You are currently browsing the monthly archive for February 2019.

Now that Google Plus is closing, the brief announcements that I used to post over there will now be migrated over to this blog.  (Some people have suggested other platforms for this also, such as Twitter, but I think for now I can use my existing blog to accommodate these sorts of short posts.)

  1. The NSF-CBMS regional research conferences are now requesting proposals for the 2020 conference series.  (I was the principal lecturer for one of these conferences back in 2005; it was a very intensive experience, but quite enjoyable, and I am quite pleased with the book that resulted from it.)
  2. The awardees for the Sloan Fellowships for 2019 have now been announced.  (I was on the committee for the mathematics awards.  For the usual reasons involving the confidentiality of letters of reference and other sensitive information, I will be unfortunately be unable to answer any specific questions about our committee deliberations.)

I have just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows“, submitted to Pure and Applied Functional Analysis. This paper continues my attempts to establish “universality” properties of the Euler equations on Riemannian manifolds {(M,g)}, as I conjecture that the freedom to set the metric {g} ought to allow one to “program” such Euler flows to exhibit a wide range of behaviour, and in particular to achieve finite time blowup (if the dimension is sufficiently large, at least).

In coordinates, the Euler equations read

\displaystyle \partial_t u^k + u^j \nabla_j u^k = - \nabla^k p \ \ \ \ \ (1)

 

\displaystyle \nabla_k u^k = 0

where {p: [0,T] \rightarrow C^\infty(M)} is the pressure field and {u: [0,T] \rightarrow \Gamma(TM)} is the velocity field, and {\nabla} denotes the Levi-Civita connection with the usual Penrose abstract index notation conventions; we restrict attention here to the case where {u,p} are smooth and {M} is compact, smooth, orientable, connected, and without boundary. Let’s call {u} an Euler flow on {M} (for the time interval {[0,T]}) if it solves the above system of equations for some pressure {p}, and an incompressible flow if it just obeys the divergence-free relation {\nabla_k u^k=0}. Thus every Euler flow is an incompressible flow, but the converse is certainly not true; for instance the various conservation laws of the Euler equation, such as conservation of energy, will already block most incompressible flows from being an Euler flow, or even being approximated in a reasonably strong topology by such Euler flows.

However, one can ask if an incompressible flow can be extended to an Euler flow by adding some additional dimensions to {M}. In my paper, I formalise this by considering warped products {\tilde M} of {M} which (as a smooth manifold) are products {\tilde M = M \times ({\bf R}/{\bf Z})^m} of {M} with a torus, with a metric {\tilde g} given by

\displaystyle d \tilde g^2 = g_{ij}(x) dx^i dx^j + \sum_{s=1}^m \tilde g_{ss}(x) (d\theta^s)^2

for {(x,\theta) \in \tilde M}, where {\theta^1,\dots,\theta^m} are the coordinates of the torus {({\bf R}/{\bf Z})^m}, and {\tilde g_{ss}: M \rightarrow {\bf R}^+} are smooth positive coefficients for {s=1,\dots,m}; in order to preserve the incompressibility condition, we also require the volume preservation property

\displaystyle \prod_{s=1}^m \tilde g_{ss}(x) = 1 \ \ \ \ \ (2)

though in practice we can quickly dispose of this condition by adding one further “dummy” dimension to the torus {({\bf R}/{\bf Z})^m}. We say that an incompressible flow {u} is extendible to an Euler flow if there exists a warped product {\tilde M} extending {M}, and an Euler flow {\tilde u} on {\tilde M} of the form

\displaystyle \tilde u(t,(x,\theta)) = u^i(t,x) \frac{d}{dx^i} + \sum_{s=1}^m \tilde u^s(t,x) \frac{d}{d\theta^s}

for some “swirl” fields {\tilde u^s: [0,T] \times M \rightarrow {\bf R}}. The situation here is motivated by the familiar situation of studying axisymmetric Euler flows {\tilde u} on {{\bf R}^3}, which in cylindrical coordinates take the form

\displaystyle \tilde u(t,(r,z,\theta)) = u^r(t,r,z) \frac{d}{dr} + u^z(t,r,z) \frac{d}{dz} + \tilde u^\theta(t,r,z) \frac{d}{d\theta}.

The base component

\displaystyle u^r(t,r,z) \frac{d}{dr} + u^z(t,r,z) \frac{d}{dz}

of this flow is then a flow on the two-dimensional {r,z} plane which is not quite incompressible (due to the failure of the volume preservation condition (2) in this case) but still satisfies a system of equations (coupled with a passive scalar field {\rho} that is basically the square of the swirl {\tilde u^\rho}) that is reminiscent of the Boussinesq equations.

