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DTSTART;TZID=America/New_York:20260908T090000
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SUMMARY:The Geometry of Machine Learning 2026
DESCRIPTION:The Geometry of Machine Learning 2026 \nDates: September 8–11\, 2026 \nLocation: Harvard CMSA\, Room G10\, 20 Garden Street\, Cambridge MA 02138 & via Zoom Webinar \nRegister to attend in person \nRegister for Zoom Webinar \nLarge language models are presently\, and will increasingly\, be complemented by other dimensions of intelligence: formal verification and energy-based optimizers\, becoming parts of larger ecosystems. Can AIs reason geometrically and can we use geometry to reveal how data is currently processed in NNs? Can AIs reveal the geometry of mathematics\, as well as studying geometry as a subject within math. This conference is intended to continue the discussion of these topics. \nConfirmed Speakers: \n\nNada Amin\, Harvard\nRandall Balestriero\, Brown\nMichael Brenner\, Harvard and Google\nBennet Chow\, UCSD\nSurya Ganguli\, Stanford\nBoris Hanin\, Princeton\nRoi Holtzman\, Oxford\nRobert Koirala\, UCSD\nDmitry Krotov\, Dynamical Mind\nSlava Krushkal\, Virginia\nJared Duker Lichtman\, Stanford\nMike Mulligan\, UCR\, Logical Intelligence\nLuca Pesce\, Harvard\nGabriel Poesia\, U Michigan (via Zoom)\nZiyang Qin\, Cornell\nMathew Vanherreweghe\, Logical Intelligence\nSean Welleck\, CMU (via Zoom)\nMattiew Wyart\, JHU (via Zoom)\n\nOrganizers: Michael R. Douglas (CMSA) and Mike Freedman (CMSA) \n  \nSchedule \nTuesday\, Sep. 8\, 2026 \n8:15–8:45 am\nBreakfast \n8:45–9:30 am\nMike Mulligan\, UCR\, Logical Intelligence\nCompression is all you need: Modeling mathematics\nAbstract: The mathematics humans discover and value (“human math”) is a vanishingly small subset of all valid deductions (“formal math”). I’ll argue that human math is distinguished by its compressibility through hierarchically nested definitions and theorems\, like a polynomial-growth space rather than the exponential-growth space one might expect when proofs are viewed as strings of symbols. The argument combines toy monoid models with an empirical analysis of MathLib\, a large Lean library of formalized mathematics we treat as a proxy for human math. I’ll close with how compression itself can serve as a measure of mathematical interest\, giving agents a sense of direction toward where human math lives. \n9:30–9:45 am\nBreak \n9:45–10:30 am\nBoris Hanin\, Princeton \n10:30–11:00 am\nBreak \n11:00–11:45 am\nSlava Krushkal\, University of Virginia\nUsing AI to study 4-manifold topology\nAbstract: Central open questions in geometric classification theory of topological 4-manifolds have a reformulation in terms of the Round Handle Problem. It asks whether a given link in the 3-sphere is slice (bounds disjoint disks) in the 4-manifold obtained by attaching certain round handles to the 4-ball. I will discuss an algebraic-combinatorial formulation of the problem\, ongoing AI-assisted work on it\, and the results obtained to date. \n11:45 am–12:00 pm\nBreak \n12:00–12:45 pm\nRobert Koirala\, UCSD; Ziyang Qin\, Cornell; and Bennett Chow\, UCSD\nAI for Ricci flow: Discovery and Formalization \n  \nWednesday\, Sep. 9\, 2026 \n8:15–8:45 am\nBreakfast \n8:45–9:30 am\nSurya Ganguli\, Stanford \n9:30–9:45 am\nBreak \n9:45–10:30 am\nMathew Vanherreweghe\, Logical Intelligence\nSparsity Before Averaging: Kolmogorov–Arnold Geometry in Language Models\nAbstract: The Kolmogorov–Arnold theorem writes any continuous multivariate function as a composition of one-dimensional functions and addition. Freedman and Mulligan recently showed that ordinary neural networks\, trained by gradient descent\, rediscover the geometry of that construction on their own\, first in regression\, then in vision. This talk asks the same question of large language models\, through the Jacobian that connects each prediction back to the model’s internal state. Measured carefully\, per context and before any averaging\, the