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The day has come.

Enter a thread on isogenies, random walks and automorphism groups.

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I will explain some of the background we used to write these papers with Ben Smith, so I encourage you to go read them. There are some results at the end of the thread


The main objects in isogeny-based cryptography are elliptic curves and isogenies, usually defined over finite fields. And, of course, isogeny graphs.

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You might want to read about elliptic curves


First of all, an isogeny is a nonconstant morphism of elliptic curves fixing the point at infinity.

So it is a group morphism, a morphism of algebraic curves, it is surjective, and it has finite kernel.
One exciting thing from over the holidays is that the Int'l Civil Aviation Organization (@ICAO) and aircraft engine manufacturers for the first time publicly released quantitative emissions indices for particle # and mass emitted per unit fuel burn! 🧵 1/

https://t.co/zXuFN66SWe


Up until now, certification data for aircraft engine particle emissions have been reported in terms of a "smoke number", which is derived from the change in reflectance of a Whatman 4 filter after collecting 16.2 kg/m² exhaust. 2/

https://t.co/6fa9Kkdupr


One challenge with the smoke number measurement is that the filter doesn't capture all of the particles -- especially small ones. Is the decrease in smoke number over time because we are trading a few large soot particles for many more smaller soot particles? 3/

While smoke numbers are not particularly helpful for quantitatively assessing particle emissions impacts on air quality & upper tropospheric composition, they have been a huge success in motivating the reduction of unsightly exhaust plumes! 4/

Photo credit: flickr/dsleeter_2000


The new EI data are much more useful for #AirQuality and #Climate modeling efforts seeking to understand the environmental impacts of aviation. Since they're collected at ground conditions, the data are most relevant to AQ, but maybe can be extrapolated to cruise conditions? 5/
Check out this special feature of #ProcB edited by @bkoskella @gut_health and me. Thanks to the authors for their excellent article contributions - and all articles are #freetoread. Hopefully there is something of interest for all #microbiome researchers here @RSocPublishing


Application of ecological and evolutionary theory to microbiome community dynamics across systems | Proceedings of the Royal Society B: Biological Sciences https://t.co/LzzUDDE7Qg @jamesemcdonald
@gut_health
@bkoskella
@RSocPublishing

Transmission efficiency drives host–microbe associations | Proceedings of the Royal Society B: Biological Sciences https://t.co/bBdGS1XdTo
@PhilipLeftwich
Matthew P. Edgington
and Tracey Chapman

Host–microbiota–insect interactions drive emergent virulence in a complex tree disease | Proceedings of the Royal Society B: Biological Sciences https://t.co/RpwpxgHgY4
@clydeandforth1
Martin Broberg
Sandra Denman
@jamesemcdonald

A game theory model for gut bacterial nutrient utilization strategies during human infancy | Proceedings of the Royal Society B: Biological Sciences https://t.co/o3kIzv0ZbR
Inga Leena Angell
Knut Rudi