A cross-section of the MIT community convened discussions on topics ranging from data centers and super pollutants to agriculture and education.

Kevin Hartnett is the author of *The Proof in the Code*.SPONSOR:---Cyber Fund built the Monastery to help founders ship products that were impossible a year ago. Applications for Batch 1 are now open.Apply now: https://cyber.fund---In 1998 Thomas Hales finished a proof of the Kepler conjecture that leaned so heavily on computer code that the Annals of Mathematics could not say whether it was true. That impasse opens Kevin Hartnett's book *The Proof in the Code*, and this documentary.Built around Tim's conversation with Hartnett, the film follows Lean from a side project at Microsoft Research to the system mathematicians now use to check their own field. Leonardo de Moura, Lean's creator, explains why everything rests on a small trusted kernel. Jeremy Avigad recalls the argument over foundations that changed Lean. Kevin Buzzard describes his fear that his corner of mathematics was becoming a house of cards, and Paul Lessard separates a correct proof from a useful one. Mathlib, Peter Scholze's Liquid Tensor challenge and Terence Tao's Equational Theories project all pass through.The second half turns to AI. AlphaProof, natural-language olympiad proofs and autoformalisation sharpen the question the book keeps asking: when a machine produces a proof nobody can read, what has actually been checked? The answer lies in the formalisation gap, the distance between the mathematics people mean and the statement a checker certifies. The film closes on the human story behind the software.---TIMESTAMPS:0:00 Cold open: proof, code and the wrong problem1:50 Kepler's oranges and the proof nobody could check6:54 How mathematicians decide a proof is true9:04 Correct is not the same as useful12:17 Sponsor: Cyber Fund12:53 What Lean is, and the small kernel it trusts15:23 From Microsoft's Z3 to Lean18:22 The argument over Lean's foundations24:52 One researcher versus a growing user base26:16 Kevin Buzzard's house of cards30:43 Mathlib as a barn raising37:16 The perfectoid-spaces stunt47:12 Scholze's proof nobody had read: Liquid Tensor53:20 Can AI learn to find proofs? AlphaProof58:43 AI mathematics in 2026 and Tao's experiment1:01:36 Lean's place in AI mathematics1:02:17 AI brute force versus the right abstraction1:05:19 Autoformalisation's Trojan horse1:07:25 The formalisation gap1:11:45 A proof is not understanding1:16:58 Integration hell and the zlib specification1:21:14 Verification in the AI software lifecycle1:22:38 The human story behind Lean1:31:09 The birthday walk1:33:05 Farewell---REFERENCES:[2:12] Thomas Hales, A proof of the Kepler conjecture (Annals of Mathematics, 2005)https://annals.math.princeton.edu/2005/162-3/p01[2:12] The Flyspeck project (the formal proof of the Kepler conjecture)https://github.com/flyspeck/flyspeck[12:53] Leanhttps://lean-lang.org/[30:43] Mathlibhttps://github.com/leanprover-community/mathlib4[30:43] Kevin Hartnett, "Building the Mathematical Library of the Future" (Quanta Magazine, 2020)https://www.quantamagazine.org/building-the-mathematical-library-of-the-future-20201001/[47:12] The Liquid Tensor Experimenthttps://github.com/leanprover-community/lean-liquid[53:20] AlphaProof (Google DeepMind): AI achieves silver-medal standard solving International Mathematical Olympiad problemshttps://deepmind.google/blog/ai-solves-imo-problems-at-silver-medal-level/[59:23] The Equational Theories Project (Terence Tao)https://github.com/teorth/equational_theories[1:18:30] lean-zip (the zlib proof)https://github.com/kim-em/lean-zipFrontierMath Erdős (Epoch AI, 2026)https://epoch.ai/benchmarks/frontiermath-erdosBook: Kevin Hartnett, *The Proof in the Code* (Quanta Books, 2026)https://www.quantabooks.org/books/the-proof-in-the-code/---RESCRIPT: https://app.rescript.info/share/8604a0cd6ea4604a368028a05f4e6be4Support MLST on Patreon: https://www.patreon.com/mlstDiscord: https://discord.gg/HNnAwSduud

AI has entered its prove-it era. This is the session on the infrastructure that decides whether you pass.

Scientists have filmed tiny magnetic vortices called skyrmions melting out of an organized lattice for the first time, revealing new ways to manipulate these unusual structures. The breakthrough could help make brain-like computers a reality, potentially enabling dramatically smaller and more energy-efficient technology.

arXiv:2610.32409v1 Announce Type: new Abstract: Event-based spiking neural networks (SNNs) encode information through temporal firing dynamics, but the same stateful computation makes their predictions sensitive to event loss, timestamp jitter, and speed variation. These perturbations alter evidence accumulation, yet late-layer firing trajectories can retain error-diagnostic structure that is absent from the final logits alone. We introduce Perturbation-Aware Firing Scaffolds (PAFS), a labeled post-training calibration method that exploits this structure without updating the pretrained backbone. PAFS associates standardized late-layer leaky integrate-and-fire trajectories with the class-probability residuals of an Adapter anchor, then applies singular value decomposition to extract compact residual-aligned channel directions. At inference, firing trajectories from the same backbone pass are projected onto this fixed low-rank scaffold and combined with anchor statistics to predict a residual logit correction. Across four matched dataset-backbone settings spanning three event datasets and two spiking Transformer architectures, PAFS improves corrupted-average and worst-condition accuracy over the corresponding Adapter anchors, with maximum gains of 3.06 and 3.65 percentage points. In the DailyAction-DVS/Spikingformer setting, the added trainable parameters amount to approximately 0.34% of the frozen backbone.

