The 2024 New Horizons in Mathematics Prize goes to

**Roland Bauerschmidt**for*outstanding contributions to probability theory and the development of renormalisation group techniques*.**Michael Groechenig**for*contributions to the theory of rigid local systems and applications of p-adic integration to mirror symmetry and the fundamental lemma*.**Angkana Rüland**for*contributions to applied analysis, in particular the analysis of microstructure in solid-solid phase transitions and the theory of inverse problems*.

The 2024 Maryam Mirzakhani New Frontiers Prize goes to

**Hannah Larson**for*advances in Brill-Noether theory and the geometry of the moduli space of curves*.**Laura Monk**for*advancing our understanding of random hyperbolic surfaces of large genus*.**Mayuko Yamashita**for*contributions to mathematical physics, index theory*.

This remarkable property led people to believe that all the coarse information about a metric space \(X\) is stored in its uniform Roe algebra \(C^*_u(X)\) and the problem is only to find a way to retrieve it (i.e., to define the corresponding invariant of C*-algebras).

One such example is the asymptotic dimension of \(X\). One of its possible definitions is as follows: \(X\) has asymptotic dimension at most \(d\) if for every \(R \gg 1\) there is a cover \(\mathcal{U}\) of \(X\) such that the members of \(\mathcal{U}\) have uniformly bounded diameter and every \(R\)-ball in \(X\) intersects at most \(d+1\) members of \(\mathcal{U}\). A candidate for the corresponding invariant of C*-algebras is nuclear dimension as introduced by Winter-Zacharias (arXiv:0903.4914), which we are not going to define here, and they proved the inequality \[\mathrm{dim}_{\mathrm{nuc}}(C^*_u(X)) \le \textrm{as-dim}(X)\,.\]

But the reverse inequality is still open. And in this post I actually want to express my opinion that, even if one can prove the reverse inequality, it would not be a natural statement for me. The reason is that the nuclear dimension is an invariant defined for any C*-algebra (though it might be infinite), but by far not every C*-algebra is the uniform Roe algebra of a (bounded geometry) metric space.

In my previous post about this topic I explained the following result by White-Willett: Let \(A\) be a C*-algebra and assume that it contains a Cartan subalgebra \(B\) with `nice’ properties. Then there exists a metric space \(Y\) of bounded geometry such that the pair \(B \subset A\) is isomorphic as C*-algebras to \(\ell^\infty(Y) \subset C^*_u(Y)\).

Therefore, a more natural theorem (than the above proposed equality of nuclear dimension of \(C^*_u(X)\) with asymptotic dimension of \(X\)) would be a comparison of asymptotic dimension of \(X\) with an invariant defined for pairs \(B \subset A\) of a Cartan subalgebra \(B\) inside a C*-algebra \(A\). And this was actually achieved at the beginning of this year by Li-Liao-Winter (arXiv:2303.16762): They defined for any pair \(D \subset A\), where \(D\) is an abelian sub-C*-algebra in \(A\), an invariant they called the diagonal dimension and proved \[\mathrm{dim}_{\mathrm{diag}}(B \subset C^*_u(X)) = \textrm{as-dim}(X)\] for every Cartan subalgebra \(B \subset C^*_u(X)\) with `nice’ properties as laid out by White-Willett.

]]>The above question is known as the Kervaire conjecture. Phrased more concretely, if \(G\) is any non-trivial group and \(r \in G \ast \mathbb{Z}\), is \((G \ast \mathbb{Z})/\langle\!\langle r\rangle\!\rangle\) again non-trivial?

This is an example of a problem which is extremely easy to state but which is still unsolved in complete generality. The best partial results so far (and that I could find by a quick search) are the following:

- It is true for finite groups, proven by Gerstenhaber and Rothaus.
- The above implies the conjecture for all residually finite groups and even all hyperlinear ones (the latter includes all sofic groups), noticed by Pestov.
- For torsion-free groups, proven by Klyachko.
- Instead of putting restrictions on the group, Klyachko and Lurye proved the conjecture under a restriction on the relation, namely that it is a proper power (i.e. \(r = w^k\) for some \(k \ge 2\)).

Since we don’t know any non-sofic group (which is one of the major open problems in group theory), a counter-example to the Kervaire conjecture currently seems to be completely out of reach.

