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Mark Keel, Tristan Roy, and I have just uploaded to the arXiv the paper “Global well-posedness for the Maxwell-Klein-Gordon equation below the energy norm“, submitted to Discrete and Continuous Dynamical Systems. This project started about eight years ago, and was in fact a partial result was essentially finished by 2002, but managed to get put on the backburner for a while due to many other priorities. Anyway, this paper applies the I-method to the Maxwell-Klein-Gordon system of equations in the Coulomb gauge (a simplified model for the hyperbolic Yang-Mills equations) in three spatial dimensions. Previously to this paper, it was known that the Cauchy problem was globally well-posed in the energy norm (which is essentially $H^1({\Bbb R}^3)$) and locally well-posed in $H^s({\Bbb R}^3)$ for $s > 1/2$, with this condition being essentially best possible except for the endpoint. Here we manage to lower the regularity threshold for global wellposedness to $s > \sqrt{3}/2 \approx 0.866$.  (The partial result alluded to earlier was for $s > 7/8 = 0.875$; at one point we had announced an improvement to $5/6 \approx 0.833$, but the argument turned out to be flawed.) This is part of what is now quite a large family of such “global well-posedness below the energy norm” results, but there are some notable technical features here which were not present in earlier works. Firstly, we can show that there is no smoothing effect in the nonlinearity, ruling out use of the Fourier truncation method. Secondly, due to our use of rescaling, supercritical quantities such as the $L^2$ norm are not under control, which necessitates some unusual treatment of the low frequency portions of the scalar and vector fields. Namely, they are estimated in $L^p$ spaces rather than $L^2$ ones. This complicates a number of tasks, ranging from controlling the elliptic theory, to understanding the coercive nature of the Hamiltonian, to establishing the nonlinear commutator estimates underlying the almost conservation law.

Richard Oberlin, Andreas Seeger, Christoph Thiele, Jim Wright, and I have just uploaded to the arXiv our paper “A variation norm Carleson theorem“, submitted to J. Europ. Math. Soc..

The celebrated Carleson-Hunt theorem asserts that if $f: {\Bbb R}/{\Bbb Z} \to {\Bbb C}$ is an $L^p$ function for some $1 < p \leq \infty$, then the partial Fourier series

$S_n f(x) := \sum_{k=-n}^n \hat f(k) e^{2\pi ikx}$

of $f$ converge to $f$ almost everywhere.  (The claim fails for $p=1$, as shown by a famous counterexample of Kolmogorov.)  The theorem follows easily from the inequality

$\| \sup_n |S_n f(x)| \|_{L^p({\Bbb R}/{\Bbb Z})} \leq C_p \| f \|_{L^p({\Bbb R}/{\Bbb Z})},$ (1)

where $1 < p < \infty$, and $C_p$ depends only on $p$.  Indeed, one first verifies Carleson’s theorem for a dense subclass of $L^p$ (e.g. the space of test functions) and then uses a standard limiting argument involving (1) (this is an example of the trick “give yourself an epsilon of room“).

The Carleson-Hunt theorem shows that $S_n f(x)$ converges as $n \to \infty$ for almost every $x$, but does not say much more about the nature of that convergence.  One way to measure the strength of the convergence is to introduce the variational norms

${\mathcal V}^r (S_n f(x))_{n = 0}^\infty := \sup_{n_1 \leq \ldots \leq n_k} (\sum_{j=1}^{k-1} |S_{n_{j+1}} f(x) - S_{n_j} f(x)|^r)^{1/r}$

for various $1 \leq r \leq \infty$.  For $r = \infty$ this is the Carleson maximal function $\sup_n |S_n f(x)|$; for $r=1$ this is the total variation of the sequence $S_n f(x)$, which one can verify to be the $\ell^1$ norm of $\hat f$.

Our main result is to obtain the following variational strengthening of (1)

$\| {\mathcal V}^r S_n f(x) \|_{L^p({\Bbb R}/{\Bbb Z})} \leq C_{p,r} \| f \|_{L^p({\Bbb R}/{\Bbb Z})}$ (2)

whenever $r > 2$ and $r/(r-1) < p < \infty$; these conditions on $p,r$ are optimal. (For those readers familiar with martingales, the relationship of (2) to (1) is analogous to the relationship between Lepingle’s inequality (a variant of the more well known Doob’s inequality) and the Hardy-Littlewood maximal inequality.)