On a fixed {d}-dimensional manifold {(M,g)}, let {{\mathcal F}} denote the space of incompressible flows {u: [0,T] \rightarrow \Gamma(TM)}, equipped with the smooth topology (in spacetime), and let {{\mathcal E} \subset {\mathcal F}} denote the space of such flows that are extendible to Euler flows. Our main theorem is

Theorem 1

  • (i) (Generic inextendibility) Assume {d \geq 3}. Then {{\mathcal E}} is of the first category in {{\mathcal F}} (the countable union of nowhere dense sets in {{\mathcal F}}).
  • (ii) (Non-rigidity) Assume {M = ({\bf R}/{\bf Z})^d} (with an arbitrary metric {g}). Then {{\mathcal E}} is somewhere dense in {{\mathcal F}} (that is, the closure of {{\mathcal E}} has non-empty interior).

More informally, starting with an incompressible flow {u}, one usually cannot extend it to an Euler flow just by extending the manifold, warping the metric, and adding swirl coefficients, even if one is allowed to select the dimension of the extension, as well as the metric and coefficients, arbitrarily. However, many such flows can be perturbed to be extendible in such a manner (though different perturbations will require different extensions, in particular the dimension of the extension will not be fixed). Among other things, this means that conservation laws such as energy (or momentum, helicity, or circulation) no longer present an obstruction when one is allowed to perform an extension (basically this is because the swirl components of the extension can exchange energy (or momentum, etc.) with the base components in a basically arbitrary fashion.

These results fall short of my hopes to use the ability to extend the manifold to create universal behaviour in Euler flows, because of the fact that each flow requires a different extension in order to achieve the desired dynamics. Still it does seem to provide a little bit of support to the idea that high-dimensional Euler flows are quite “flexible” in their behaviour, though not completely so due to the generic inextendibility phenomenon. This flexibility reminds me a little bit of the flexibility of weak solutions to equations such as the Euler equations provided by the “{h}-principle” of Gromov and its variants (as discussed in these recent notes), although in this case the flexibility comes from adding additional dimensions, rather than by repeatedly adding high-frequency corrections to the solution.

The proof of part (i) of the theorem basically proceeds by a dimension counting argument (similar to that in the proof of Proposition 9 of these recent lecture notes of mine). Heuristically, the point is that an arbitrary incompressible flow {u} is essentially determined by {d-1} independent functions of space and time, whereas the warping factors {\tilde g_{ss}} are functions of space only, the pressure field is one function of space and time, and the swirl fields {u^s} are technically functions of both space and time, but have the same number of degrees of freedom as a function just of space, because they solve an evolution equation. When {d>2}, this means that there are fewer unknown functions of space and time than prescribed functions of space and time, which is the source of the generic inextendibility. This simple argument breaks down when {d=2}, but we do not know whether the claim is actually false in this case.

The proof of part (ii) proceeds by direct calculation of the effect of the warping factors and swirl velocities, which effectively create a forcing term (of Boussinesq type) in the first equation of (1) that is a combination of functions of the Eulerian spatial coordinates {x^i} (coming from the warping factors) and the Lagrangian spatial coordinates {a^\beta} (which arise from the swirl velocities, which are passively transported by the flow). In a non-empty open subset of {{\mathcal F}}, the combination of these coordinates becomes a non-degenerate set of coordinates for spacetime, and one can then use the Stone-Weierstrass theorem to conclude. The requirement that {M} be topologically a torus is a technical hypothesis in order to avoid topological obstructions such as the hairy ball theorem, but it may be that the hypothesis can be dropped (and it may in fact be true, in the {M = ({\bf R}/{\bf Z})^d} case at least, that {{\mathcal E}} is dense in all of {{\mathcal F}}, not just in a non-empty open subset).

 

Just a quick post to advertise two upcoming events sponsored by institutions I am affiliated with:

  1. The 2019 National Math Festival will be held in Washington D.C. on May 4 (together with some satellite events at other US cities).  This festival will have numerous games, events, films, and other activities, which are all free and open to the public.  (I am on the board of trustees of MSRI, which is one of the sponsors of the festival.)
  2. The Institute for Pure and Applied Mathematics (IPAM) is now accepting applications for its second Industrial Short Course for May 16-17 2019, with the topic of “Deep Learning and the Latest AI Algorithms“.  (I serve on the Scientific Advisory Board of this institute.)  This is an intensive course (in particular requiring active participation) aimed at industrial mathematicians involving both the theory and practice of deep learning and neural networks, taught by Xavier Bresson.   (Note: space is very limited, and there is also a registration fee of $2,000 for this course, which is expected to be in high demand.)

[This post is collectively authored by the ICM structure committee, whose membership includes myself, and is listed in full in the post below – T.]

The International Congress of Mathematicians (ICM) is widely considered to be the premier conference for mathematicians.  It is held every four years; for instance, the 2018 ICM was held in Rio de Janeiro, Brazil, and the 2022 ICM is to be held in Saint Petersburg, Russia.  The most high-profile event at the ICM is the awarding of the 10 or so prizes of the International Mathematical Union (IMU) such as the Fields Medal, and the lectures by the prize laureates; but there are also approximately twenty plenary lectures from leading experts across all mathematical disciplines, several public lectures of a less technical nature, about 180 more specialised invited lectures divided into about twenty section panels, each corresponding to a mathematical field (or range of fields), as well as various outreach and social activities, exhibits and satellite programs, and meetings of the IMU General Assembly; see for instance the program for the 2018 ICM for a sample schedule.  In addition to these official events, the ICM also provides more informal networking opportunities, in particular allowing mathematicians at all stages of career, and from all backgrounds and nationalities, to interact with each other.

