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Last week, we had Peter Scholze give an interesting distinguished lecture series here at UCLA on “Prismatic Cohomology”, which is a new type of cohomology theory worked out by Scholze and Bhargav Bhatt. (Video of the talks will be available shortly; for now we have some notes taken by two note–takers in the audience on that web page.) My understanding of this (speaking as someone that is rather far removed from this area) is that it is progress towards the “motivic” dream of being able to define cohomology 
- Singular cohomology, which roughly speaking works when the domain ring
or
, but can allow for arbitrary coefficients
- de Rham cohomology, which roughly speaking works as long as the coefficient ring
-adic cohomology, which is a remarkably powerful application of étale cohomology, but only works well when the coefficient ring
is localised around a prime
of the domain ring
- Crystalline cohomology, in which the domain ring is a field
of some finite characteristic
There are various relationships between the cohomology theories, for instance de Rham cohomology coincides with singular cohomology for smooth varieties in the limiting case 
The new prismatic cohomology of Bhatt and Scholze unifies many of these cohomologies in the “neighbourhood” of the point 
To define prismatic cohomology rings 
(And yes, Peter confirmed that he and Bhargav were inspired by the Dark Side of the Moon album cover in selecting the terminology.)
There was an abstract definition of prismatic cohomology (as being the essentially unique cohomology arising from prisms that obeyed certain natural axioms), but there was also a more concrete way to view them in terms of coordinates, as a “
prismatic cohomology in coordinates can be computed using a “
![\displaystyle d_q (t^n) = [n]_q t^{n-1} d_q t](https://cdn.statically.io/img/s0.wp.com/latex.php?latex=%5Cdisplaystyle+d_q+%28t%5En%29+%3D+%5Bn%5D_q+t%5E%7Bn-1%7D+d_q+t&bg=ffffff&fg=000000&s=0&c=20201002)
where
![\displaystyle [n]_q = \frac{q^n-1}{q-1} = 1 + q + \dots + q^{n-1}](https://cdn.statically.io/img/s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Bn%5D_q+%3D+%5Cfrac%7Bq%5En-1%7D%7Bq-1%7D+%3D+1+%2B+q+%2B+%5Cdots+%2B+q%5E%7Bn-1%7D&bg=ffffff&fg=000000&s=0&c=20201002)
is the “
This week at UCLA, Pierre-Louis Lions gave one of this year’s Distinguished Lecture Series, on the topic of mean field games. These are a relatively novel class of systems of partial differential equations, that are used to understand the behaviour of multiple agents each individually trying to optimise their position in space and time, but with their preferences being partly determined by the choices of all the other agents, in the asymptotic limit when the number of agents goes to infinity. A good example here is that of traffic congestion: as a first approximation, each agent wishes to get from A to B in the shortest path possible, but the speed at which one can travel depends on the density of other agents in the area. A more light-hearted example is that of a Mexican wave (or audience wave), which can be modeled by a system of this type, in which each agent chooses to stand, sit, or be in an intermediate position based on his or her comfort level, and also on the position of nearby agents.
Under some assumptions, mean field games can be expressed as a coupled system of two equations, a Fokker-Planck type equation evolving forward in time that governs the evolution of the density function 
Due to lack of time and preparation, I was not able to transcribe Lions’ lectures in full detail; but I thought I would describe here a heuristic derivation of the mean field game equations, and mention some of the results that Lions and his co-authors have been working on. (Video of a related series of lectures (in French) by Lions on this topic at the Collége de France is available here.)
To avoid (rather important) technical issues, I will work at a heuristic level only, ignoring issues of smoothness, convergence, existence and uniqueness, etc.
In his final lecture, Prof. Margulis talked about some of the ideas around the theory of unipotent flows on homogeneous spaces, culminating in the orbit closure, equidsitribution, and measure classification theorems of Ratner in the subject. Margulis also discussed the application to metric theory of Diophantine approximation which was not covered in the preceding lecture.
Today, Prof. Margulis continued his lecture series, focusing on two specific examples of homogeneous dynamics applications to number theory, namely counting lattice points on algebraic varieties, and quantitative versions of the Oppenheim conjecture. (Due to lack of time, the third application mentioned in the previous lecture, namely metric theory of Diophantine approximation, was not covered.)
The final distinguished lecture series for the academic year here at UCLA is being given this week by Gregory Margulis, who is giving three lectures on “homogeneous dynamics and number theory”. In his first lecture, Prof. Margulis surveyed some classical problems in number theory that turn out, rather surprisingly, to have more or less equivalent counterparts in homogeneous dynamics – the theory of dynamical systems on homogeneous spaces 
As usual, any errors in this post are due to my transcription of the talk.
In the third of the Distinguished Lecture Series given by Eli Stein here at UCLA, Eli presented a slightly different topic, which is work in preparation with Alex Nagel, Fulvio Ricci, and Steve Wainger, on algebras of singular integral operators which are sensitive to multiple different geometries in a nilpotent Lie group.
In the second of the Distinguished Lecture Series given by Eli Stein here at UCLA, Eli expanded on the themes in the first lecture, in particular providing more details as to the recent (not yet published) results of Lanzani and Stein on the boundedness of the Cauchy integral on domains in several complex variables.
The first Distinguished Lecture Series at UCLA for this academic year is given by Elias Stein (who, incidentally, was my graduate student advisor), who is lecturing on “Singular Integrals and Several Complex Variables: Some New Perspectives“. The first lecture was a historical (and non-technical) survey of modern harmonic analysis (which, amazingly, was compressed into half an hour), followed by an introduction as to how this theory is currently in the process of being adapted to handle the basic analytical issues in several complex variables, a topic which in many ways is still only now being developed. The second and third lectures will focus on these issues in greater depth.
As usual, any errors here are due to my transcription and interpretation of the lecture.
[Update, Oct 27: The slides from the talk are now available here.]
Avi Wigderson‘s final talk in his Distinguished Lecture Series on “Computational complexity” was entitled “Arithmetic computation“; the complexity theory of arithmetic circuits rather than boolean circuits.
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