abstract: Let $\gamma:[0,1]\rightarrow\mathbb{S}^2$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det(\gamma(\theta),\gamma'(\theta),\linebreak\gamma" (\theta))\neq 0$. For each $\theta\in[0,1]$, let $V_\theta=\gamma(\theta)^\perp$ and let $\pi_\theta:\mathbb{R}^3\rightarrow V_\theta$ be the orthogonal projections. We prove that if $A\subset\mathbb{R}^3$ is a Borel set, then for a.e. $\theta\in [0,1]$ we have $\dim(\pi_\theta(A))=\min\{2,\dim A\}$. More generally, we prove an exceptional set estimate. For $A\subset\mathbb{R}^3$ and $0\le s\le 2$, define $E_s(A):=\{\theta\in[0,1]:\dim(\pi_\theta(A))2$, then for a.e. $\theta\in[0,1]$ we have $\mathcal{H}^2(\pi_\theta(A))>0$.
abstract: The proofs of Theorem 1.1 and Theorem 1.5\,(2) in the authors' paper \emph{The Hasse norm principle for abelian extensions} are incorrect. We point out the mistakes and provide correct proofs, using techniques of the original paper.
abstract: We introduce a coarse perspective on relations of the $\operatorname{SU}(2)$-Witten-Reshetikhin-Turaev TQFT, the Weil-Petersson geometry of the Teichm\"uller space, and volumes of hyperbolic 3-manifolds. Using data from the asymptotic expansions of the curve operators in the skein theoretic version of the $\operatorname{SU}(2)$-TQFT, we define the quantum intersection number between pants decompositions of a closed surface. We show that the quantum intersection number admits two sided bounds in terms of the geometric intersection number and we use it to obtain a metric on the pants graph of surfaces. Using work of Brock we show that the pants graph equipped with this metric is quasi-isometric to the Teichm\"uller space with the Weil-Petersson metric and that the translation length of our metric provides two sided linear bounds on the volume of hyperbolic fibered manifolds. We briefly discuss how these relations are interpreted from the view point of $\operatorname{SU}(2)$-character varieties of 3-manifolds. We also obtain a characterization of pseudo-Anosov mapping classes in terms of asymptotics of the quantum intersection number under iteration in the mapping class group and relate these asymptotics with stretch factors. We also discuss how these results fit with a conjecture of Andersen, Masbaum and Ueno about quantum representations of mapping class groups.
The theta correspondence for the dual pair (Uen,Uep,q) (where covers of Un and Up,q respectively) is used to find nice bases for a collection of unitary highest weight modules of Uep,q, and also for the highest weight vectors with respect to the obvious maximal compact subgroup. The techniques used involve signed Hibi rings, an extension of the concept of Hibi ring, which has been identified as an underlying structure in constructing standard monomial theories, and in finding straightening laws in certain rings. An extension of the Littlewood-Richardson rule to tensor products of arbitrary rational representations of GLn(C) is required for finding the desired bases, and is also provided. Uen and Uep,q are 2-fold
Let m >= 3, we prove that (alpha n(theta) mod 1)n>0 has Poissonian m-point correlation for all alpha > 0, provided theta
Zygmund dilations are a group of dilations lying in between the standard product the(delta 1x1,delta 2x2,delta 1 delta 2x3). The dyadic multiresolution analysis and the related dyadic-probabilistic methods have been very impactful in the modern product singular integral theory. However, multiresolution analysis has not been understood in the Zygmund dilation setting or in other modified product space and justify its usefulness by bounding, on weighted spaces, a general class of singular integrals that Zygmund kernels showcasing the optimality of our kernel assumptions for weighted estimates.
We present a variant of the current widely-used method initiated by Choie and Kohnen in the study of the location of the first sign change of the Fourier coefficients of a holomorphic cusp form when all the coefficients are real. This version circumvents the use of Atkin-Lehner theory of cuspidal newforms, instead utilizing the Eisenstein series, and it applies directly to cases including integral weight cusp forms on the congruence subgroup Gamma 0(N) of any level N as well as half-integral weight cusp forms.
