# Mazzocco’s Inaugural Lecture (Birmingham)

Professor Marta Mazzocco *Crooked Surfaces and Chewing Gum,* Inaugural Lecture, University of Birmingham, 27 March 2019

From the UoB Newsletter:

Professor Mazzocco is a specialist in the area of integrable systems … often motivated by mathematical physics. … Marta’s research brings geometry, quantum algebra and analysis together to describe and tackle problems which have so far resisted all other methods. … “I look forward to start a new group in Geometry and Mathematical Physics … ”.

… “I believe that … it is paramount to nurture interdisciplinary research as well as to raise the profile of the unique contribution that Mathematical Sciences brings to the overall research base, the economy and society”.

Marta aimed her lecture at her friends and family, on the grounds that mathematicians could always knock on her door (or read her papers). Thus it provided a rare bridge between the technical and the everyday, which prompted lots of musings in at least this audience member. So sorry if I missed much of what she actually said.

Some contemporaneous jottings:

- Euclid’s ‘Axioms’ are more properly called ‘Postulates’.
- I can think of many conversations with both technical and nontechnical people who are not specialist mathematicians where the use of the term ‘postulate’ rather than ‘axiom’ might have avoided some confusion.

- Many of life’s problems concern holes in spaces, giving rise to (mathematical representations with) negative Euler characteristics.
- This chimes with my view that many of life’s current problems are where conventional mainstream rationalities and logics are not at all, or perhaps not straightforwardly, applicable.

- If you take a chewing gum with a hole and stretch it, as topologists do then in the limit the gum will snap, and sometimes some properties are continuous across the limit.
- Wow! Can such a seemingly vital insight really have eluded generations of mathematical physicists? Isn’t it obvious when you think about it?

- In particular (complex) superimposed states in quantum physics can be thought of as combinations of nearby (simpler) base states.
- I can think of 1,000s of hours of conversations with mathematical physicists, formal and informal, that might have been more productive with this idea as an element.

- Quantization and non-commutativity in quantum mechanics arises as the result of constraints.
- With broader implications? (See below).

- Bordered Cusped Riemann Surfaces (somehow) show the limits of statistical topology.
- Something to look up!

Had I not had a train to catch, I might have commented – as is my wont – “Perhaps Turing was right”. This would have been largely out of habit, but there seems to me some sort of connection with his theory of morphogenesis. I also wondered about a link to the fictional work of C.L. Dodgson . (Only as I write this have I recalled that, like Mazzocco, Dodgson linked geometry and algebra!)

The link to my own interests is perhaps more tenuous, but perhaps something to think about. Conventional probability theory rests on the identity:

P(A^B) = P(B).P(A|B).

This leads to Bayes’ theorem via the implicit identity:

P(A^B) = P(B^A).

My main ‘beef’ about probability theory is that – much as I find with much quantum mechanics – I find the notation leads one into errors that only the greatest care can avoid, so that no-one (least of all me) seems able to compute probabilistically ‘live’ in a way that appears to naive observers to be sufficiently fluent to demonstrate any degree of competence. (How do people get through their vivas?) Interestingly (to me) Marta had not expected to have a white board available, and struggled to do the a very straightforward bit of algebra to demonstrate a seemingly obvious result: the familiar notation that she was using kept triggering inappropriate computational habits. But this only made her presentation the more enlightening, I thought.

Mazzocco’s approach also made me think of Whitehead’s process theory and in particular the idea that what we seem to be ‘observing’ or ‘measuring’ are only ‘emergent properties‘ that only indirect relate to properties of some underlying ‘stuff’, such that apparent discontinuities are artefacts of the (simplified) model, not breakdowns of reality.

Back to probability theory: for some time I have been wondering about value of developing a non-commutative variant. Now I think I have my justification. (At least for me.)