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Just a brief update to the previous post. Gerhard Paseman and I have now set up a web site for the Short Communication Satellite (SCS) for the virtual International Congress of Mathematicians (ICM), which will be an experimental independent online satellite event in which short communications on topics relevant to one or two of the sections of the ICM can be submitted, reviewed by peers, and (if appropriate for the SCS event) displayed in a virtual “poster room” during the Congress on July 6-14 (which, by the way, has recently released its schedule and list of speakers). Our plan is to open the registration for this event on April 5, and start taking submissions on April 20; we are also currently accepting any expressions of interest in helping out with the event, for instance by serving as a reviewer. For more information about the event, please see the overview page, the guidelines page, and the FAQ page of the web site. As viewers will see, the web site is still somewhat under construction, but will be updated as we move closer to the actual Congress.

The comments section of this post would be a suitable place to ask further questions about this event, or give any additional feedback.

UPDATE: for readers who have difficulty accessing the links above, here are backup copies of the overview page and guidelines page.

[As with previous posts regarding ICM satellite events, I am authoring this post as an individual, and not in my capacity as chair of the ICM Structure Committee, which does not have any advisory or supervisory role over ICM satellite events – T.]

One of the traditional features of the International Congress of Mathematicians are the “short communications”, organized by the local organizing committee (as opposed to the International Mathematical Union), which allows participants at the congress to present either a poster or a short talk (typically 15 minutes or so) during the congress. For instance, here are the titles of the short communications and posters from the 2018 ICM, and here are the short communications and posters from the 2014 ICM. While not as high profile as other events of the ICM such as the plenary lectures, sectional lectures, or prize lectures, the short communications and posters can offer a chance for academics from a quite diverse range of institutions worldwide (and even a few independent mathematicians) be able to present their work to a mathematical audience.

There has been some volunteer effort to try to replicate some form of this event for the upcoming virtual ICM this July as a semi-official “satellite” event of the virtual ICM; it would technically not be part of the core ICM program, but I expect it would be recognized by the IMU as an independently organized satellite. Due to lack of time, funding, and technical expertise, we will not be able to offer any video, audio, or physical hosting for such an event, but we believe that a modest virtual event is possible involving submission of either a PDF “poster” or a PDF “slide deck”, together with other metadata such as author, title, abstract, and external links (e.g., to an externally hosted video presentation of the poster or slides), with some reviewing to ensure a certain minimum level of quality of approved submissions (we are thinking about setting guidelines similar to those required for a submission to the arXiv), and some ability to offer feedback on each submission. (For instance, we are thinking of hosting the event on a MediaWiki, with each communication being given a separate page which can attract discussion and responses to queries from the author(s).) We are also thinking of grouping the poster or slides according to the 20 sections of the 2022 ICM. We would then promote these communications during the virtual ICM, for instance on this blog or on the unofficial ICM Discord. Perhaps some of the other proposed online experiments for virtual events discussed in this previous post could also be implemented experimentally on this satellite event to demonstrate proof-of-concept. (If the event turns out to be successful, one could hope that it could serve as a pilot project for a longer-term and better funded platform for virtual short communications to accompany other conferences, but for now we would like to focus just on the virtual ICM satellite event.)

As one of our first actions, we would like to survey the level of interest in such an event, both among potential submitters of posters or slides, and also potential volunteers to help organize the event (in particular we may need some assistance in manually reviewing submissions, though we do plan to enlist peer reviewers by requiring submitters to rate and comment on other submissions in the same section). We have therefore created a form to (very unscientifically) gauge this level in order to decide on the scale of this project (or whether to attempt it at all). All readers of this blog are welcome to offer feedback through that form, or as a comment to this blog.

EDIT (Mar 29): a formal announcement will be made soon, but you can view a draft of the announcement here.

[Note: while I happen to be the chair of the ICM Structure Committee, I am authoring this blog post as an individual, and not as a representative of that committee or of the IMU, as they do not have jurisdiction over satellite conferences. -T.]

The International Mathematical Union (IMU) has just released some updates on the status of the 2022 International Congress of Mathematicians (ICM), which was discussed in this previous post:

• The General Assembly will take place in person in Helsinki, Finland, on July 3-4.
• The IMU award ceremony will be held in the same location on July 5.
• The ICM will take place virtually (with free participation) during the hours 9:00-18:00 CEST of July 6-14, with talks either live or pre-recorded according to speaker preference.

Due to the limited time and resources available, the core ICM program will be kept to the bare essentials; the lectures will be streamed but without opportunity for questions or other audience feedback. However, the IMU encourages grassroots efforts to supplement the core program with additional satellite activities, both “traditional” and “non-traditional”. Examples of such satellite activities include:

A more updated list of these events can be found here.

I will also mention the second Azat Miftakov Days, which are unaffiliated with the ICM but held concurrently with the beginning of the congress (and the prize ceremony).

Strictly speaking, satellite events are not officially part of the Congress, and not directly subject to IMU oversight; they also receive no funding or support from the IMU, other than sharing of basic logistical information, and recognition of the satellite conferences on the ICM web site. Thus this (very exceptional and sui generis) congress will differ in format from previous congresses, in that many of the features of the congress that traditionally were managed by the local organizing committee will be outsourced this one time to the broader mathematical community in a highly decentralized fashion.

