Title: No Free Conversion: What Mathematical Shorthand Hides

Speaker: Hanyu Li (Peking University)

Time: Sep. 8th (Tuesday) 15:10 - 18:00

Location: Teaching Building No.2 (二教)   411

Abstract:

A single mathematical problem often gives rise to several computational tasks: deciding whether a solution exists, finding one, verifying one, counting solutions, or finding a best one. For familiar problems, standard reductions make these tasks feel interchangeable. This convenience can be misleading: shared names and notation may hide a genuine proof obligation when an argument moves from one task to another. 

In this talk, I will explain why such conversions cannot in general be taken for free, and how an unnoticed change of task can leave an otherwise convincing proof incomplete. I will also describe an exploratory search by an LLM agent that uncovered examples of this pattern across optimization, databases, approximate counting, quantum computation, and complexity theory, including papers at several leading theory venues. These gaps are often only a line or two long and locally quite plausible. The broader message is simple: mathematical shorthand is essential for communication, but sometimes what it quietly erases is exactly what the proof still needs. 

Three years ago, Dongchen Li and I noticed this phenomenon while working on complexity theory, and left it there. Now, today’s LLMs have made returning to it both possible and hard to resist.