Because a sequence with finite $r$-variation for some finite $r$ is necessarily convergent, this leads to a new proof of the Carleson-Hunt theorem without the need for a dense subclass.  In particular, we obtain ergodic theory analogues of this result, in the case where no obvious dense subclass is available; more precisely, we obtain a new (and more “quantitative”) proof of a Wiener-Wintner-type theorem (first obtained by Lacey and Terwilleger), namely that given any measure-preserving group $(T_t)_{t \in {\Bbb R}}$ on a measure space $X$, and a function $f \in L^p(X)$ for some $p>1$, one has for almost every $x \in X$ that for every real number $\theta$, the integrals $\int_{\varepsilon \leq |t| \leq N} T^t f(x) e^{i\theta t}/t\ dt$ converge as $\varepsilon \to 0, N \to \infty$ for every $\theta$ (not merely almost every $\theta$).

The estimate (2) also provides a new proof of a result of Christ and Kiselev on the almost everywhere boundedness of eigenfunctions of Schrodinger operators with $L^p$ potentials with $p<2$.  Unfortunately, due to various endpoint issues, this barely fails to settle the endpoint case $p=2$, a conjecture known as the nonlinear Carleson conjecture (discussed in this previous post).

The approach here follows the Lacey-Thiele approach to Carleson’s theorem (which is in turn based on an earlier approach of Fefferman), based on linearising the Carleson maximal function by picking the integer $n = n(x)$ which attains the supremum $\sup_n |S_n f(x)|$, dividing phase space into “tiles”, and organising these tiles into “trees” and then into “forests” based on the distribution of the phase space “energy” of $f$, together with the “mass” distribution of the graph of the function $x \mapsto n(x)$.  One then needs to combine various “Bessel” type bounds on the energy, “Vitali-type” bounds on the mass, and “Calderon-Zygmund” type estimates on the trees together to obtain the result.

In our setting, the main new difficulty is that there are multiple integers $n_1(x),\ldots,n_{k(x)}(x)$ associated to each point rather than one, which requires a more detailed analysis of the “multiplicity” of forests that was not present in earlier work.  (Also, the Calderon-Zygmund estimates need to be replaced with Lepingle type estimates, though this is a relatively standard change, being first introduced in a paper of Bourgain.)

Israel Gelfand, who made profound and prolific contributions to many areas of mathematics, including functional analysis, representation theory, operator algebras, and partial differential equations, died on Monday, age 96.

Gelfand’s beautiful theory of ${C^*}$-algebras and related spaces made quite an impact on me as a graduate student in Princeton, to the point where I was seriously considering working in this area; but there was not much activity in operator algebras at the time there, and I ended up working in harmonic analysis under Eli Stein instead. (Though I am currently involved in another operator algebras project, of which I hope to be able to discuss in the near future. The commutative version of Gelfand’s theory is discussed in these lecture notes of mine.)

I met Gelfand only once, in one of the famous “Gelfand seminars” at the IHES in 2000. The speaker was Tim Gowers, on his new proof of Szemerédi’s theorem. (Endre Szemerédi, incidentally, was Gelfand’s student.) Gelfand’s introduction to the seminar, on the subject of Banach spaces which both mathematicians contributed so greatly to, was approximately as long as Gowers’ talk itself!

There are far too many contributions to mathematics by Gelfand to name here, so I will only mention two. The first are the Gelfand-Tsetlin patterns, induced by an ${n \times n}$ Hermitian matrix ${A}$. Such matrices have ${n}$ real eigenvalues ${\lambda_{n,1} \leq \ldots \leq \lambda_{n,n}}$. If one takes the top ${n-1 \times n-1}$ minor, this is another Hermitian matrix, whose ${n-1}$ eigenvalues ${\lambda_{n-1,1} \leq \ldots \leq \lambda_{n-1,n-1}}$ intersperse the ${n}$ eigenvalues of the original matrix: ${\lambda_{n,i} \leq \lambda_{n-1,i} \leq \lambda_{n,i+1}}$ for every ${1 \leq i \leq n-1}$. This interspersing can be easily seen from the minimax characterisation

$\displaystyle \lambda_{n,i} = \inf_{\hbox{dim}(V)=i} \sup_{v \in V: \|v\|=1} \langle Av, v \rangle$

of the eigenvalues of ${A}$, with the eigenvalues of the ${n-1 \times n-1}$ minor being similar but with ${V}$ now restricted to a subspace of ${{\mathbb C}^{n-1}}$ rather than ${{\mathbb C}^n}$.