In order to coordinate the various grassroots efforts to establish such satellite events, Alexei Borodin, Martin Hairer, and myself have set up a satellite coordination group to share information and advice on these events. (It should be noted that while Alexei, Martin and myself serve on either the structure committee or the program committee of the ICM, we are acting here as individuals rather than as official representatives of the IMU.) Anyone who is interested in organizing, hosting, or supporting such an event is welcome to ask to join the group (though I should say that most of the discussion concerns boring logistical issues). Readers are also welcome to discuss broader issues concerning satellites, or the congress as a whole, in the comments to this post. I will also use this space to announce details of satellite events as they become available (most are currently still only in the early stages of planning).

Jan Grebik, Rachel Greenfeld, Vaclav Rozhon and I have just uploaded to the arXiv our preprint “Measurable tilings by abelian group actions“. This paper is related to an earlier paper of Rachel Greenfeld and myself concerning tilings of lattices ${{\bf Z}^d}$, but now we consider the more general situation of tiling a measure space ${X}$ by a tile ${A \subset X}$ shifted by a finite subset ${F}$ of shifts of an abelian group ${G = (G,+)}$ that acts in a measure-preserving (or at least quasi-measure-preserving) fashion on ${X}$. For instance, ${X}$ could be a torus ${{\bf T}^d = {\bf R}^d/{\bf Z}^d}$, ${A}$ could be a positive measure subset of that torus, and ${G}$ could be the group ${{\bf R}^d}$, acting on ${X}$ by translation.

If ${F}$ is a finite subset of ${G}$ with the property that the translates ${f+A}$, ${f \in F}$ of ${A \subset X}$ partition ${X}$ up to null sets, we write ${F \oplus A =_{a.e.} X}$, and refer to this as a measurable tiling of ${X}$ by ${A}$ (with tiling set ${F}$). For instance, if ${X}$ is the torus ${{\bf T}^2}$, we can create a measurable tiling with ${A = [0,1/2]^2 \hbox{ mod } {\bf Z}^2}$ and ${F = \{0,1/2\}^2}$. Our main results are the following:

• By modifying arguments from previous papers (including the one with Greenfeld mentioned above), we can establish the following “dilation lemma”: a measurable tiling ${F \oplus A =_{a.e.} X}$ automatically implies further measurable tilings ${rF \oplus A =_{a.e.} X}$, whenever ${r}$ is an integer coprime to all primes up to the cardinality ${\# F}$ of ${F}$.
• By averaging the above dilation lemma, we can also establish a “structure theorem” that decomposes the indicator function ${1_A}$ of ${A}$ into components, each of which are invariant with respect to a certain shift in ${G}$. We can establish this theorem in the case of measure-preserving actions on probability spaces via the ergodic theorem, but one can also generalize to other settings by using the device of “measurable medial means” (which relates to the concept of a universally measurable set).
• By applying this structure theorem, we can show that all measurable tilings ${F \oplus A = {\bf T}^1}$ of the one-dimensional torus ${{\bf T}^1}$ are rational, in the sense that ${F}$ lies in a coset of the rationals ${{\bf Q} = {\bf Q}^1}$. This answers a recent conjecture of Conley, Grebik, and Pikhurko; we also give an alternate proof of this conjecture using some previous results of Lagarias and Wang.
• For tilings ${F \oplus A = {\bf T}^d}$ of higher-dimensional tori, the tiling need not be rational. However, we can show that we can “slide” the tiling to be rational by giving each translate ${f + A}$ of ${A}$ a “velocity” ${v_f \in {\bf R}^d}$, and for every time ${t}$, the translates ${f + tv_f + A}$ still form a partition of ${{\bf T}^d}$ modulo null sets, and at time ${t=1}$ the tiling becomes rational. In particular, if a set ${A}$ can tile a torus in an irrational fashion, then it must also be able to tile the torus in a rational fashion.
• In the two-dimensional case ${d=2}$ one can arrange matters so that all the velocities ${v_f}$ are parallel. If we furthermore assume that the tile ${A}$ is connected, we can also show that the union of all the translates ${f+A}$ with a common velocity ${v_f = v}$ form a ${v}$-invariant subset of the torus.
• Finally, we show that tilings ${F \oplus A = {\bf Z}^d \times G}$ of a finitely generated discrete group ${{\bf Z}^d \times G}$, with ${G}$ a finite group, cannot be constructed in a “local” fashion (we formalize this probabilistically using the notion of a “factor of iid process”) unless the tile ${F}$ is contained in a single coset of ${\{0\} \times G}$. (Nonabelian local tilings, for instance of the sphere by rotations, are of interest due to connections with the Banach-Tarski paradox; see the aforementioned paper of Conley, Grebik, and Pikhurko. Unfortunately, our methods seem to break down completely in the nonabelian case.)

In this post I would like to collect a list of resources that are available to mathematicians displaced by conflict. Here are some general resources:

There are also resources specific to the current crisis:

Finally, there are a number of institutes and departments who are willing to extend visiting or adjunct positions to such displaced mathematicians:

If readers have other such resources to contribute (or to update the ones already listed), please do so in the comments and I will modify the above lists as appropriate.

As with the previous post, any purely political comment not focused on such resources will be considered off-topic and thus subject to deletion.