by this perspective, we revisit line search methods and introduce improved strategies, including curve search techniques, that better align with the intrinsic dynamics of the nonlinear system. We further show that nonlinear preconditioning can be naturally interpreted as a transformation that reduces stiffness and restores balance, leading to improved robustness and convergence.

Numerical examples from nonlinear PDEs illustrate how this framework not only enhances performance, but also provides a unified viewpoint for understanding globalization, acceleration, and stabilization in Newton-type methods.

Abstract

The Kadomtsev-Petviashvili II (KPII) equation is one of the few physically relevant integrable systems in more than one spatial dimension. In this talk, we present an overview of the inverse scattering theory and the stationary phase method, and explain how these tools are used to derive the long-time asymptotic behavior of solutions.

Abstract
Artificial intelligence (AI) has become increasingly integrated into modern radiology, offering tools that support image interpretation, quantitative analysis, workflow prioritization, and clinical decision-making. Recent advances in machine learning and deep learning have enabled the detection of subtle imaging patterns and the efficient analysis of large-scale imaging data, helping to enhance diagnostic accuracy and improve workflow efficiency. In current radiology practice, AI is already being applied in areas such as lesion detection, organ segmentation, triage, and risk prediction.
Pancreatic cancer remains one of the most lethal malignancies, in part because of its subtle imaging features and the difficulty of early diagnosis on routine CT. In this talk, we will review the emerging role of AI in pancreatic cancer detection using CT imaging, with a focus on recent advances in AI-based detection models and their potential to facilitate earlier diagnosis. We will also briefly discuss the broader real- world applications of AI in radiology and consider the challenges and opportunities for translating these technologies into clinical practice.

Abstract

Consider the set of positive integers representing the number of spanning trees in simple graphs with n vertices. How quickly can this set grow as a function of n? In this talk, we discuss a proof of the exponential growth of this set, which resolves an open problem of Sedlacek from 1966. The proof uses a connection with continued fractions and advances towards Zaremba’s conjecture in number theory. This is joint work with Alex Kontorovich and Igor Pak. This talk is intended for general audience.

Abstract
One will present a two expansions of the spectral information to yield the pointwise structure of the Green;s function.