-
On the generic part of the cohomology of non-compact unitary Shimura varieties | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2024-03-05 Ana Caraiani, Peter Scholze
We prove that the generic part of the $\mathrm{mod}\, \ell$ cohomology of Shimura varieties associated to quasi-split unitary groups of even dimension is concentrated above the middle degree, extending our previous work to a non-compact case. The result applies even to Eisenstein cohomology classes coming from the locally symmetric space of the general linear group, and has been used in joint work
-
Prime number theorem for analytic skew products | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2024-03-05 Adam Kanigowski, Mariusz Lemańczyk, Maksym Radziwiłł
We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus $\mathbb{T}^2$. More precisely, for every irrational $\alpha$ and every $1$-periodic real analytic $g:\mathbb{R}\to\mathbb{R}$ of zero mean, let $T_{\alpha,g} : \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be defined by $(x,y) \mapsto (x+\alpha,y+g(x))$. We prove that if $T_{\alpha, g}$ is uniquely ergodic
-
Generalized soap bubbles and the topology of manifolds with positive scalar curvature | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2024-03-05 Otis Chodosh, Chao Li
We prove that for $n\in \{4,5\}$, a closed aspherical $n$-manifold does not admit a Riemannian metric with positive scalar curvature. Additionally, we show that for $n\leq 7$, the connected sum of a $n$-torus with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When combined with contributions by Lesourd–Unger–Yau, this proves that the Schoen–Yau Liouville theorem
-
Pointwise convergence of the non-linear Fourier transform | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2024-03-05 A. Poltoratski
We prove pointwise convergence for the scattering data of a Dirac system of differential equations. Equivalently, we prove an analog of Carleson’s theorem on almost everywhere convergence of Fourier series for a version of the non-linear Fourier transform. Our proofs are based on the study of resonances of Dirac systems using families of meromorphic inner functions, generated by a Riccati equation
-
Wilkie’s conjecture for Pfaffian structures | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2024-03-05 Gal Binyamini, Dmitry Novikov, Benny Zak
We prove an effective form of Wilkie’s conjecture in the structure generated by restricted sub-Pfaffian functions: the number of rational points of height $H$ lying in the transcendental part of such a set grows no faster than some power of $\log H$. Our bounds depend only on the Pfaffian complexity of the sets involved. As a corollary we deduce Wilkie’s original conjecture for $\mathbb{R}_{\rm exp}$
-
Canonical representations of surface groups | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2024-03-05 Aaron Landesman, Daniel Litt
Let $\Sigma _{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group $\mathrm {Mod}_{g,n}$ of $\Sigma _{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if $$\rho : \pi _1(\Sigma _{g,n})\to \mathrm {GL}_r(\mathbb {C})$$ is a representation whose conjugacy class has finite orbit under $\mathrm {Mod}_{g,n}$, and $r\lt \sqrt {g+1}$
-
Oka properties of complements of holomorphically convex sets | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2024-03-05 Yuta Kusakabe
Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in $\mathbb{C}^{n}$ $(n>1)$ is Oka. Furthermore, we obtain new examples of non-elliptic Oka manifolds which
-
The asymptotics of $r(4,t)$ | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2024-03-05 Sam Mattheus, Jacques Verstraete
For integers $s,t \geq 2$, the Ramsey number $r(s,t)$ denotes the minimum $n$ such that every $n$-vertex graph contains a clique of order $s$ or an independent set of order $t$. In this paper we prove \[ r(4,t) = \Omega\Bigl(\frac{t^3}{\log^4 \! t}\Bigr) \quad \quad \mbox{ as }t \rightarrow \infty,\] which determines $r(4,t)$ up to a factor of order $\log^2 \! t$, and solves a conjecture of Erdős.
