Tonight, my great and good friends at the Secret Science Club hosted a Zoom lecture featuring mathemetician Dr Jordan Elleberg of the University of Wisconsin, Madison. Dr Ellenberg's newly published book is Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else.
Dr Ellebberg began his lecture by noting that everything is connected, showing a 'map' of connections of topics in his book. Ronald Ross was a physician who determined that malaria was transmitted by mosquitos. Ross was an indifferent doctor, but had a love for mathematics, and he applied mathematical models to epidemiology. Ross wanted to formulate a theory of phenomena, starting with epidemics. His work in this field was the beginning of mathematical modeling. He applied it to the problem of malaria... eliminating malaria would involve eliminating mosquitos, which is impossible. Mosquitos can be temporarily eliminated from an area- how long would it take for them to repopulate an area. Mosquitos do not move in predetermined fashion, they move largely at random. Ross enlisted mathemetician Karl Pearson to couch a model of mosquito repopulation of an area in neutral terms, removing references to insects- The Problem of the Random Walk.
Botanist Robert Brown noticed the movement of particles in a medium, which became known as Brownian motion- he wondered if is it a vital life principle, noticing it in pollen first. Brown tested it on organic and non-organic materials (including a 'fragment of the Sphinx'). Albert Einstein noted that the molecules are colliding, causing this movement. This motion can be figured only on a basis of probability.
Russian mathematician Andrey Markov had a reputation for being furious- he was angry that Tolstoy was excommunicated while he was not, so he ended up being excommunicated as well. He approached the problem of the Law of Large Numbers, which basically states that if one were to flip a coin numerous times, the more times it is flipped, the probability of heads and tails approach fifty percent increases. Flip a coin ten times, there is a good chance there will be six heads and four tails... flipping one thousand coins, having six hundred heads and four hundred tails would be less probable. Markov formulated the concept of the Markov Chain. He applied the Markov chain to determine the sequence of vowels and consonants in Pushkin's poem Eugene Onegin
Dr Ellengram then played around with bigrams- what letters are likely to follow other letters? He mentioned playing with a computer game called AI Dungeon which can be used to generate texts. He presented an artificial intelligence generated text about geometry- not quite convincing, but with an occasional flash of brilliance such as: "But squares aren't just shapes, they're also numbers!" Can machines replace humans? There is a line of difficulty from, say Tic Tac To to a perfect Go game- computers aren't smarter if they can beat humans at chess or Go, it's a one dimensional difficulty issue... a robot may beat a human at chess, but it can't fold a shirt. Machines will be great collaborators for us- we must determine which tasks they can outperform us in. Dr Ellenberg hopes they can be capable partners.
The lecture was followed by a Q&A session, which began with a question about gerrymandering- new districts are going to be drawn, this gives the people who draw these lines great power over who gets elected. In Wisconsin, the current legislators draw the maps, which is a problem. Legislators are given the keys to thwarting the electorate. Districting is a geometric problem- there are districts which look like 'polyamorous octopuses'. Mathematical tests can determine how bad gerrymandering is.
How many holes in a straw? It depends.
Dr Ellenberg criticized a mathematical approach which separates the subject into discrete courses of study- mathematical fields are connected.
The Random Walk is a probability problem, but also a geometry problem. Dr Ellenberg sees most math as having a geometric component.
Was Lewis Carroll aware of Bigrams? Jabberwocky seems to hint that he was... his fake words sound plausible, but Dr Ellenberg wasn't sure if he were aware of bigrams... it would be a great fake theory to promulgate, Dr Ellenberg joked.
Squaring the circle- problem for the ancient Greeks, could a square be created with the same area as a circle? It became a symbol of a difficult problem. Lincoln, a geometry enthusiast, used this metaphor to express difficulty.
Mathematics is built one the one hand on rigid reasoning and on the other hand on intuition. Geometry is based on our bodies, our two-dimensional field of perception and our three-dimensional space.
Regarding internet searches, the search engines use a random walk process to determine the priority of search results.
Regarding the use of math to map pandemics, Dr Ellenberg referred back to Ross attempt to formulate a theory of phenomena. People move, pathogens are transmitted, this is a geometric problem.
Once again, the Secret Science Club has dished out another fantastic lecture, a humorous deep dive into esoteric topics. Kudos to Dr Ellenberg, Margaret and Dorian. For a small taste of the Secret Science Club experience, here is the Good Doctor speaking on the subject of his new book:
Pour yourself a nice beverage, sit back, and soak in that SCIENCE!!!