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DTSTART;TZID=America/New_York:20210407T150000
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DTSTAMP:20240515T193258Z
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SUMMARY:Type theory from the perspective of artificial intelligence
DESCRIPTION:Speaker: David McAllester – Toyota Technological Institute at Chicago \nTitle: Type theory from the perspective of artificial intelligence \nAbstract: This talk will discuss dependent type theory from the perspective of artificial intelligence and cognitive science. From an artificial intelligence perspective it will be argued that type theory is central to defining the “game” of mathematics — an action space and reward structure for pure mathematics. From a cognitive science perspective type theory provides a model of the grammar of the colloquial (natural) language of mathematics. Of particular interest is the notion of a signature-axiom structure class and the three fundamental notions of equality in mathematics — set-theoretic equality between structure elements\, isomorphism between structures\, and Birkoff and Rota’s notion of cryptomorphism between structure classes. This talk will present a version of type theory based on set-theoretic semantics and the 1930’s notion of structure and isomorphism given by the Bourbaki group of mathematicians. It will be argued that this “Bourbaki type theory” (BTT) is more natural and accessible to classically trained mathematicians than Martin-Löf type theory (MLTT). BTT avoids the Curry-Howard isomorphism and axiom J of MLTT. The talk will also discuss BTT as a model of MLTT. The BTT model is similar to the groupoid model in that propositional equality is interpreted as isomorphism but different in various details. The talk will also briefly mention initial thoughts in defining an action space and reward structure for a game of mathematics.
URL:https://cmsa.fas.harvard.edu/event/4-7-2021-new-technologies-in-mathematics-seminar/
CATEGORIES:New Technologies in Mathematics Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-New-Technologies-in-Mathematics-04.07.21.png
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20210414T150000
DTEND;TZID=America/New_York:20210414T160000
DTSTAMP:20240517T194855Z
CREATED:20240126T064644Z
LAST-MODIFIED:20240517T194855Z
UID:10001384-1618412400-1618416000@cmsa.fas.harvard.edu
SUMMARY:A Bayesian neural network predicts the dissolution of compact planetary systems
DESCRIPTION:Speaker: Miles Cranmer – Princeton University \nTitle: A Bayesian neural network predicts the dissolution of compact planetary systems \nAbstract: Despite over three hundred years of effort\, no solutions exist for predicting when a general planetary configuration will become unstable. I will discuss our deep learning architecture (arxiv:2101.04117) which pushes forward this problem for compact systems. While current machine learning algorithms in this area rely on scientist-derived instability metrics\, our new technique learns its own metrics from scratch\, enabled by a novel internal structure inspired from dynamics theory. The Bayesian neural network model can accurately predict not only if\, but also when a compact planetary system with three or more planets will go unstable. Our model\, trained directly from short N-body time series of raw orbital elements\, is more than two orders of magnitude more accurate at predicting instability times than analytical estimators\, while also reducing the bias of existing machine learning algorithms by nearly a factor of three. Despite being trained on three-planet configurations\, the model demonstrates robust generalization to five-planet systems\, even outperforming models designed for that specific set of integrations. I will also discuss some work on recovering symbolic representations of such models using arxiv:2006.11287.
URL:https://cmsa.fas.harvard.edu/event/4-14-2021-new-technologies-in-mathematics/
LOCATION:Virtual
CATEGORIES:New Technologies in Mathematics Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-New-Technologies-in-Mathematics-04.14.21.png
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20210421T150000
DTEND;TZID=America/New_York:20210421T160000
DTSTAMP:20240515T192602Z
CREATED:20240126T065403Z
LAST-MODIFIED:20240515T192602Z
UID:10001388-1619017200-1619020800@cmsa.fas.harvard.edu
SUMMARY:Homotopy type theory and the quest for extensionality
DESCRIPTION:Speaker: Michael Shulman – Dept. of Mathematics\, University of San Diego \nTitle: Homotopy type theory and the quest for extensionality \nAbstract: Over the past decades\, dependent type theory has proven to be a powerful framework for verified software and formalized mathematics.  However\, its treatment of equality has always been somewhat uncomfortable.  Recently\, homotopy type theory has made progress towards a more useful notion of equality\, which natively implements both isomorphism-invariance in mathematics and representation-independence in programming. This progress is based on ideas from abstract homotopy theory and higher category theory\, and with the development of cubical type theories it can be implemented as a true programming language.  In this talk\, I will survey these developments and their potential applications\, and suggest some directions for further improvement. \n 
URL:https://cmsa.fas.harvard.edu/event/4-21-2021-new-tech-in-math-seminar/
LOCATION:Virtual
CATEGORIES:New Technologies in Mathematics Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-New-Technologies-in-Mathematics-04.21.21.png
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