-
A reverse Minkowski theorem | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-12-29 Oded Regev, Noah Stephens-Davidowitz
We prove a conjecture due to Dadush, showing that if $\mathcal{L} \subset \mathbb{R}^n$ is a lattice such that $\mathrm{det}(\mathcal{L}’)\ge 1$ for all sublattices $\mathcal{L}’ \subseteq \mathcal{L}$, then $\sum_{\mathbf{y}\in \mathcal{L}} e^{-\pi t^2 \|\mathbf{y} \|^2} \le 3/2$, where $t := 10(\log n + 2)$. From this we derive bounds on the number of short lattice vectors, which can be viewed as
-
Purity for flat cohomology | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-12-29 Kęstutis Česnavičius, Peter Scholze
We establish the flat cohomology version of the Gabber–Thomason purity for étale cohomology: for a complete intersection Noetherian local ring $(R, \mathfrak {m})$ and a commutative, finite, flat $R$-group $G$, the flat cohomology $H^i_\mathfrak {m}(R, G)$ vanishes for for $i \le \mathrm{dim}(R)$. For small $i$, this settles conjectures of Gabber that extend the Grothendieck–Lefschetz theorem and give
-
Proof of the simplicity conjecture | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-12-29 Daniel Cristofaro-Gardiner, Vincent Humilière, Sobhan Seyfaddini
In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for \hbox $n \ge 3$, asked if the same statement holds in dimension two. We show that the group of compactly supported area-preserving homeomorphisms of the two-disc is not simple. This settles what is known as the “simplicity conjecture” in the affirmative. In fact, we
-
Log-concave polynomials II: High-dimensional walks and an FPRAS for counting bases of a matroid | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-12-29 Nima Anari, Kuikui Liu, Shayan Oveis Gharan, Cynthia Vinzant
We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where $0\lt q\lt 1$. Consequently, we can sample random spanning forests in a graph and estimate the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani
-
Nonabelian level structures, Nielsen equivalence, and Markoff triples | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-12-29 William Y. Chen
In this paper we establish a congruence on the degree of the map from a component of a Hurwitz space of covers of elliptic curves to the moduli stack of elliptic curves. Combinatorially, this can be expressed as a congruence on the cardinalities of Nielsen equivalence classes of generating pairs of finite groups. Building on the work of Bourgain, Gamburd, and Sarnak, we apply this congruence to show
-
Motivic invariants of birational maps | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-12-29 Hsueh-Yung Lin, Evgeny Shinder
We construct invariants of birational maps with values in the Kontsevich–Tschinkel group and in the truncated Grothendieck groups of varieties. These invariants are morphisms of groupoids and are well-suited to investigating the structure of the Grothendieck ring and L-equivalence. Building on known constructions of L-equivalence, we prove new unexpected results about Cremona groups.
-
Erratum to “Special subvarieties of non-arithmetic ball quotients\ and Hodge theory” | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-12-29 Gregorio Baldi, Emmanuel Ullmo
No abstract available for this article
-
Existence of infinitely many minimal hypersurfaces in closed manifolds | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-03-23 Antoine Song
Using min-max theory, we show that in any closed Riemannian manifold of dimension at least $3$ and at most $7$, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds on the methods developed by F. C. Marques and A. Neves.
-
Potential automorphy over CM fields | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-03-23 Patrick B. Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao V. Le Hung, James Newton, Peter Scholze, Richard Taylor, Jack A. Thorne
Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato–Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic
-
The Hasse principle for random Fano hypersurfaces | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-03-23 Tim Browning, Pierre Le Boudec, Will Sawin
It is known that the Brauer–Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thélène that the Brauer–Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational
-
Stable minimal hypersurfaces in $\mathbb {R}^{N+1+\ell }$ with singular set an arbitrary closed $K\subset \{0\}\times \mathbb {R}^{\ell }$ | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-03-23 Leon Simon
With respect to a $C^{\infty }$ metric which is close to the standard Euclidean metric on $\mathbb {R}^{N+1+\ell }$, where $N\ge 7$ and $\ell \ge 1$ are given, we construct a class of embedded $(N+\ell )$-dimensional hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset $K\subset \{0\}\times \mathbb {R}^{\ell
-
On Frobenius exact symmetric tensor categories (with Appendix A by Alexander Kleshchev) | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-03-23 Kevin Coulembier, Pavel Etingof, Victor Ostrik
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper
-
Comptage des systémes locaux $\ell$-adiques sur une courbe | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-02-03 Hongjie Yu