The origin of Kervaire’s conjecture is his paper on the classification of high-dimensional knot groups. He proved the following: *Given \(n \ge 3\). The group \(\pi\) is isomorphic to \(\pi_1(S^{n+2} – f(S^n))\) for some differential imbedding \(f\colon S^n \to S^{n+2}\) if and only if \(\pi/\pi^\prime \cong \mathbb{Z}\), the weight of \(\pi\) is \(1\), and \(H_2(\pi) = 0\).*

In the above statement \(\pi^\prime\) denotes the commutator subgroup of \(\pi\), the weight of \(\pi\) is the smallest integer \(k\) with the property that there exists a set of \(k\) elements in \(\pi\) whose normal closure equals \(\pi\), and \(H_2(\pi)\) is the second homology group of \(\pi\) with integral coefficients and trivial action of \(\pi\) on \(\mathbb{Z}\).

At the end of his paper, Kervaire shows that the conditions \(\pi/\pi^\prime \cong \mathbb{Z}\) and \(H_2(\pi) = 0\) do not imply that the weight of \(\pi\) is \(1\), i.e. that the latter is not a redundant condition in the theorem. His example is of the form \(\pi = G \ast \mathbb{Z}\). Hence the question whether a non-trivial free product \(G \ast \mathbb{Z}\) can ever be of weight \(1\) arose.

]]>In April of this year a preprint titled **Can you hear your location on a manifold?** was put on the arXiv by Wyman and Xi (arXiv:2304.04659). A bit more concretely, the question is this: *You stand on a compact Riemannian manifold, make a single sharp “snap” sound, and then listen intently to its reverberations. If you have perfect hearing and perfect knowledge of the shape of the manifold, can you deduce your location (up to isometries) within it?*

Mathematically the question means the following: Let \((e_j)_{j \in \mathbb{N}}\) be an orthonormal basis of eigenfunctions of the Laplace operator (with Dirichlet or Neumann boundary conditions in case the manifold has non-empty boundary) an denote by \(\lambda_j\) the corresponding eigenvalues. For a point \(x\) of the manifold the Weyl counting function is defined as \[N_x(\lambda) := \sum_{\lambda_j \le \lambda} |e_j(x)|^2\,.\] Can one now recover (up to isometries) the point \(x\) from the knowledge of the function \(N_x\), i.e. if \(N_x = N_y\) must there exist an isometry mapping \(x\) to \(y\)?

The result of the above mentioned preprint is now the following: *On any compact, smooth manifold without boundary and of dimension at least 2 a generic Riemannian metric has the property that \(N_x = N_y\) implies \(x=y\)*. Generic means here that the set of these metrics has meager complement in the \(C^\infty\)-topology on all Riemannian metrics.

In a recent preprint (arXiv:2305.17743) this drawback was overcome now: A new shape (a modification of the previous one) that tiles the plane aperiodically using only translations and rotations, and even if you allow reflections it can not tile periodically.

Here is a webpage (by one of the authors) with some resources about this and from where I got the picture below: link. And here is another blog post with some information: link.

]]>Per Enflo is famous for several fundamental results in functional analysis (see the English Wikipedia article linked above for an overview). The one related to this post is his counter-example (the first one to be found) to the invariant subspace problem in Banach spaces: He constructed a Banach space together with a bounded linear operator on it such that this operator does not have any non-trivial invariant closed subspace.

Enflo came up with this counter-example in 1975 and since then, of course, more have been found, even on the Banach space \(\ell^1\). But up to now no counter-example is known on a reflexive Banach space; and especially, not on a separable Hilbert space (on non-separable Hilbert spaces every bounded linear operator has a non-trivial invariant closed subspace for trivial reasons).

Enflo’s arXiv preprint from today proposes a proof that actually every bounded linear operator acting on a separable Hilbert space has a non-trivial invariant closed subspace, i.e. a positive solution to the invariant subspace problem in separable Hilbert spaces! If his arguments turn out to be correct, it will be quite a big deal!

]]>Let us define the main player of this post, the uniform Roe algebra: We consider bounded operators \(a\) on \(\ell^2(X)\) and think of them as \(X\text{-by-}X\)-matrices \(a=(a_{xy})_{x,y \in X})\). We define the propagation of \(a\) by \[\mathrm{prop}(a) := \sup\{d(x,y)\colon a_{xy} \not= 0\} \in [0,\infty]\,.\] The uniform Roe algebra \(C_u^*(X)\) is defined as the norm closure of the \({}^*\)-algebra of all bounded operators on \(\ell^2(X)\) of finite propagation.

Note that the uniform Roe algebra \(C^*_u(X)\) naturally contains \(\ell^\infty(X)\), considered as diagonal operators (i.e. operators with zero propagation). If \(X\) is the metric space of a finitely generated group \(G\) with a word metric, then we have a canonical isomorphism \(C^*_u(X) \cong \ell^\infty(G) \rtimes_r G\) to the corresponding reduced crossed product.

