The lecture was amazing, both in content and presentation. It was engaging for high school students, and stimulating for experts. I wish I had a video or a copy of the slides. Krieger is well known to some through her Numberphile videos on YouTube. Here are a few things I learned in the lecture.
Mathematics is the language of relationships and patterns.
We forget how even the concept of numbers is abstract. The notion of functions is even more so.
An underlying theme of the lecture was that of emergence: a simple rule describing the interactions between the components of a system lead to collective behaviour (complexity) of the whole system.
Examples were given from biological systems that raise the question: how does the system know to do this?
Swarms of starlings were shown in the short film, The art of flying by Jan van IJken.
How do they move in concert when there is no leader?
Other examples included ant bridges, an experiment with a slime mould that was able to replicate the Japanese transport network (here is the Canadian version), stripes and spots on animals (pattern formation explained with coupled reaction-diffusion equations by Alan Turing).
To illustrate how simple rules lead to complex behaviour, several cellular automata were demonstrated starting with Pascal's triangle and Sierpinski triangle. The latter was connected to biology through the pattern on the shell of a (poisonous) cone snail.
Rule 30 produces patterns similar to those found on the shell. It has periodic patterns such as stripes and aperiodic chaotic patterns.
It seems the new Cambridge train station also has this pattern!
Rule 184 can describe traffic including jamming for medium traffic densities.
The occurrence of a traffic jam does not depend on the initial state or a particular car, but only depends on the density of cars and the interaction (rule) between cars.
A nice video was shown of a traffic shockwave.
When water flows from a tap (faucet) and hits a flat sink bottom at right angles it may produce a "hydraulic jump" such as that shown below.
That is just the first half of the lecture. I may blog later about the second half which concerned chaos, defined as small initial changes leading to significant changes in outcome.
One of the most interesting things for me about the lecture was Krieger's claim that "Emergent complexity isn't everywhere. It can be hard to detect or confirm.'' i.e., just because we see complex behaviour (patterns) does not mean that it is due to emergence. In question time she said that this was in response to some of Wolfram's grand claims in A New Kind of Science, along the lines that everything (consciousness, gravity, continuity, free will, ...) could be explained in terms of discrete computational models such as cellular automata.
I think a more nuanced view is necessary. I agree, along with many others, that Wolfram's grand claims are not justified. But, I do not equate emergent complexity solely with simple rule-based computational models such as cellular automata. Different people do define emergence differently. For example, Sophia Kivelson and Steve Kivelson propose the following definition.
An emergent behavior of a physical system is a qualitative property that can only occur in the limit that the number of microscopic constituents tends to infinity.This would rule out classifying most of the phenomena described in the lecture as emergent. I disagree with this definition. On the other hand, I am not sure I agree with Krieger's claim. I do think almost anything interesting is emergent: consciousness, critical phenomena, the vacuum in quantum field theory, superconductivity, ...
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