歷史演講 PAST TALKS

Abstract：

傅立葉分析起源於兩百多年前對熱傳導問題的研究，如今已成為現代數學、物理及工程科學中最廣泛而深刻的數學工具之一。其核心思想是將複雜的函數與現象分解為基本的振盪成分，藉此揭示其內在結構與規律。

本次演講將從傅立葉級數與傅立葉變換在微分方程中的經典應用出發，逐步介紹傅立葉分析在波動傳播、光學繞射與成像、現代訊號處理，以及量子計算中的量子傅立葉變換與量子相位估計。

演講的一個核心主題是從分解到重建（From Decomposition to Reconstruction）：傳統傅立葉分析著重於將函數分解為基本振盪模式，而許多現代逆問題則試圖從不完整、間接或缺乏相位資訊的測量資料中，重建隱藏的幾何與物理結構。

沿著這條思路，我們將探討逆散射（Inverse Scattering）、相位恢復（Phase Retrieval），以及 MUSIC、ESPRIT 等高解析度次空間估計方法（Subspace Estimation）。

最後，結合我們近期的研究成果，說明如何運用傅立葉分析、球諧函數（Spherical Harmonics）與次空間結構，研究逆問題中的唯一性、穩定性與解析度。

本演講將著重於不同領域之間的數學思想與內在聯繫，而非繁複的技術細節，適合不同專業背景的數學研究者與學生參與。

20261022演講_范姜群健教授 (Department of Mathematics, UC Davis)

Abstract：

Modern generative models can capture rich and complex data distributions, but practical applications often require more than unconditional generation: we may want samples that satisfy a desired property, achieve high reward, or reach rare regions of the model distribution. Retraining or fine-tuning the model for every new objective can be expensive, while gradient-based guidance may be unavailable or ineffective when rewards are black-box, non-differentiable, or highly multimodal.

In this talk, I will present our recent work on test-time steering, where a pretrained generative model is kept fixed and control is performed entirely during inference. I will first introduce Source Parallel Tempering, which moves the steering problem to the simple source distribution of continuous generative models and uses parallel tempering to explore multimodal reward landscapes without requiring reward gradients. I will then discuss how similar ideas extend to masked discrete diffusion models, where reward-aware remask-and-denoise updates are combined with parallel tempering to efficiently explore separated high-reward modes in applications such as biological sequence design. Together, these works suggest a broader perspective: test-time steering can be viewed as a sampling problem over reward-tilted generative distributions, where effective control requires balancing reward optimization, model fidelity, and exploration. I will conclude with some open directions toward test-time control for generative models.

20261015演講_王士欣助理教授 (國立臺灣大學資訊工程學系)