Similarly, the eigenvalues ${\lambda_{n-2,1} \leq \ldots \leq \lambda_{n-2,n-2}}$ of the top ${n-2 \times n-2}$ minor of ${A}$ intersperse those of the previous minor. Repeating this procedure one eventually gets a pyramid of real numbers of height and width ${n}$, with the numbers in each row interspersing the ones in the row below. Such a pattern is known as a Gelfand-Tsetlin pattern. The space of such patterns forms a convex cone, and (if one fixes the initial eigenvalues ${\lambda_{n,1},\ldots,\lambda_{n,n}}$) becomes a compact convex polytope. If one fixes the initial eigenvalues ${\lambda_{n,1},\ldots,\lambda_{n,n}}$ of ${A}$ but chooses the eigenvectors randomly (using the Haar measure of the unitary group), then the resulting Gelfand-Tsetlin pattern is uniformly distributed across this polytope; the ${n=2}$ case of this observation is essentially the classic observation of Archimedes that the cross-sectional areas of a sphere and a circumscribing cylinder are the same. (Ultimately, the reason for this is that the Gelfand-Tsetlin pattern almost turns the space of all ${A}$ with a fixed spectrum (i.e. the co-adjoint orbit associated to that spectrum) into a toric variety. More precisely, there exists a mostly diffeomorphic map from the co-adjoint orbit to a (singular) toric variety, and the Gelfand-Tsetlin pattern induces a complete set of action variables on that variety.) There is also a “quantum” (or more precisely, representation-theoretic) version of this observation, in which one can decompose any irreducible representation of the unitary group ${U(n)}$ into a canonical basis (the Gelfand-Tsetlin basis), indexed by integer-valued Gelfand-Tsetlin patterns, by first decomposing this representation into irreducible representations of ${U(n-1)}$, then ${U(n-2)}$, and so forth.

The structure, symplectic geometry, and representation theory of Gelfand-Tsetlin patterns was enormously influential in my own work with Allen Knutson on honeycomb patterns, which control the sums of Hermitian matrices and also the structure constants of the tensor product operation for representations of ${U(n)}$; indeed, Gelfand-Tsetlin patterns arise as the degenerate limit of honeycombs in three different ways, and we in fact discovered honeycombs by trying to glue three Gelfand-Tsetlin patterns together. (See for instance our Notices article for more discussion. The honeycomb analogue of the representation-theoretic properties of these patterns was eventually established by Henriques and Kamnitzer, using ${gl(n)}$ crystals and their Kashiwara bases.)

The second contribution of Gelfand I want to discuss is the Gelfand-Levitan-Marchenko equation for solving the one-dimensional inverse scattering problem: given the scattering data of an unknown potential function ${V(x)}$, recover ${V}$. This is already interesting in and of itself, but is also instrumental in solving integrable systems such as the Korteweg-de Vries equation, because the Lax pair formulation of such equations implies that they can be linearised (and solved explicitly) by applying the scattering and inverse scattering transforms associated with the Lax operator. I discuss the derivation of this equation below the fold.

Let ${d}$ be a natural number. A basic operation in the topology of oriented, connected, compact, ${d}$-dimensional manifolds (hereby referred to simply as manifolds for short) is that of connected sum: given two manifolds ${M, N}$, the connected sum ${M \# N}$ is formed by removing a small ball from each manifold and then gluing the boundary together (in the orientation-preserving manner). This gives another oriented, connected, compact manifold, and the exact nature of the balls removed and their gluing is not relevant for topological purposes (any two such procedures give homeomorphic manifolds). It is easy to see that this operation is associative and commutative up to homeomorphism, thus ${M \# N \cong N \# M}$ and ${(M \# N) \# O \cong M \# (N \# O)}$, where we use ${M \cong N}$ to denote the assertion that ${M}$ is homeomorphic to ${N}$.

