从陶哲轩与AI的一段长对话谈起：顶级数学家面对陌生公式时，究竟在想什么？

TL;DR - A commentary essay analyzing a public Terence Tao–ChatGPT conversation about the Jacobian Conjecture, using it to illustrate how a top mathematician converts brute computation into structural understanding — and how AI fits into that workflow. It matters as a grounded counter-narrative to "AI solves math," showing the human role stays in question-selection and modeling.

• The worked example is an explicit 3-variable polynomial map with constant Jacobian (locally invertible everywhere) yet three distinct preimages of one point; Tao's drive is not verifying these facts but explaining why the cancellations occur.
• Key moves traced: substituting recurring expression blocks as new coordinates (revealing the constant Jacobian as two coordinate-change volume factors cancelling), assigning scaling weights to expose weighted homogeneity, and reinterpreting the 3-to-1 fiber as "choose one of a cubic's three simple roots," normalized via a resultant condition on the linear×quadratic factorization.
• Non-properness is localized at infinity: when two roots collide, the derivative at the root vanishes, sending two sheets of the normalized factorization to infinity while one stays finite — explaining why everywhere-nonzero local Jacobian doesn't force global invertibility.
• On AI usage: ChatGPT handled algebraic expansion, substitutions, bookkeeping, and rapid recomputation, while Tao retained verification, direction-setting, and distinguishing verified identities from structural conjecture — the article explicitly declines to frame it as "AI cracked a famous conjecture."

TL;DR - ByteDance's Doubao app has shipped what it claims is the first consumer "omni-modal full-duplex" video calling experience, powered by a new native audio-video model called SeedRealtime. It matters because it extends real-time voice interaction (as in OpenAI's GPT-Live) with continuous vision, so the assistant can watch, listen, and speak simultaneously.

• New base model: SeedRealtime, a native audio-video full-duplex LLM, replaces the prior stack; the article contrasts it with GPT-Live, which OpenAI documents as launching without video/screen-sharing support (audio-only full duplex), and with Thinking Machines' omni-modal work still at the research stage.
• Interaction directionality: the model reportedly fuses speaker identity, face orientation, gaze, context and scene relations to decide whether an utterance is addressed to it — the author observed it stayed silent during side conversations with a friend, and that this ability disappeared when the camera was covered.
• Turn-taking and proactivity: it waits through mid-sentence pauses and self-corrections instead of barging in after 2–3 seconds of silence, and conversely speaks up unprompted — e.g. flagging an item sorted into the wrong box under user-defined rules, with persistent task context and no repeated wake-up needed.
• Concurrent multimodal grounding: while co-watching a podcast clip, it attributed statements to individual speakers (Papi酱, 罗翔, LKS), identified people by clothing, separated background audio (a hair dryer) from speech, and answered live questions without mistaking on-screen voices for the user.
• Note: all claims are first-person product-trial impressions from a WeChat tech account, not benchmarked results.