Let $X_{1}$ be a projective, smooth and geometrically connected curve over $\mathbb{F}_{q}$ with $q=p^{n}$ elements where $p$ is a prime number, and let $X$ be its base change to an algebraic closure of $\mathbb{F}_{q}$. We give a formula for the number of irreducible $\ell$-adic local systems ($\ell\neq p$) with a fixed rank over $X$ fixed by the Frobenius endomorphism. We prove that this number behaves
-
Feral curves and minimal sets | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-02-03 Joel W. Fish, Helmut Hofer
We prove that for each Hamiltonian function $H\in \mathcal {C}^\infty (\mathbb {R}^4, \mathbb {R})$ defined on the standard symplectic $(\mathbb {R}^4, \omega _0)$, for which $M:=H^{-1}(0)$ is a non-empty compact regular energy level, the Hamiltonian flow on $M$ is not minimal. That is, we prove there exists a closed invariant subset of the Hamiltonian flow in $M$ that is neither $\emptyset $ nor all
-
Higher uniformity of bounded multiplicative functions in short intervals on average | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2023-02-03 Kaisa Matomäki, Maksym Radziwi{ł}{ł}, Terence Tao, Joni Teräväinen, Tamar Ziegler
Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$ \int^{2X}_X \sup_{\begin{smallmatrix} P(Y)\in\mathbb{R}[Y] \ \mathrm{deg}{P} \leq k\end{smallmatrix}} \left| \sum_{x\leq n \leq x+H} \lambda(n) e(-P(n))\right| dx=o (XH) $$ for all fixed $k$ and $X^{\theta} \leq H \leq X$ with $0 < \theta < 1$ fixed but arbitrarily small. Previously this was only established for
-
There is no Enriques surface over the integers | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-11-23 Stefan Schröer
We show that there is no family of Enriques surfaces over the ring of integers. This extends non-existence results of Minkowski for families of finite étale schemes, of Tate and Ogg for families of elliptic curves, and of Fontaine and Abrashkin for families of abelian varieties and more general smooth proper schemes with certain restrictions on Hodge numbers. Our main idea is to study the local system
-
Smooth mixing Anosov flows in dimension three are exponentially mixing | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-11-23 Masato Tsujii, Zhiyuan Zhang
We show that a topologically mixing $C^\infty $ Anosov flow on a 3-dimensional compact manifold is exponentially mixing with respect to any equilibrium measure with Hölder potential.
-
Special subvarieties of non-arithmetic ball quotients and Hodge theory | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-11-22 Gregorio Baldi, Emmanuel Ullmo
Let $\Gamma \subset \mathrm {PU}(1,n)$ be a lattice and $S_\Gamma $ be the associated ball quotient. We prove that, if $S_\Gamma $ contains infinitely many maximal complex totally geodesic subvarieties, then $\Gamma $ is arithmetic. We also prove an Ax–Schanuel Conjecture for $S_\Gamma $, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to
-
Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-11-22 Sven M{ö}ller, Nils~R. Scheithauer
We prove a dimension formula for the weight-$1$ subspace of a vertex operator algebra $V^{\mathrm {orb}(g)}$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge $24$ with a finite-order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $\mathrm {Aut}(V)$. Then we show that the
-
On the Brumer–Stark conjecture | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-11-22 Samit Dasgupta, Mahesh Kakde
Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer–Stark conjecture states that the Stickelberger element $\Theta ^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $\mathrm {Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring
-
Infinite volume and infinite injectivity radius | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-11-22 Mikolaj Fraczyk, Tsachik Gelander
We prove the following conjecture of Margulis. Let $G$ be a higher rank simple Lie group, and let $\Lambda \le G$ be a discrete subgroup of infinite covolume. Then, the locally symmetric space $\Lambda \backslash G/K$ admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups
-
Zimmer’s conjecture: Subexponential growth, measure rigidity, and strong property (T) | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-10-30 Aaron Brown, David Fisher, Sebastian Hurtado
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 $n-1$
-
Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-10-30 Aaron Brown, Federico Rodriguez Hertz, Zhiren Wang
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
-
Universal optimality of the $E_8$ and Leech lattices and interpolation formulas | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-10-30 Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska
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
-
One can hear the shape of ellipses of small eccentricity | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-10-30 Hamid Hezari, Steve Zelditch
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
-
Prisms and prismatic cohomology | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-10-30 Bhargav Bhatt, Peter Scholze
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
-
Redshift and multiplication for truncated Brown–Peterson spectra | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-10-30 Jeremy Hahn, Dylan Wilson
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.
