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”.

## Schedule

**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**10-11-22**(*Alexander Oldenziel*) Sufficient statistics and the Koopman-Pitman-Darmois theorem (video, pocket).**24-11-22**(*Ben Gerraty*) Renormalisation II**8-12-22**(*Calin Lazaroiu*) RG flows and partial localisation.**15-12-22**(*Daniel Murfet*) Resolution II: Embedded resolution of a quadratic form