arXiv:2610.27710v3 Announce Type: replace-cross Abstract: Diffusion models have emerged as the foundation of modern generative systems, yet their high memorization capacity raises privacy concerns. While differentially private (DP) training provides formal guarantees, it comes at a cost in sample quality. In this work, we study a complementary question: how much privacy loss is incurred during the sampling process itself? Directly analyzing modern neural denoisers is challenging because their sensitivity to a single training example is generally intractable. To overcome this challenge, we construct and analyze a clipped privacy-accountable empirical denoiser, a training-free estimator of the Bayes-optimal denoiser, whose per-step sensitivity can be computed explicitly, allowing each denoising step to be a Gaussian mechanism. This perspective enables privacy accounting under Gaussian DP when each training example contributes to each denoising step, and under $f$-DP trade-off functions when each generated sample uses a single randomly subsampled block of the training data. We report every $(\varepsilon, \delta)$ guarantee as a certified upper bound from a privacy loss distribution accountant, and use a cumulant-based Edgeworth approximation as a fast diagnostic, characterizing where it is reliable and where it becomes anti-conservative. We also analyze the privacy loss along the denoising trajectory and show that it grows steeply near the critical window in which semantic content is formed. This observation motivates hybrid samplers that use the private empirical denoiser through the formative steps where it remains a faithful proxy for the neural denoiser, then switch to a public neural denoiser for the remaining refinement. Our results on CIFAR-10 and CelebA-64 show that the privacy of diffusion models depends not only on the training procedure, but also on sampling-time design choices such as subsampling, stopping time, and hybrid denoising.

arXiv:2609.15248v2 Announce Type: replace-cross Abstract: In bandits with probing, a learner selects at most $k$ of $n$ arms per round, earns the largest of their rewards, and competes with the best fixed single arm. For independent arms with i.i.d. rewards in $[0,1]$ and $2\le k<n$, the minimax regret is $\Theta\bigl(\frac{n-k}{n}\min\{T,n/k\}\bigr)$, the cost of locating an unknown good arm, and observing only the maximum suffices to attain it. In particular, two probes give $\Theta(\min\{T,n\})$ regret, which resolves two open questions of Bhaskara et al. (2023) with weaker feedback. Our policy turns the probing surplus, the excess of the maximum over each probed arm, into exploration capital, and a permanent random core yields the sharp dependence on $k$. On arbitrary fixed reward arrays, same-round contrasts between blocks of arms attain the same rate, because the surplus pays for the stability of mirror descent. Winner feedback, the maximum together with its winning label, instead incurs an additional learning cost under joint i.i.d. rewards and on fixed arrays. We characterize this cost even when values are revealed only intermittently, using policies that know neither the horizon nor the revelation rate. For three independent arms and two probes, exact maxima, Bernoulli-rounded maxima with winning labels, and winning labels alone have minimax regret $\Theta(1)$, $\Theta(\sqrt T)$, and $\Theta(T)$: rounding preserves conditional means but not the rate.

arXiv:2606.04031v3 Announce Type: replace-cross Abstract: Leader-follower gradient iterations, including bilevel and two-time-scale methods, can exhibit large finite-time excursions despite asymptotic stability. For cascaded iterations with block matrix J = [[A,0],[C,D]], symmetric diagonal blocks of norm at most gamma < 1, and coupling norm at most c, we prove an exact, dimension-free minimax formula: at every horizon t, the worst-case amplification equals the spectral norm of [[gamma^t,0],[c t gamma^(t-1),gamma^t]], attained by an aligned two-dimensional quadratic. Consequently, the worst transient is on the order of 1 + c/(1-gamma), and eigenvalues alone cannot predict finite-time gain. Under martingale noise, the aligned system has stationary follower variance sigma^2 c^2 (1+gamma^2)/(1-gamma^2)^3, establishing cubic inverse-gap scaling. For quadratic followers, the coupling is linked to follower sensitivity R by C = (I-D)R. When the leader is isotropic and the follower map is positive semidefinite, amplification is at most 1 plus the operator norm of R, independently of the spectral gaps. A look-ahead Newton update eliminates follower memory and is minimax-optimal among linear updates that track the leader exactly. Experiments validate the bounds and show that accelerated or inexact follower solvers can exceed the sensitivity bound when their effective maps have negative eigenvalues. Code and experiment scripts: https://github.com/Axym-Labs/transient-amplification

arXiv:2512.07875v2 Announce Type: replace-cross Abstract: Kolmogorov--Arnold Networks (KANs) offer a promising path toward interpretable machine learning: their learnable activations can be studied individually while collectively fitting complex data accurately. In practice, however, trained activations often lack *symbolic fidelity*, learning pathological decompositions with no meaningful correspondence to interpretable forms. We propose Softly Symbolified Kolmogorov--Arnold Networks (S2KAN), which integrate symbolic primitives directly into training. Each activation draws from a dictionary of symbolic and dense terms, with learnable gates that sparsify the representation. Crucially, this sparsification is differentiable, enabling end-to-end optimization, and uses an optimization objective inspired by minimum description length. S2KAN can discover interpretable activation functions while retaining dense splines for flexible approximation. We introduce a quantitative measure of symbolic fidelity, assessing how closely learned activations can be approximated by short expressions from a specified symbolic language. Across real-world prediction tasks, dynamical-system forecasting, and synthetic symbolic benchmarks, S2KAN generally achieves higher symbolic fidelity and often far smaller models, while maintaining competitive predictive accuracy on real-world data and low prediction errors on most symbolic benchmark configurations. We also examine the models' learned responses on scientific data, including the contributions of their symbolic components. Together, these results show that incorporating symbolic structure during training can yield compact, accurate KAN models whose learned responses can be examined through their symbolic components.