Duality in random phenomena.

19.07.2023, 14:00  –  Campus Golm, Building 9, Room 2.22 and via Zoom

Adrian Gonzalez Casanova (UNAM, Mexico and UC Berkeley, USA), Maite Wilke Berenguer (HU Berlin)

14:00 Adrian Gonzalez Casanova (UNAM, Mexico and UC Berkeley, USA)

14:45 Tea and Coffee Break

15:15 Maite Wilke Berenguer (HU Berlin)

Adrian Gonzalez Casanova (UNAM, Mexico and UC Berkeley, USA): Duality in the context of random processes.

Abstract: Heuristically, two processes are dual if one can find a function to study one process by using the other. Sampling duality is a duality which uses a duality function S(n,x) of the form "what is the probability that all the members of a sample of size n are of a certain type, given that the number (or frequency) of that type of individuals is x". Implicitly, this technique can be traced back to the work of Blaise Pascal. Explicitly, it was studied in a paper of Martin Möhle in 1999 in the context of population genetics. We will discuss examples for which this technique proves to be useful, including applications to biology and physics.

Maite Wilke Berenguer (HU Berlin): A (very) brief introduction to population genetics.

Abstract: Mathematical population genetics resulted from trying to "elevate" biology to the standards of the physical sciences by basing it on mathematical modeling and empirical testing. At the heart of this idea is the Wright-Fisher model  - a rather simple, yet as we will see, powerful model describing the (random!) evolution of a population. Starting with an individual based model given by a random graph, we derive two relevant processes, both of which are so-called Markov chains and lead to beautiful (stochastic) processes when appropriately rescaled in time and space. The frequency leads to a Markov process  (called the Wright-Fisher Diffusion)  and the genealogy yields a partition-valued Markov chain (called Kingman's coalescent). These two very different objects are closely related through Markovian duality, which allows us to control the more complex object by using the simpler one. This powerful set-up is preserved when we (adequately) include phenomena such as dormancy or selection!
This talk, which has close ties to the preceding talk by Adrian Gonzalez-Casanova, is nevertheless self-contained.


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