-
On the Chowla and twin primes conjectures over $\mathbb F_q[T]$ | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-06-28 Will Sawin, Mark Shusterman
Using geometric methods, we improve on the function field version of the Burgess bound and show that, when restricted to certain special subspaces, the Möbius function over $\mathbb {F}_q[T]$ can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to $1$ for the Möbius function in arithmetic progressions and resolve Chowla’s $k$-point correlation conjecture
-
Finite generation for valuations computing stability thresholds and applications to {K}-stability | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-06-28 Yuchen Liu, Chenyang Xu, Ziquan Zhuang
We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac {n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly $\mathrm {K}$-stable (resp. reduced uniformly $\mathrm {K}$-stable) if and only if it is $\mathrm {K}$-stable (resp
-
On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-06-28 Frank Merle, Pierre Raphaël, Igor Rodnianski, Jeremie Szeftel
In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the
-
On the implosion of a compressible fluid II: Singularity formation | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-06-28 Frank Merle, Pierre Raphaël, Igor Rodnianski, Jeremie Szeftel
In this paper, which continues our investigation of strong singularity formation in compressible fluids, we consider the compressible three-dimensional Navier-Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and
-
Proof of the satisfiability conjecture for large $k$ | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-05-27 Jian Ding, Allan Sly, Nike Sun
We establish the satisfiability threshold for random k-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_{\mathrm{SAT}}(k)$ such that a random k-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha\lt\alpha_{\mathrm{SAT}}$, and unsatisfiable for $\alpha>\alpha_{\mathrm{SAT}}$. We show that the threshold $\al
-
The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-05-26 Misha Bialy, Andrey E. Mironov
In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $\mathcal A$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. For the original Birkhoff-Poritsky formulation
-
Non-uniqueness of Leray solutions of the forced Navier-Stokes equations | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-05-26 Dallas Albritton, Elia Brué, Maria Colombo
In a seminal work, Leray (1934) demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a “background” solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly
-
On the Hofer-Zehnder conjecture | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-04-29 Egor Shelukhin
We prove that if a Hamiltonian diffeomorphism of a closed monotone symplectic manifold with semisimple quantum homology has more contractible fixed points, counted homologically, than the total dimension of the homology of the manifold, then it must have an infinite number of contractible periodic points. This constitutes a higher-dimensional homological generalization of a celebrated result of Franks
-
Global group laws and equivariant bordism rings | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-04-29 Markus Hausmann
For every abelian compact Lie group $A$, we prove that the homotopical $A$-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the $A$-equivariant Lazard ring, introduced by Cole–Greenlees–Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian $2$-groups. Our results generalize classical theorems
-
Rigid local systems and the multiplicative eigenvalue problem | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-04-29 Prakash Belkale
We give a construction that produces irreducible complex rigid local systems on $\mathbb {P}_{\mathbb {C}}^1-\{p_1,…,p_s\}$ via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups $\operatorname {SU}(n)$ (i.e., determination of the possible
-
Pointwise ergodic theorems for non-conventional bilinear polynomial averages | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-04-29 Ben Krause, Mariusz Mirek, Terence Tao
We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages \[ A_N(f,g)(x) := \frac{1}{N} \sum_{n=1}^{N} f(T^nx) g(T^{P(n)}x) $$\] as $N \to \infty $, where $T\colon X\to X$ is a measure-preserving transformation of a $\sigma $-finite measure space $(X,\mu )$, $P(\mathrm {n}) \in \mathbb Z[\mathrm {n}]$
-
The stable Adams conjecture and higher associative structures on Moore spectra | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-02-28 Prasit Bhattacharya, Nitu Kitchloo
In this paper, we provide a new proof of the stable Adams conjecture. Our proof constructs a canonical null-homotopy of the stable J-homomorphism composed with a virtual Adams operation, by applying the K-theory functor to a multinatural transformation. We also point out that the original proof of the stable Adams conjecture is incorrect and present a correction. This correction is crucial to our main
-
Measures of maximal entropy for surface diffeomorphisms | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-02-28 Jérôme Buzzi, Sylvain Crovisier, Omri Sarig
We show that $C^\infty $-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal entropy in general, and exactly one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist. To do this we generalize Smale’s spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms
-
