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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:

20190312_145136

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:

20190312_145157

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:

20190313_152011

(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.

 

Every four years at the International Congress of Mathematicians (ICM), the Fields Medal laureates are announced.     Today, at the 2018 ICM in Rio de Janeiro, it was announced that the Fields Medal was awarded to Caucher Birkar, Alessio Figalli, Peter Scholze, and Akshay Venkatesh.

After the two previous congresses in 2010 and 2014, I wrote blog posts describing some of the work of each of the winners.  This time, though, I happened to be a member of the Fields Medal selection committee, and as such had access to a large number of confidential letters and discussions about the candidates with the other committee members; in order to have the opinions and discussion as candid as possible, it was explicitly understood that these communications would not be publicly disclosed.  Because of this, I will unfortunately not be able to express much of a comment or opinion on the candidates or the process as an individual (as opposed to a joint statement of the committee).  I can refer you instead to the formal citations of the laureates (which, as a committee member, I was involved in crafting, and then signing off on), or the profiles of the laureates by Quanta magazine; see also the short biographical videos of the laureates by the Simons Foundation that accompanied the formal announcements of the winners. I am sure, though, that there will be plenty of other mathematicians who will be able to present the work of each of the medalists (for instance, there was a laudatio given at the ICM for each of the winners, which should eventually be made available at this link).

I know that there is a substantial amount of interest in finding out more about the inner workings of the Fields Medal selection process.  For the reasons stated above, I as an individual will unfortunately be unable to answer any questions about this process (e.g., I cannot reveal any information about other nominees, or of any comparisons between any two candidates or nominees).  I think I can safely express the following two personal opinions though.  Firstly, while I have served on many prize committees in the past, the process for the Fields Medal committee was by far the most thorough and deliberate of any I have been part of, and I for one learned an astonishing amount about the mathematical work of all of the shortlisted nominees, which was an absolutely essential component of the deliberations, in particular giving the discussions a context which would have been very difficult to obtain for an individual mathematician not in possession of all the confidential letters, presentations, and other information available to the committee (in particular, some of my preconceived impressions about the nominees going into the process had to be corrected in light of this more complete information).  Secondly, I believe the four medalists are all extremely deserving recipients of the prize, and I fully stand by the decision of the committee to award the Fields medals this year to these four.

I’ll leave the comments to this post open for anyone who wishes to discuss the work of the medalists.  But, for the reasons above, I will not participate in the discussion myself.

[Edit, Aug 1: looks like the ICM site is (barely) up and running now, so links have been added.  At this time of writing, there does not seem to be an online announcement of the composition of the committee, but this should appear in due course. -T.]

[Edit, Aug 9: the composition of the Fields Medal Committee for 2018 (which included myself) can be found here. -T.]

 

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