newest language models turn out to have grown this geometry on their own to a degree. We then show that this geometry can be installed intentionally\, cheaply\, and without significant cost on downstream tasks\, and examine what this offers for interpretability. Joint work with Michael Freedman and Michael Mulligan. \n10:30–11:00 am\nBreak \n11:00–11:45 am\nMichael Brenner\, Harvard and Google \n11:45 am–12:00 pm\nBreak \n12:00–12:45 pm\nMattiew Wyart\, JHU (via Zoom)\nLearn from your own latents\, not from tokens\nAbstract: Language models need more than a hundred thousand times the data a child does. One explanation is that predicting raw tokens is simply the wrong level: methods like data2vec and JEPA instead train a network to predict its own internal representations\, with strong empirical results but no theory of why. Using a hierarchical grammar that models that language and images have a hidden hierarchical structure\, we quantify the gain exactly. Token-level learning needs a number of examples growing exponentially with the depth of the hierarchy; latent prediction needs a number independent of it. We also show data2vec performs this hierarchical prediction implicitly\, which suggests that explicitly stacking levels – as in H-JEPA – yields little. \n  \nThursday\, Sep. 10\, 2026 \n8:15–8:45 am\nBreakfast \n8:45–9:30 am\nDmitry Krotov\, Dynamical Mind\nDense Associative Memory: Physical systems for novel AI architectures\nAbstract: Dense Associative Memories are recurrent neural networks with fixed-point attractor states that are described by an energy function. In contrast to conventional Hopfield Networks\, which were popular in the 1980s\, Dense Associative Memories have a very large information storage capacity\, making them appealing tools for many problems in AI. In this talk\, I will provide an intuitive understanding and mathematical framework for this class of models and give examples of problems in AI that can be tackled using these new ideas. Specifically\, I will explore the relationship between Dense Associative Memories and transformers. I will present a neural network called the Energy Transformer\, which unifies energy-based modeling\, associative memories\, and transformers in a single architecture. I will demonstrate how Energy Transformers can be used for challenging tasks in image processing\, solve partial differential equations\, and serve as computational modules for energy-based language modeling. I will also discuss an exciting possibility of mapping these models onto analog hardware accelerators\, which could enable much more energy-efficient inference compared to GPUs. \n9:30–9:45 am\nBreak \n9:45–10:30 am\nRoi Holtzman\, Oxford \n10:30–11:00 am\nBreak \n11:00–11:45 am\nSean Welleck\, CMU (via Zoom)\nThe Problem is the Problem: Towards Scalable Mathematical Discovery\nAbstract: If we give AI a mathematical problem\, it can often help us find a solution. However\, research and discovery also involve choosing which problems to solve in the first place. In this talk\, I will describe Find\, Attempt\, and Recommend\, an agentic pipeline that finds open problems in the literature\, attempts to solve them\, and recommends promising problem-resolution pairs for human review. I will discuss a pilot study in combinatorics that found resolutions to several open conjectures\, along with strategies for allocating a budget of model attempts in order to maximize different discovery objectives. \n11:45 am–12:00 pm\nBreak \n12:00–12:45 pm\nGabriel Poesia\, University of Michigan (via Zoom) \nFriday\, Sep. 11\, 2026 \n8:15–8:45 am\nBreakfast \n8:45–9:30 am\nNada Amin\, Harvard\nCompiling Programs to Neurons\nAbstract: Neural networks are ordinarily programmed indirectly: we specify architectures\, objectives\, and data\, and rely on learning to discover a computation. I will focus on Cajal\, a typed\, higher-order linear programming language whose programs compile correctly to linear neurons and\, with iteration\, to recurrent