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DTSTART;TZID=America/New_York:20251001T140000
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SUMMARY:Tropicalized quantum field theory
DESCRIPTION:New Technologies in Mathematics Seminar \nSpeaker: Michael Borinsky\, Perimeter Institute  \nTitle: Tropicalized quantum field theory \nAbstract: Quantum field theory (QFT) is one of the most accurate methods for making phenomenological predictions in physics\, but it has a significant drawback: obtaining concrete predictions from it is computationally very demanding. The standard perturbative approach expands an interacting QFT around a free QFT\, using Feynman diagrams. However\, the number of these diagrams grows superexponentially\, making the approach quickly infeasible. \nI will talk about arXiv:2508.14263\, which introduces an intermediate layer between free and interacting field theories: a tropicalized QFT. Often\, this tropicalized QFT can be solved exactly. The exact solution manifests as a non-linear recursion equation fulfilled by the expansion coefficients of the quantum effective action. Geometrically\, this recursion computes volumes of moduli spaces of metric graphs and is thereby analogous to Mirzakhani’s volume recursions on the moduli space of curves. Building on this exact solution\, an algorithm can be constructed that samples points from the moduli space of graphs approximately proportional to their perturbative contribution. Via a standard Monte Carlo approach we can evaluate the original QFT using this algorithm. Remarkably\, this algorithm requires only polynomial time and memory\, suggesting that perturbative quantum field theory computations actually lie in the polynomial-time complexity class\, while all known algorithms for evaluating individual Feynman integrals are at least exponential in time and memory. The (potential) capabilities of this approach are remarkable: For instance\, we can compute perturbative expansions of massive scalar D=3 phi^3 and D=4 phi^4 quantum field theories up to loop orders between 20 and 50 using a basic proof-of-concept implementation. These perturbative orders are completely inaccessible using a naive approach.
URL:https://cmsa.fas.harvard.edu/event/newtech_10125/
LOCATION:Virtual
CATEGORIES:New Technologies in Mathematics Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-NTM-Seminar-10.1.2025.docx-1-scaled.png
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DTSTART;TZID=America/New_York:20251008T140000
DTEND;TZID=America/New_York:20251008T150000
DTSTAMP:20260504T205600
CREATED:20250930T181425Z
LAST-MODIFIED:20251009T195959Z
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SUMMARY:Understanding Optimization in Deep Learning with Central Flows
DESCRIPTION:New Technologies in Mathematics Seminar \nSpeaker: Alex Damian\, Harvard \nTitle: Understanding Optimization in Deep Learning with Central Flows \nAbstract: Traditional theories of optimization cannot describe the dynamics of optimization in deep learning\, even in the simple setting of deterministic training. The challenge is that optimizers typically operate in a complex\, oscillatory regime called the “edge of stability.” In this paper\, we develop theory that can describe the dynamics of optimization in this regime. Our key insight is that while the *exact* trajectory of an oscillatory optimizer may be challenging to analyze\, the *time-averaged* (i.e. smoothed) trajectory is often much more tractable. To analyze an optimizer\, we derive a differential equation called a “central flow” that characterizes this time-averaged trajectory. We empirically show that these central flows can predict long-term optimization trajectories for generic neural networks with a high degree of numerical accuracy. By interpreting these central flows\, we are able to understand how gradient descent makes progress even as the loss sometimes goes up; how adaptive optimizers “adapt” to the local loss landscape; and how adaptive optimizers implicitly navigate towards regions where they can take larger steps. Our results suggest that central flows can be a valuable theoretical tool for reasoning about optimization in deep learning. \n 
URL:https://cmsa.fas.harvard.edu/event/newtech_10825/
LOCATION:Hybrid – G10
CATEGORIES:New Technologies in Mathematics Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-NTM-Seminar-10.8.2025-scaled.png
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20251022T140000
DTEND;TZID=America/New_York:20251022T150000
DTSTAMP:20260504T205600
CREATED:20251008T132005Z
LAST-MODIFIED:20251008T133142Z
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SUMMARY:The Carleson project: A collaborative formalization
DESCRIPTION:New Technologies in Mathematics Seminar \nSpeaker: María Inés de Frutos Fernández\, Mathematical Institute\, University of Bonn \nTitle: The Carleson project: A collaborative formalization \nAbstract: A well-known result in Fourier analysis establishes that the partial Fourier sums of a smooth periodic function $f$ converge uniformly to $f$\, but the situation is a lot more subtle for e.g. continuous functions. However\, in 1966 Carleson proved that they do converge at almost all points for $L^2$ periodic functions on the real line. Carleson’s proof is famously hard to read\, and there are no known easy proofs of this theorem. As a large collaborative project\, we have formalized in Lean a generalization of Carleson’s theorem in the setting of doubling metric measure spaces (proven in 2023)\, and Carleson’s original result as a corollary. In this talk I will give an overview of the project\, with a focus on how the collaboration was organized. \n 
URL:https://cmsa.fas.harvard.edu/event/newtech_102225/
LOCATION:Virtual
CATEGORIES:New Technologies in Mathematics Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-NTM-Seminar-10.22.2025-scaled.png
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