The uniform Roe algebra has the following remarkable property: Two metric spaces \(X\) and \(Y\) are coarsely equivalent if and only if their uniform Roe algebras \(C^*_u(X)\) and \(C^*_u(Y)\) are Morita equivalent (a notion slightly more general than being isomorphic as \(C^*\)-algebras). This was finally proven, after several preliminary results by other authors, by Baudier-Braga-Farah-Khukhro-Vignati-Willett (arXiv:2106.11391).

The above reminds one of the classical Gelfand duality stating that, say, two compact Hausdorff spaces \(X\) and \(Y\) are homeomorphic if and only if their \(C^*\)-algebras of continuous functions \(C(X)\) and \(C(Y)\) are isomorphic as \(C^*\)-algebras.

But Gelfand duality actually says more. For example, it also says that every unital, commutative \(C^*\)-algebra \(A\) arises as \(C(Z)\) for some compact Hausdorff space \(Z\). But not every arbitrary (unital) \(C^*\)-algebra is the uniform Roe algebra of a metric space. Therefore, to go towards an honest duality between metric spaces (of bounded geometry) with coarse equivalences and \(C^*\)-algebras with Morita equivalence, one has to characterize those \(C^*\)-algebras which actually do arise as uniform Roe algebras of metric spaces.

This was achieved by White-Willett (arXiv:1808.04410) who recognized the importance of the sub-\(C^*\)-algebra \(\ell^\infty(X)\) inside \(C^*_u(X)\). This is a Cartan subalgebra with certain special properties and abstracting these they arrived at the following theorem: Let \(A\) be a \(C^*\)-algebra and assume that it contains a Cartan subalgebra \(B\) with `nice’ properties (these are of course made precise by White-Willett). Then there exists a metric space \(Y\) of bounded geometry such that the pair \(B \subset A\) is isomorphic as \(C^*\)-algebras to \(\ell^\infty(Y) \subset C^*_u(Y)\).

]]>Another problem is also to distinguish positive sectional curvature from positive Ricci curvature. If we only count those results which hit into the above mentioned distinction between positive and nonnegative sectional curvature and which at the same time do not also apply to positive Ricci curvature, we arrive at exactly **two **theorems:

- (Synge 1936) Even-dimensional manifolds of positive curvature are simply connected if and only if they are orientable; especially, the only nontrivial fundamental group can be \(\small \mathbb{Z}/2\mathbb{Z}\). In the odd-dimensional case positively curved manifolds must be orientable.
- (Schoen 2023) In \(\small 4k\)-dimensional manifolds of positive curvature the only possible non-trivial element of \(\pi_1\) must act trivially on \(\pi_2\). In odd-dimensional manifolds of positive curvature no element of \(\pi_1\) can reverse the orientation of a nontrivial element of \(\pi_2\).

Synge’s result rules out positive curvature on the manifold \(\small RP^n \times RP^n\), whereas Schoen’s recent result rules out positive curvature on \(\small RP^2 \times S^{4k-2}\). Note that these products admit metrics of nonnegative sectional curvature.

What about the case of, say, \(\small S^2 \times S^2\)? Well, this is still open and known as Hopf’s conjecture. This conjecture can be actually extended to the following version: There are no positively curved metrics on the product of two closed manifold.

]]>Since many standard tools from functional analysis like the Hahn-Banach theorem or the Baire category theorem fail without Choice, one has to be very careful in proving even the basic facts; and this is on top of the fact that, for example, fundamental notions like *infinite *have different definitions without Choice.

Some exotic phenomena that can happen now, are the following:

- There exist sets \(X\) (with strange set-theoretic properties without Choice) such that the Hilbert space \(H = \ell^2(X)\) is infinite-dimensional but every closed subset of it has either finite dimension or finite codimension. Further, the C*-algebra of all bounded, linear operator \(B(H)\) is in this case equal to \(K(H) + \mathbb{C} \cdot \mathrm{id}\).
- There exist sets \(X\) (with strange set-theoretic properties without Choice) such that the spectrum of each bounded linear operator on \(H = \ell^2(X)\) consists of only finitely many points which are all eigenvalues.
- In certain models of ZF the commutative C*-algebra \(\ell^\infty(\mathbb{N}) / c_0(\mathbb{N})\)
- … has no non-trivial bounded linear functionals,
- … has no non-trivial representation on a Hilbert space,
- … is not isomorphic to \(C(X)\) for a compact Hausdorff space \(X\).