This is a working seminar, where we try to figure out the relationship between the following three sets of ideas:

• John Wentworth’s Natural Abstraction: there exist abstractions (relatively low-dimensional summaries which capture information relevant for prediction) which are “natural” in the sense that we should expect a wide variety of cognitive systems to converge on using them (taken from here). The original post seems to be this.
• Resolution of singularities in algebraic geometry, which exist by a famous theorem of Hironaka.
• Renormalisation of physical theories, typically quantum (field) theories, a procedure by which we move in the space of theories / Hamiltonians, ideally removing degrees of freedom that are “irrelevant” to the phenomena that we wish to study.

Some of the topics to be explored:

• Renormalisation of Ising models and diagonalisation of Hamiltonians (= finding a presentation of the dynamics that emphasises creation and annihilation operators for degrees of freedom that are “important”, i.e. have large eigenvalues).
• PCA (= finding a presentation of data that emphasises degrees of freedom that are “important” i.e. such that a scalar projection has large variance)

References:

• Emch-Liu “The Logic of Thermostatistical Physics” Section 13.2 The renormalisation program.
• Di Francesco-Mathieu-Sénéchal “Conformal Field Theory” Ch. 3, statistical mechanics.
• A tour of renormalisation by Simon Dedeo.
• C. Bergbauer, R. Brunetti, D. Kreimer “Renormalization and resolution of singularities” 2009.
• A. Pagano “Lecture notes on statistical mechanics”.
• M. Shanahan “Talking about large language models” 2022.

Schedule

• TBD (Calin Lazaroiu) RG flows and partial localisation.
• TBD (Daniel Murfet) Resolution II: Embedded resolution of a quadratic form

Past talks

• 29-9-22 (Alexander Oldenziel) Natural Abstraction I (video), notes.
• 13-10-22 (Daniel Murfet) Resolution I: Blowing up (video).
• 27-10-22 (Ben Gerraty) Renormalisation I: The Ising model (video)
• 10-11-22 (Alexander Oldenziel) Sufficient statistics and the Koopman-Pitman-Darmois theorem (video, pocket).
• 24-11-22 (Ben Gerraty) Renormalisation II (video)