Representations of complex semi-simple Lie groups and Lie algebras
Pages 383-429 by
From volume 85-3
On compact subgroups of the diffeomorphism groups of Kervaire spheres
Pages 359-369 by
From volume 85-3
Redshift and multiplication for truncated Brown–Peterson spectra
We equip $\mathrm {BP} \langle n \rangle $ with an $\mathbb {E}_3$-$\mathrm{BP}$-algebra structure for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is of chromatic height exactly $n+1$, and the map $\mathrm {K}(\mathrm{BP}\langle n \rangle )_{(p)} \to \mathrm {L}_{n+1}^{f} \mathrm {K}(\mathrm {BP}\langle n\rangle )_{(p)}$ has bounded above fiber.
Pages 1277-1351 by
From volume 196-3
Prisms and prismatic cohomology
We introduce the notion of a prism, which may be regarded as a “deperfection” of the notion of a perfectoid ring. Using prisms, we attach a ringed site — the prismatic site — to a $p$-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral $p$-adic cohomology theories.
As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of $q$-de Rham cohomology as conjectured by the second author, and settle a vanishing conjecture for the $p$-adic Tate twists $\mathbf {Z}_p(n)$ introduced in our previous joint work with Morrow.
Pages 1135-1275 by
From volume 196-3
One can hear the shape of ellipses of small eccentricity
We show that if the eccentricity of an ellipse is sufficiently small, then up to isometries it is spectrally unique among all smooth domains. We do not assume any symmetry, convexity, or closeness to the ellipse, on the class of domains.
In the course of the proof we also show that for nearly circular domains, the lengths of periodic orbits that are shorter than the perimeter of the domain must belong to the singular support of the wave trace. As a result we also obtain a Laplace spectral rigidity result for the class of axially symmetric nearly circular domains using a similar result of De Simoi, Kaloshin, and Wei concerning the length spectrum of such domains.
Pages 1083-1134 by
From volume 196-3
Universal optimality of the $E_8$ and Leech lattices and interpolation formulas
We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions eight and twenty-four, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions.
The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\hat f$ at the radii $\sqrt{2n}$ for integers $n\ge 1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska’s work on sphere packing and placing it in the context of a more conceptual theory.
The supplemental computer-assisted proof of kernel inequalities for this paper is available at the following locations:
https://doi.org/10.4007/annals.2022.196.3.3.code and https://dspace.mit.edu/handle/1721.1/141226
Pages 983-1082 by
From volume 196-3
Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds
We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is smaller than $r(G)$, then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most $m(G)$, we show there is a quasi-invariant measure on the manifold such that the action is measurably isomorphic to a relatively measure-preserving action over a standard boundary action.
Pages 941-981 by
From volume 196-3
Zimmer’s conjecture: Subexponential growth, measure rigidity, and strong property (T)
We prove several cases of Zimmer’s conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if $\Gamma$ is a cocompact lattice in $\mathrm{SL}(n,\mathbb{R})$, $M$ is a compact manifold, and $\omega$ a volume form on $M$, we show that any homomorphism $\alpha : \Gamma \rightarrow \mathrm{Diff}(M)$ has finite image if the dimension of $M$ is less than $n-1$ and that any homomorphism $\alpha : \Gamma \rightarrow \mathrm{Diff}(M,\omega)$ has finite image if the dimension of $M$ is less than $n$. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Lafforgue’s strong property \rm (T) to establish the existence of an invariant Riemannian metric.
Pages 891-940 by
From volume 196-3
On the bound of the dimensions of the isometry groups of all possible riemannian metrics on an exotic sphere
Pages 351-358 by
From volume 85-2
Boundedness of translation invariant operators on Hölder spaces and $L^p$-spaces
Pages 337-349 by
From volume 85-2
Hecke polynomials as congruence $\zeta$ functions in elliptic modular case
Pages 267-295 by
From volume 85-2
Construction of class fields and zeta functions of algebraic curves
Pages 58-159 by
From volume 85-1
Analysis in matrix spaces and some new representations of $\mathrm{SL}(N,\mathbf{C})$
Pages 461-490 by
From volume 86-3
On the algebraic curves uniformized by arithmetical automorphic functions
Pages 449-460 by
From volume 86-3
A $\Pi_\ast$-module structure for $\Theta_\ast$ and applications to transformation groups
Pages 434-448 by
From volume 86-3
Existence theorems for analytic linear partial differential equations
Pages 246-270 by
From volume 86-2
On the non-vanishing of the Jacobian of a homeomorphism of harmonic gradients
Pages 518-529 by
From volume 88-3
A Lefschetz fixed point formula for elliptic complexes. II. Applications
Pages 451-491 by
From volume 88-3
On the set of non-locally flat points of a submanifold of codimension one
Pages 281-290 by
From volume 88-2
The word problem in fundamental groups of sufficiently large irreducible 3-manifolds
Pages 272-280 by
From volume 88-2
Algebraic cycles on abelian varieties of complex multiplication type
Pages 161-180 by
From volume 88-2
Representations of uniformly hyperfinite algebras and their associated von Neumann rings
Pages 138-171 by
From volume 86-1
Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure
Pages 321-391 by
From volume 87-2
A $p$-adic proof of Weil’s conjectures
The first portion of this paper appeared in the preceding issue of this Journal.
Pages 195-255 by
From volume 87-1
A $p$-adic proof of Weil’s conjectures
Due to the length of this paper, it is being published in two parts. The second part will appear at the beginning of the next issue of this journal.
Pages 105-194 by
From volume 87-1
Determination of the differentiably simple rings with a minimal ideal
Pages 433-459 by
From volume 90-3