(It is important that the orientation is preserved; if, for instance, ${d=3}$, and ${M}$ is a chiral 3-manifold which is chiral (thus ${M \not \cong -M}$, where ${-M}$ is the orientation reversal of ${M}$), then the connect sum ${M \# M}$ of ${M}$ with itself is also chiral (by the prime decomposition; in fact one does not even need the irreducibility hypothesis for this claim), but ${M \# -M}$ is not. A typical example of an irreducible chiral manifold is the complement of a trefoil knot. Thanks to Danny Calegari for this example.)

The ${d}$-dimensional sphere ${S^d}$ is an identity (up to homeomorphism) of connect sum: ${M \# S^d \cong M}$ for any ${M}$. A basic result in the subject is that the sphere is itself irreducible:

Theorem 1 (Irreducibility of the sphere) If ${S^d \cong M \# N}$, then ${M, N \cong S^d}$.

For ${d=1}$ (curves), this theorem is trivial because the only connected ${1}$-manifolds are homeomorphic to circles. For ${d=2}$ (surfaces), the theorem is also easy by considering the genus of ${M, N, M \# N}$. For ${d=3}$ the result follows from the prime decomposition. But for higher ${d}$, these ad hoc methods no longer work. Nevertheless, there is an elegant proof of Theorem 1, due to Mazur, and known as Mazur’s swindle. The reason for this name should become clear when one sees the proof, which I reproduce below.

Suppose ${M \# N \cong S^d}$. Now consider the infinite connected sum

$\displaystyle (M \# N) \# (M \# N) \# (M \# N) \# \ldots.$

This is an infinite connected sum of spheres, and can thus be viewed as a half-open cylinder, which is topologically equivalent to a sphere with a small ball removed; alternatively, one can contract the boundary at infinity to a point to recover the sphere ${S^d}$. On the other hand, by using the associativity of connected sum (which will still work for the infinite connected sum, if one thinks about it carefully), the above manifold is also homeomorphic to

$\displaystyle M \# (N \# M) \# (N \# M) \# \ldots$

which is the connected sum of ${M}$ with an infinite sequence of spheres, or equivalently ${M}$ with a small ball removed. Contracting the small balls to a point, we conclude that ${M \cong S^d}$, and a similar argument gives ${N \cong S^d}$.

A typical corollary of Theorem 1 is a generalisation of the Jordan curve theorem: any locally flat embedded copy of ${S^{d-1}}$ in ${S^d}$ divides the sphere ${S^d}$ into two regions homeomorphic to balls ${B^d}$. (Some sort of regularity hypothesis, such as local flatness, is essential, thanks to the counterexample of the Alexander horned sphere. If one assumes smoothness instead of local flatness, the problem is known as the Schönflies problem, and is apparently quite subtle, especially in the four-dimensional case ${d=4}$.)

One can ask whether there is a way to prove Theorem 1 for general ${d}$ without recourse to the infinite sum swindle. I do not know the complete answer to this, but some evidence against this hope can be seen by noting that if one works in the smooth category instead of the topological category (i.e. working with smooth manifolds, and only equating manifolds that are diffeomorphic, and not merely homeomorphic), then the exotic spheres in five and higher dimensions provide a counterexample to the smooth version of Theorem 1: it is possible to find two exotic spheres whose connected sum is diffeomorphic to the standard sphere. (Indeed, in five and higher dimensions, the exotic sphere structures on ${S^d}$ form a finite abelian group under connect sum, with the standard sphere being the identity element. The situation in four dimensions is much less well understood.) The problem with the swindle here is that the homeomorphism generated by the infinite number of applications of the associativity law is not smooth when one identifies the boundary with a point.

The basic idea of the swindle – grouping an alternating infinite sum in two different ways – also appears in a few other contexts. Most classically, it is used to show that the sum ${1-1+1-1+\ldots}$ does not converge in any sense which is consistent with the infinite associative law, since this would then imply that ${1=0}$; indeed, one can view the swindle as a dichotomy between the infinite associative law and the presence of non-trivial cancellation. (In the topological manifold category, one has the former but not the latter, whereas in the case of ${1-1+1-1+\ldots}$, one has the latter but not the former.) The alternating series test can also be viewed as a variant of the swindle.

