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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 notetakers 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 ${H^i(X/\overline{A}, A)}$ for varieties ${X}$ (or similar objects) defined over arbitrary commutative rings ${\overline{A}}$, and with coefficients in another arbitrary commutative ring ${A}$. Currently, we have various flavours of cohomology that only work for certain types of domain rings ${\overline{A}}$ and coefficient rings ${A}$:

• Singular cohomology, which roughly speaking works when the domain ring ${\overline{A}}$ is a characteristic zero field such as ${{\bf R}}$ or ${{\bf C}}$, but can allow for arbitrary coefficients ${A}$;
• de Rham cohomology, which roughly speaking works as long as the coefficient ring ${A}$ is the same as the domain ring ${\overline{A}}$ (or a homomorphic image thereof), as one can only talk about ${A}$-valued differential forms if the underlying space is also defined over ${A}$;
• ${\ell}$-adic cohomology, which is a remarkably powerful application of étale cohomology, but only works well when the coefficient ring ${A = {\bf Z}_\ell}$ is localised around a prime ${\ell}$ that is different from the characteristic ${p}$ of the domain ring ${\overline{A}}$; and
• Crystalline cohomology, in which the domain ring is a field ${k}$ of some finite characteristic ${p}$, but the coefficient ring ${A}$ can be a slight deformation of ${k}$, such as the ring of Witt vectors of ${k}$.

There are various relationships between the cohomology theories, for instance de Rham cohomology coincides with singular cohomology for smooth varieties in the limiting case ${A=\overline{A} = {\bf R}}$. The following picture Scholze drew in his first lecture captures these sorts of relationships nicely:

The new prismatic cohomology of Bhatt and Scholze unifies many of these cohomologies in the “neighbourhood” of the point ${(p,p)}$ in the above diagram, in which the domain ring ${\overline{A}}$ and the coefficient ring ${A}$ are both thought of as being “close to characteristic ${p}$” in some sense, so that the dilates ${pA, pA'}$ of these rings is either zero, or “small”. For instance, the ${p}$-adic ring ${{\bf Z}_p}$ is technically of characteristic ${0}$, but ${p {\bf Z}_p}$ is a “small” ideal of ${{\bf Z}_p}$ (it consists of those elements of ${{\bf Z}_p}$ of ${p}$-adic valuation at most ${1/p}$), so one can think of ${{\bf Z}_p}$ as being “close to characteristic ${p}$” in some sense. Scholze drew a “zoomed in” version of the previous diagram to informally describe the types of rings ${A,A'}$ for which prismatic cohomology is effective:

To define prismatic cohomology rings ${H^i_\Delta(X/\overline{A}, A)}$ one needs a “prism”: a ring homomorphism from ${A}$ to ${\overline{A}}$ equipped with a “Frobenius-like” endomorphism ${\phi: A \to A}$ on ${A}$ obeying some axioms. By tuning these homomorphisms one can recover existing cohomology theories like crystalline or de Rham cohomology as special cases of prismatic cohomology. These specialisations are analogous to how a prism splits white light into various individual colours, giving rise to the terminology “prismatic”, and depicted by this further diagram of Scholze:

(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 “${q}$-deformation” of de Rham cohomology. Whereas in de Rham cohomology one worked with derivative operators ${d}$ that for instance applied to monomials ${t^n}$ by the usual formula

$\displaystyle d(t^n) = n t^{n-1} dt,$

prismatic cohomology in coordinates can be computed using a “${q}$-derivative” operator ${d_q}$ that for instance applies to monomials ${t^n}$ by the formula

$\displaystyle d_q (t^n) = [n]_q t^{n-1} d_q t$

where

$\displaystyle [n]_q = \frac{q^n-1}{q-1} = 1 + q + \dots + q^{n-1}$

is the “${q}$-analogue” of ${n}$ (a polynomial in ${q}$ that equals ${n}$ in the limit ${q=1}$). (The ${q}$-analogues become more complicated for more general forms than these.) In this more concrete setting, the fact that prismatic cohomology is independent of the choice of coordinates apparently becomes quite a non-trivial theorem.

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 ${m}$ of the agents, and a Hamilton-Jacobi (or Hamilton-Jacobi-Bellman) type equation evolving backward in time that governs the computation of the optimal path for each agent. The combination of both forward propagation and backward propagation in time creates some unusual “elliptic” phenomena in the time variable that is not seen in more conventional evolution equations. For instance, for Mexican waves, this model predicts that such waves only form for stadiums exceeding a certain minimum size (and this phenomenon has apparently been confirmed experimentally!).

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 $G/\Gamma$.

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.

On Thursday, Avi Wigderson continued his Distinguished Lecture Series here at UCLA on computational complexity with his second lecture “Expander Graphs – Constructions and Applications“. As in the previous lecture, he spent some additional time after the talk on an “encore”, which in this case was how lossless expanders could be used to obtain rapidly decodable error-correcting codes.

The talk was largely based on these slides. Avi also has a recent monograph with Hoory and Linial on these topics. (For a brief introduction to expanders, I can also recommend Peter Sarnak’s Notices article. I also mention expanders to some extent in my third Milliman lecture.)