Rough solutions of the $3$-D compressible Euler equations | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-02-28 Qian Wang
We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \mathfrak{w}) \in H^s\times H^s\times H^{s’}$, $22$. At the opposite extreme, in the incompressible case, i.e., with a constant density, the result is known to hold for $\mathfrak{w}\in H^s$, $s>3/2$ and fails for $s\le 3/2$
-
Keel’s base point free theorem and quotients in mixed characteristic | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-02-28 Jakub Witaszek
We develop techniques of mimicking the Frobenius action in the study of universal homeomorphisms in mixed characteristic. As a consequence, we show a mixed characteristic Keel’s base point free theorem obtaining applications towards the mixed characteristic Minimal Model Program, we generalise Kollár’s theorem on the existence of quotients by finite equivalence relations to mixed characteristic, and
-
The Chow $t$-structure on the $\infty $-category of motivic spectra | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-02-28 Tom Bachmann, Hana Jia Kong, Guozhen Wang, Zhouli Xu
We define the Chow $t$-structure on the $\infty$-category of motivic spectra $\mathcal{SH}(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $\mathcal{SH}(k)^{c\heartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $\mathcal{SH}(k)^{\mathrm{cell}, c\heartsuit}$ as the category of even graded
-
Lebesgue measure of Feigenbaum Julia sets | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2021-12-21 Artur Avila, Mikhail Lyubich
We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family $P_c: z \mapsto z^2+c$ the corresponding set of parameters $c$ is shown to have positive Hausdorff dimension. Our examples include renormalization fixed points, and the corresponding quadratic polynomials in their stable manifold are the first known rational maps for which the
-
The Gaussian Double-Bubble and Multi-Bubble Conjectures | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-01-01 Emanuel Milman, Joe Neeman
We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb {R}^n$ into $q$ cells of prescribed (positive) Gaussian measure when $2 \leq q \leq n+1$, is to use a “simplicial cluster,” obtained from the Voronoi cells of $q$ equidistant points. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers (up to null-sets).
-
Affine Beilinson-Bernstein localization at the critical level for $\mathrm {GL}_2$ | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-01-01 Sam Raskin
We prove the rank 1 case of a conjecture of Frenkel-Gaitsgory: critical level Kac-Moody representations with regular central characters localize onto the affine Grassmannian. The method uses an analogue in local geometric Langlands of the existence of Whittaker models for most representations of $\mathrm {GL}_2$ over a non-Archimedean field.
-
Erratum: Euclidean triangles have no hot spots | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-01-01 Chris Judge, Sugata Mondal
Original article: https://doi.org/10.4007/annals.2020.191.1.3
-
Erratum: A corrected proof of the scale recurrence lemma from the paper “Stable intersections of regular Cantor sets with large Hausdorff dimensions” | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2022-01-01 Carlos Gustavo T. de A. Moreira, Alex Mauricio Zamudio
This is an erratum for the paper “Stable intersections of regular Cantor sets with large Hausdorff dimensions” by Moreira and Yoccoz. We show how to fix a flaw — a bad choice of parameters — in the proof of the scale recurrence lemma. This lemma is an important step towards establishing the main theorem. Original article: https://doi.org/10.2307/3062110
-
Polynomial structure of Gromov–Witten potential of quintic $3$-folds | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2021-11-02 Huai-Liang Chang, Shuai Guo, Jun Li
We prove two structure theorems for the Gromov-Witten theory of the quintic threefolds, which together give an effective algorithm for the all genus Gromov-Witten potential functions of quintics. By using these structure theorems, we prove Yamaguchi-Yau’s Polynomial Ring Conjecture in this paper and prove Bershadsky-Cecotti-Ooguri-Vafa’s Feynman rule conjecture in the subsequent paper.
-
Finite-time singularity formation for $C^{1,\alpha }$ solutions to the incompressible Euler equations on $\mathbb{R}^3$ | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2021-11-02 Tarek M. Elgindi
It has been known since work of Lichtenstein and Gunther in the 1920s that the 3D incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.
-
Equiangular lines with a fixed angle | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2021-11-02 Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, Yufei Zhao
Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix $0 < \alpha < 1$. Let $N_\alpha (d)$ denote the maximum number of lines through the origin in $\mathbb {R}^d$ with pairwise common angle arccos$\alpha$. Let $k$ denote the minimum number (if it
-
Improved bounds for the sunflower lemma | Annals of Mathematics Ann. Math. (IF 4.9) Pub Date : 2021-11-02 Ryan Alweiss, Shachar Lovett, Kewen Wu, Jiapeng Zhang
A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all of them. Erdős and Rado proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must contain a sunflower with $r$ petals. The famous sunflower conjecture states that the bound on the number of sets can be improved to