Another variant of the swindle arises in the proof of the Cantor–Bernstein–Schröder theorem. Suppose one has two sets ${A, B}$, together with injections from ${A}$ to ${B}$ and from ${B}$ to ${A}$. The first injection leads to an identification ${B \cong C \uplus A}$ for some set ${C}$, while the second injection leads to an identification ${A \cong D \uplus B}$. Iterating this leads to identifications

$\displaystyle A \cong (D \uplus C \uplus D \uplus \ldots) \uplus X$

and

$\displaystyle B \cong (C \uplus D \uplus C \uplus \ldots) \uplus X$

for some additional set ${X}$. Using the identification ${D \uplus C \cong C \uplus D}$ then yields an explicit bijection between ${A}$ and ${B}$.

(Thanks to Danny Calegari for telling me about the swindle, while we were both waiting to catch an airplane.)

[Update, Oct 7: See the comments for several further examples of swindle-type arguments.]

This week I was in my home town of Adelaide, Australia, for the 2009 annual meeting of the Australian Mathematical Society. This was a fairly large meeting (almost 500 participants). One of the highlights of such a large meeting is the ability to listen to plenary lectures in fields adjacent to one’s own, in which speakers can give high-level overviews of a subject without getting too bogged down in the technical details. From the talks here I learned a number of basic things which were well known to experts in that field, but which I had not fully appreciated, and so I wanted to share them here.

The first instance of this was from a plenary lecture by Danny Calegari entitled “faces of the stable commutator length (scl) ball”. One thing I learned from this talk is that in homotopy theory, there is a very close relationship between topological spaces (such as manifolds) on one hand, and groups (and generalisations of groups) on the other, so that homotopy-theoretic questions about the former can often be converted to purely algebraic questions about the latter, and vice versa; indeed, it seems that homotopy theorists almost think of topological spaces and groups as being essentially the same concept, despite looking very different at first glance. To get from a space ${X}$ to a group, one looks at homotopy groups ${\pi_n(X)}$ of that space, and in particular the fundamental group ${\pi_1(X)}$; conversely, to get from a group ${G}$ back to a topological space one can use the Eilenberg-Maclane spaces ${K(G,n)}$ associated to that group (and more generally, a Postnikov tower associated to a sequence of such groups, together with additional data). In Danny’s talk, he gave the following specific example: the problem of finding the least complicated embedded surface with prescribed (and homologically trivial) boundary in a space ${X}$, where “least complicated” is measured by genus (or more precisely, the negative component of Euler characteristic), is essentially equivalent to computing the commutator length of the element in the fundamental group ${\pi(X)}$ corresponding to that boundary (i.e. the least number of commutators one is required to multiply together to express the element); and the stable version of this problem (where one allows the surface to wrap around the boundary ${n}$ times for some large ${n}$, and one computes the asymptotic ratio between the Euler characteristic and ${n}$) is similarly equivalent to computing the stable commutator length of that group element. (Incidentally, there is a simple combinatorial open problem regarding commutator length in the free group, which I have placed on the polymath wiki.)

This theme was reinforced by another plenary lecture by Ezra Getzler entitled “${n}$-groups”, in which he showed how sequences of groups (such as the first ${n}$ homotopy groups ${\pi_1(X),\ldots,\pi_n(X)}$) can be enhanced into a more powerful structure known as an ${n}$-group, which is more complicated to define, requiring the machinery of simplicial complexes, sheaves, and nerves. Nevertheless, this gives a very topological and geometric interpretation of the concept of a group and its generalisations, which are of use in topological quantum field theory, among other things.

Mohammed Abuzaid gave a plenary lecture entitled “Functoriality in homological mirror symmetry”. One thing I learned from this talk was that the (partially conjectural) phenomenon of (homological) mirror symmetry is one of several types of duality, in which the behaviour of maps into one mathematical object ${X}$ (e.g. immersed or embedded curves, surfaces, etc.) are closely tied to the behaviour of maps out of a dual mathematical object ${\hat X}$ (e.g. functionals, vector fields, forms, sections, bundles, etc.). A familiar example of this is in linear algebra: by taking adjoints, a linear map into a vector space ${X}$ can be related to an adjoint linear map mapping out of the dual space ${X^*}$. Here, the behaviour of curves in a two-dimensional symplectic manifold (or more generally, Lagrangian submanifolds in a higher-dimensional symplectic manifold), is tied to the behaviour of holomorphic sections on bundles over a dual algebraic variety, where the precise definition of “behaviour” is category-theoretic, involving some rather complicated gadgets such as the Fukaya category of a symplectic manifold. As with many other applications of category theory, it is not just the individual pairings between an object and its dual which are of interest, but also the relationships between these pairings, as formalised by various functors between categories (and natural transformations between functors). (One approach to mirror symmetry was discussed by Shing-Tung Yau at a distinguished lecture at UCLA, as transcribed in this previous post.)

