陽明交通大學應用數學系

Abstract
A Boolean function is a function from the discrete hypercube to {-1, 1} or {0, 1}. Such a function can be viewed as a voting system: each person has either opinion -1 or 1, and the function tells us the result after knowing all the opinions. In this talk, we will focus on whether a Boolean function is noise sensitive, in the sense that the result can be easily interrupted by a small amount of noise. We will then talk about Boolean functions that arise from percolation, a model in statistical physics, and an interesting implication of noise sensitivity of this model.

Abstract. The celebrated Kirchhoff matrix-tree theorem connects a combinatorial quantity---the number of spanning trees of a graph---with an algebraic one---the determinant of its reduced Laplacian matrix. This theorem has analogues for Eulerian digraphs and embedded graphs, where determinants of certain matrices count Eulerian tours and quasi-trees, respectively. We will explore the interplay between these two settings and see how the corresponding matrices lead to Jacobian groups that naturally act on Eulerian tours and quasi-trees.

Abstract
Machine learning often asks how two representations of the same data relate: an image encoder and a text encoder, a teacher and a student, or the observed signals of two coupled dynamical systems. Persistent homology offers a multiscale summary of such representations. The usual practice, however, is to match persistence diagrams with a symmetric distance, which leaves two questions open: what should align, and in which direction?
The first part of the talk asks what should align. The death edges of zero-dimensional persistent homology are exactly the edges of a minimum spanning tree, so topology records which samples are linked, not only when components merge. Using this identity, our recent work revisits the Platonic Representation Hypothesis across 204 vision–language model pairs. After calibration, the relational structure of representations converges at both local and global scales. Agreement on exact distances, by contrast, weakens quickly and loses its dependence on model capacity.
The second part asks in which direction alignment runs. We introdcue persistent cross entropy (PCE) that extends persistent entropy to pairs of diagrams. One diagram induces a probability on the points of the other, and an explicit event collects what it fails to explain. The excess over persistent entropy is a Kullback–Leibler divergence and is stable in the Wasserstein distance. We present several numerical examples that shows that PCE is directional and it recovers causal direction in dynamical systems without a joint attractor, and in a first distillation experiment it serves as a teacher-to-student topology loss.

A 2-edge-colouring of a graph G is called locally complete if for each vertex v, the vertices adjacent to v through edges of the same colour induce a complete subgraph in G. Contreras-Balbuena et al characterized locally complete 2-edge-coloured graphs which have alternating Hamiltonian cycles. Chvátal and Sbihi proved that graphs which are locally complete 2-edge-colourable are one of two types of claw-free perfect graphs indecomposable via clique-cutsets. Maffray and Reed gave a forbidden subgraph characterization of these graphs.
We compare locally complete 2-edge-colourable graphs with proper interval graphs and proper circular-arc graphs. We characterize proper interval graphs and proper circular-arc graphs which are locally complete 2-edge-colourable by forbidden subgraphs.

Graphs can be viewed as (non-archimedean) analogs of Riemann surfaces. For example, there is a notion of Jacobians for graphs. More classically, graphs can be viewed as electrical networks. I will explain the interplay between these points of view and some recent applications in algebraic and arithmetic geometry.

Abstract
Fluid-fluid interfaces laden with discrete particles behave like continuous elastic sheets, leading to their applications in emulsion and foam stabilization. Although current continuum models can qualitatively describe the elastic buckling of granular particle-laden interfaces—often referred to as granular rafts—under compression, the connection between their macroscopic collective properties and the microscopic behavior of individual particles remains unclear. Here, by combining systematic experiments with first-principle modeling, we aim to reveal how the macroscopic mechanical properties of particle rafts emerge from particle-scale interactions and predict the failure modes of a granular raft under compression. Our study highlights the dual nature of particle rafts and exemplifies how collective dynamics can arise from discrete components with simple interactions.

Abstract. The use of large language models has changed the world we live in, from communicating in different languages to student use in homework. Recently it has proved to be effective on research level mathematics problems. In this talk, I will discuss the role of the researcher/professor in this newly emerging world.