neurons. This allows discrete programming structures such as conditionals and iteration to coexist with gradient-based learning. Experiments show that connecting compiled neurons with learned networks can improve learning speed and data efficiency. This work is led by PhD student Joey Velez-Ginorio and is joint with his advisors at UPenn\, Konrad Kording and Steve Zdancewic. I will close with a complementary direction from my work: using language models together with formal verification to generate programs and proofs with machine-checkable guarantees. Together\, these projects explore how programming languages can provide structure and control at the boundary between programs and learned systems. \n9:30–9:45 am\nBreak \n9:45–10:30 am\nRandall Balestriero\, Brown University\nCounterfactual World Models for Real World Deployment \n10:30–11:00 am\nBreak \n11:00–11:45 am\nLuca Pesce\, Harvard CMSA\nA spiked perspective on feature learning with gradient-based methods\nAbstract: Depth is widely believed to give neural networks a clear computational advantage over shallow models\, and making this belief precise is a central problem in learning theory. We study a controlled high-dimensional setting where this can be done. The targets are hierarchical: the relevant structure is distributed across latent subspaces of decreasing dimension\, so that a shallow model must resolve all of it simultaneously\, while a deep network does not. Each layer forms an intermediate representation\, allowing learning to proceed in stages\, with every stage reducing the effective dimension of the remaining problem. We analyze this staged mechanism through the gradient descent dynamics of a deep network yielding a sharp separation in sample complexity between shallow and deep architectures. The result is a concrete account of why depth allows such functions to be learned from substantially fewer samples than shallow methods require. \n11:45 am–12:00 pm\nBreak \n12:00–12:45 pm\nJared Duker Lichtman\, Stanford \n  \nSupport provided by Logical Intelligence. \n \n  \n 
URL:https://cmsa.fas.harvard.edu/event/gml_2026/
LOCATION:CMSA 20 Garden Street Cambridge\, Massachusetts 02138 United States
CATEGORIES:Conference,Event
ATTACH;FMTTYPE=image/jpeg:https://cmsa.fas.harvard.edu/media/GML2026-Poster.4.jpg
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DTSTART;TZID=America/New_York:20260909T140000
DTEND;TZID=America/New_York:20260909T150000
DTSTAMP:20260729T181924Z
CREATED:20260729T181924Z
LAST-MODIFIED:20260729T181924Z
UID:10003893-1788962400-1788966000@cmsa.fas.harvard.edu
SUMMARY:New Technologies in Mathematics Seminar
DESCRIPTION:New Technologies in Mathematics Seminar \nSpeaker: tba
URL:https://cmsa.fas.harvard.edu/event/newtech_9926/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:New Technologies in Mathematics Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20260909T163000
DTEND;TZID=America/New_York:20260909T173000
DTSTAMP:20260904T190049Z
CREATED:20260706T171637Z
LAST-MODIFIED:20260904T190049Z
UID:10003960-1788971400-1788975000@cmsa.fas.harvard.edu
SUMMARY:Working with LLMs to do high quality math
DESCRIPTION:AI for the Working Mathematician \nSpeaker: Daniel Litt\, University of Toronto \nTitle: Working with LLMs to do high quality math \nAbstract: Much hay has been made of the capabilities of LLMs to do math autonomously\, and indeed frontier models have resolved some long-standing\, interesting open questions. But our goal is not to produce papers\, but rather to do high quality mathematics. I’ll discuss some experiments over the past year in trying to get LLMs to help with producing high quality mathematics\, non-autonomously\, and speculate a bit about the future.
URL:https://cmsa.fas.harvard.edu/event/ai_9926/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:AI for the Working Mathematician,Colloquia & Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-AI-for-Mathematicians-Seminar-9.9.2026.docx-scaled.png
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