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DTSTART;TZID=America/New_York:20221202T110000
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DTSTAMP:20240229T111029Z
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SUMMARY:Compactness and Anticompactness Principles in Set Theory
DESCRIPTION:Member Seminar \nSpeaker: Alejandro Poveda \nTitle: Compactness and Anticompactness Principles in Set Theory \nAbstract: Several fundamental properties in Topology\, Algebra or Logic are expressed in terms of Compactness Principles.For instance\, a natural algebraic question is the following: Suppose that G is an Abelian group whose all small subgroups are free – Is the group G free? If the answer is affirmative one says that compactness holds; otherwise\, we say that compactness fails. Loosely speaking\, a compactness principle is anything that fits the following slogan: Suppose that M is a mathematical structure (a group\, a topological space\, etc) such that all of its small substructures N have certain property $\varphi$; then the ambient structure M has property $\varphi$\, as well. Oftentimes when these questions are posed for infinite sets the problem becomes purely set-theoretical and axiom-sensitive. In this talk I will survey the most paradigmatic instances of compactness and present some related results of mine. If time permits\, I will hint the proof of a recent result (joint with Rinot and Sinapova) showing that stationary reflection and the failure of the Singular Cardinal Hypothesis can co-exist. These are instances of two antagonist set-theoretic principles: the first is a compactness principle while the second is an anti-compactness one. This result solves a question by M. Magidor from 1982.
URL:https://cmsa.fas.harvard.edu/event/member-seminar-12222/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Member Seminar
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DTSTART;TZID=America/New_York:20221207T140000
DTEND;TZID=America/New_York:20221207T150000
DTSTAMP:20240116T060930Z
CREATED:20230808T185642Z
LAST-MODIFIED:20240116T060930Z
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SUMMARY:How do Transformers reason? First principles via automata\, semigroups\, and circuits
DESCRIPTION:New Technologies in Mathematics Seminar \nSpeaker: Cyril Zhang\, Microsoft Research \nTitle: How do Transformers reason? First principles via automata\, semigroups\, and circuits \nAbstract: The current “Transformer era” of deep learning is marked by the emergence of combinatorial and algorithmic reasoning capabilities in large sequence models\, leading to dramatic advances in natural language understanding\, program synthesis\, and theorem proving. What is the nature of these models’ internal representations (i.e. how do they represent the states and computational steps of the algorithms they execute)? How can we understand and mitigate their weaknesses\, given that they resist interpretation? In this work\, we present some insights (and many further mysteries) through the lens of automata and their algebraic structure. \nSpecifically\, we investigate the apparent mismatch between recurrent models of computation (automata & Turing machines) and Transformers (which are typically shallow and non-recurrent). Using tools from circuit complexity and semigroup theory\, we characterize shortcut solutions\, whereby a shallow Transformer with only o(T) layers can exactly replicate T computational steps of an automaton. We show that Transformers can efficiently represent these shortcuts in theory; furthermore\, in synthetic experiments\, standard training successfully finds these shortcuts. We demonstrate that shortcuts can lead to statistical brittleness\, and discuss mitigations. \nJoint work with Bingbin Liu\, Jordan Ash\, Surbhi Goel\, and Akshay Krishnamurthy.
URL:https://cmsa.fas.harvard.edu/event/nt-12722/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:New Technologies in Mathematics Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/12.07.2022.png
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