There was a related theme in a talk by Dennis Gaitsgory entitled “The geometric Langlands program”. From my (very superficial) understanding of the Langlands program, the behaviour of specific maps into a reductive Lie group ${G}$, such as representations in ${G}$ of a fundamental group, étale fundamental group, class group, or Galois group of a global field, is conjecturally tied to specific maps out of a dual reductive Lie group ${\hat G}$, such as irreducible automorphic representations of ${\hat G}$, or of various structures (such as derived categories) attached to vector bundles on ${\hat G}$. There are apparently some tentatively conjectured links (due to Witten?) between Langlands duality and mirror symmetry, but they seem at present to be fairly distinct phenomena (one is topological and geometric, the other is more algebraic and arithmetic). For abelian groups, Langlands duality is closely connected to the much more classical Pontryagin duality in Fourier analysis. (There is an analogue of Fourier analysis for nonabelian groups, namely representation theory, but the link from this to the Langlands program is somewhat murky, at least to me.)

Related also to this was a plenary talk by Akshay Venkatesh, entitled “The Cohen-Lenstra heuristics over global fields”. Here, the question concerned the conjectural behaviour of class groups of quadratic fields, and in particular to explain the numerically observed phenomenon that about ${75.4\%}$ of all quadratic fields ${{\Bbb Q}[\sqrt{d}]}$ (with $d$ prime) enjoy unique factorisation (i.e. have trivial class group). (Class groups, as I learned in these two talks, are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of the full fundamental group.) One thing I learned here was that there was a canonical way to randomly generate a (profinite) abelian group, by taking the product of randomly generated finite abelian ${p}$-groups for each prime ${p}$. The way to canonically randomly generate a finite abelian ${p}$-group is to take large integers ${n, D}$, and look at the cokernel of a random homomorphism from ${({\mathbb Z}/p^n{\mathbb Z})^d}$ to ${({\mathbb Z}/p^n{\mathbb Z})^d}$. In the limit ${n,d \rightarrow \infty}$ (or by replacing ${{\mathbb Z}/p^n{\mathbb Z}}$ with the ${p}$-adics and just sending ${d \rightarrow \infty}$), this stabilises and generates any given ${p}$-group ${G}$ with probability

$\displaystyle \frac{1}{|\hbox{Aut}(G)|} \prod_{j=1}^\infty (1 - \frac{1}{p^j}), \ \ \ \ \ (1)$

where ${\hbox{Aut}(G)}$ is the group of automorphisms of ${G}$. In particular this leads to the strange identity

$\displaystyle \sum_G \frac{1}{|\hbox{Aut}(G)|} = \prod_{j=1}^\infty (1 - \frac{1}{p^j})^{-1} \ \ \ \ \ (2)$

where ${G}$ ranges over all ${p}$-groups; I do not know how to prove this identity other than via the above probability computation, the proof of which I give below the fold.

Based on the heuristic that the class group should behave “randomly” subject to some “obvious” constraints, it is expected that a randomly chosen real quadratic field ${{\Bbb Q}[\sqrt{d}]}$ has unique factorisation (i.e. the class group has trivial ${p}$-group component for every ${p}$) with probability

$\displaystyle \prod_{p \hbox{ odd}} \prod_{j=2}^\infty (1 - \frac{1}{p^j}) \approx 0.754,$

whereas a randomly chosen imaginary quadratic field ${{\Bbb Q}[\sqrt{-d}]}$ has unique factorisation with probability

$\displaystyle \prod_{p \hbox{ odd}} \prod_{j=1}^\infty (1 - \frac{1}{p^j}) = 0.$

The former claim is conjectural, whereas the latter claim follows from (for instance) Siegel’s theorem on the size of the class group, as discussed in this previous post. Ellenberg, Venkatesh, and Westerland have recently established some partial results towards the function field analogues of these heuristics.