For each Congress, a Program Committee (together with subcommittees for each section) is entrusted with the task of selecting who will give the lectures of the ICM (excluding the lectures by prize laureates, which are selected by separate prize committees); they also have decided how to appropriately subdivide the entire field of mathematics into sections.   Given the prestigious nature of invitations from the ICM to present a lecture, this has been an important and challenging task, but one for which past Program Committees have managed to fulfill in a largely satisfactory fashion.

Nevertheless, in the last few years there has been substantial discussion regarding ways in which the process for structuring the ICM and inviting lecturers could be further improved, for instance to reflect the fact that the distribution of mathematics across various fields has evolved over time.   At the 2018 ICM General Assembly meeting in Rio de Janeiro, a resolution was adopted to create a new Structure Committee to take on some of the responsibilities previously delegated to the Program Committee, focusing specifically on the structure of the scientific program.  On the other hand, the Structure Committee is not involved with the format for prize lectures, the selection of prize laureates, or the selection of plenary and sectional lecturers; these tasks are instead the responsibilities of other committees (the local Organizing Committee, the prize committees, and the Program Committee respectively).

The first Structure Committee was constituted on 1 Jan 2019, with the following members:

As one of our first actions, we on the committee are using this blog post to solicit input from the mathematical community regarding the topics within our remit.  Among the specific questions (in no particular order) for which we seek comments are the following:

  1. Are there suggestions to change the format of the ICM that would increase its value to the mathematical community?
  2. Are there suggestions to change the format of the ICM that would encourage greater participation and interest in attending, particularly with regards to junior researchers and mathematicians from developing countries?
  3. What is the correct balance between research and exposition in the lectures?  For instance, how strongly should one emphasize the importance of good exposition when selecting plenary and sectional speakers?  Should there be “Bourbaki style” expository talks presenting work not necessarily authored by the speaker?
  4. Is the balance between plenary talks, sectional talks, and public talks at an optimal level?  There is only a finite amount of space in the calendar, so any increase in the number or length of one of these types of talks will come at the expense of another.
  5. The ICM is generally perceived to be more important to pure mathematics than to applied mathematics.  In what ways can the ICM be made more relevant and attractive to applied mathematicians, or should one not try to do so?
  6. Are there structural barriers that cause certain areas or styles of mathematics (such as applied or interdisciplinary mathematics) or certain groups of mathematicians to be under-represented at the ICM?  What, if anything, can be done to mitigate these barriers?

Of course, we do not expect these complex and difficult questions to be resolved within this blog post, and debating these and other issues would likely be a major component of our internal committee discussions.  Nevertheless, we would value constructive comments towards the above questions (or on other topics within the scope of our committee) to help inform these subsequent discussions.  We therefore welcome and invite such commentary, either as responses to this blog post, or sent privately to one of the members of our committee.  We would also be interested in having readers share their personal experiences at past congresses, and how it compares with other major conferences of this type.   (But in order to keep the discussion focused and constructive, we request that comments here refrain from discussing topics that are out of the scope of this committee, such as suggesting specific potential speakers for the next congress, which is a task instead for the 2022 ICM Program Committee.)

While talking mathematics with a postdoc here at UCLA (March Boedihardjo) we came across the following matrix problem which we managed to solve, but the proof was cute and the process of discovering it was fun, so I thought I would present the problem here as a puzzle without revealing the solution for now.

The problem involves word maps on a matrix group, which for sake of discussion we will take to be the special orthogonal group SO(3) of real 3 \times 3 matrices (one of the smallest matrix groups that contains a copy of the free group, which incidentally is the key observation powering the Banach-Tarski paradox).  Given any abstract word w of two generators x,y and their inverses (i.e., an element of the free group {\bf F}_2), one can define the word map w: SO(3) \times SO(3) \to SO(3) simply by substituting a pair of matrices in SO(3) into these generators.  For instance, if one has the word w = x y x^{-2} y^2 x, then the corresponding word map w: SO(3) \times SO(3) \to SO(3) is given by

\displaystyle w(A,B) := ABA^{-2} B^2 A

for A,B \in SO(3).  Because SO(3) contains a copy of the free group, we see the word map is non-trivial (not equal to the identity) if and only if the word itself is nontrivial.

Anyway, here is the problem:

Problem. Does there exist a sequence w_1, w_2, \dots of non-trivial word maps w_n: SO(3) \times SO(3) \to SO(3) that converge uniformly to the identity map?

To put it another way, given any \varepsilon > 0, does there exist a non-trivial word w such that \|w(A,B) - 1 \| \leq \varepsilon for all A,B \in SO(3), where \| \| denotes (say) the operator norm, and 1 denotes the identity matrix in SO(3)?

As I said, I don’t want to spoil the fun of working out this problem, so I will leave it as a challenge. Readers are welcome to share their thoughts, partial solutions, or full solutions in the comments below.

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