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55:22 Institut des Hautes Études Scientifiques (IHÉS) English 2015

Probability sheaves

In the Articel "The dawning of the age of stochasticity", Tao observes that the probability theory concerns itself with properties that are \preserved with respect to extension of the underlying sample space", in much the same way that modern geometry concerns itself with properties that are invariant with respect to underlying symmetries. Reformulating this in category-theoretic language, probabilistic concepts organise themselves into presheaves over a category of sample spaces. In this talk, I observe that they further form sheaves, and I consider ramications of this observation. As a suitable category of sample spaces, I take the category of measure-preserving measurable maps (modulo almost sure equality) between standard (a.k.a. Lebesgue-Rokhlin) probability spaces. In this category, every cospan completes to a commutative square enjoying a universal conditional independence property. As a consequence, the category carries an atomic Grothendieck topology, whose sheaves can themselves be characterised in terms of conditional independence. Examples of such probability sheaves include sheaf representations of standard probability spaces (given by representables), sheaves of random variables, sheaves of probability measures (given by a general coend construction), and sheaves of orbits of ergodic group actions. In general, I argue that the resulting atomic topos of probability sheaves is a natural category of generalised probabilistic concepts. Moreover, as a boolean topos, it models a mathematical universe in which random variable occurs as a primitive rather than derived mathematical notion. I believe this model has the potential to inform the development of an alternative approach to probability theory founded on primitive random variables, somewhat along the lines envisaged by Mumford in.
  • Published: 2015
  • Publisher: Institut des Hautes Études Scientifiques (IHÉS)
  • Language: English
36:15 Heidelberg Laureate Forum Foundation English 2018

6th HLF – Interviews with mathematics and computer science laureates : Richard Manning Karp

Laureates at the 6th HLF sit down with Tom Geller, Tom Geller Productions, to discuss their career, mentoring and their experience at the Heidelberg Laureate Forum (HLF). These renowned scientists have been honored with most prestigious awards in mathematics and computer science: Abel Prize, ACM A.M. Turing Award, ACM Prize in Computing, Fields Medal and Nevanlinna Prize. The opinions expressed in this video do not necessarily reflect the views of the Heidelberg Laureate Forum Foundation or any other person or associated institution involved in the making and distribution of the video. Background: The Heidelberg Laureate Forum Foundation (HLFF) annually organizes the Heidelberg Laureate Forum (HLF), which is a networking event for mathematicians and computer scientists from all over the world. The HLFF was established and is funded by the German foundation the Klaus Tschira Stiftung (KTS), which promotes natural sciences, mathematics and computer science. The HLF is strongly supported by the award-granting institutions, the Association for Computing Machinery (ACM: ACM A.M. Turing Award, ACM Prize in Computing), the International Mathematical Union (IMU: Fields Medal, Nevanlinna Prize), and the Norwegian Academy of Science and Letters (DNVA: Abel Prize). The Scientific Partners of the HLFF are the Heidelberg Institute for Theoretical Studies (HITS) and Heidelberg University.
  • Published: 2018
  • Publisher: Heidelberg Laureate Forum Foundation
  • Language: English
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