We give the first example of an open manifold with positive Ricci curvature and a non-proper Busemann function at a point. This provides counterexamples to a longtime well-known open question whether the Busemann function at a point of an open manifold with nonnegative Ricci curvature is proper.
The purpose of this paper is to derive volume and other geometric information for three-dimensional complete manifolds with positive scalar curvature. In the case that the Ricci curvature is nonnegative, it is shown that the volume of the manifold must be of linear growth when the scalar curvature is bounded from below by a positive constant. This answers a question of Gromov in the affirmative for dimension three. Volume growth estimates are also obtained for the case when scalar curvature decays to zero. In fact, results of similar nature are established for the more general case that the Ricci curvature is asymptotically nonnegative.
We prove that for any twist rigid compact $p$-adic analytic group $G$, its twist representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If $G$ is moreover a pro-$p$ group, we prove that its twist representation zeta function is rational in $p^{-s}$. To establish these results we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass. Second part of arXiv:2007.10694.
abstract: We prove the local Gan--Gross--Prasad conjecture for generic $L$-packets of real unitary groups. The proof is to reduce the conjecture to the tempered case which has been treated in our previous paper.
Motivated by the Lipschitz rigidity problem in scalar curvature geometry, we prove that if a closed smooth spin manifold admits a distance decreasing continuous map of non-zero degree to a sphere, then either the scalar curvature is strictly less than the sphere somewhere or the map is a distance isometry. Moreover, the property also holds for continuous metrics with scalar curvature lower bound in some weak sense. This extends a result in the recent work of Cecchini-Hanke-Schick and answers a question of Gromov. The method is based on studying the harmonic map heat flow coupled with the Ricci flow from rough initial data to reduce the case to smooth metrics and smooth maps so that results by Llarull can be applied.
We study the zeros of sections of the form $T_k s_k$ of a large power $L^{\otimes k} \to M$ of a holomorphic positive Hermitian line bundle over a compact K\''ahler manifold $M$, where $s_k$ is a random holomorphic section of $L^{\otimes k}$ and $T_k$ is a Berezin-Toeplitz operator, in the limit $k \to +\infty$. In particular, we compute the second order approximation of the expectation of the distribution of these zeros. In a ball of radius of order $k^{-\frac{1}{2}}$ around $x \in M$, assuming that the principal symbol $f$ of $T_k$ is real-valued and vanishes transversally, we show that this expectation exhibits two drastically different behaviors depending on whether $f(x) = 0$ or $f(x) \neq 0$. These different regimes are related to a similar phenomenon about the convergence of the normalized Fubini-Study forms associated with $T_k$: they converge to the K\''ahler form in the sense of currents as $k\rightarrow + \infty$, but not as differential forms (even pointwise). This contrasts with the standard case $f=1$, in which the convergence is in the $\mathscr{C}^{\infty}$-topology. From this, we are able to recover the zero set of $f$ from the zeros of $T_k s_k$.
We extend the recent work of Chong et al., (2022) to the critical case. More precisely, we prove global in time, uniform in $N$ estimates for the solutions $\phi$, $\Lambda$ and $\Gamma$ of a coupled system of Hartree--Fock--Bogoliubov type with interaction potential $\frac1NV_N(x-y)=N^{2}v(N(x-y))$. We assume that the potential $v$ is small which satisfies some technical conditions, and the initial conditions have finite energy. The main ingredient is a sharp estimate for the linear Schr\"odinger equation with potential in 6+1 dimension, which may be of interest in its own right.
The unknotting number of knots is a difficult quantity to compute, and even its behavior under basic satelliting operations is not understood. We establish a lower bound on the unknotting number of cable knots and iterated cable knots purely in terms of the winding number of the pattern. The proof uses Alishahi-Eftekhary's bounds on unknotting number from knot Floer homology together with Hanselman-Watson's computation of the knot Floer homology of cables in terms of immersed curves in the punctured torus.
In this paper, we prove interior gradient estimates for the Lagrangian mean curvature equation, if the Lagrangian phase is critical and supercritical and $C^{2}$. Combined with the a priori interior Hessian estimates proved in [Bha21, Bha22], this solves the Dirichlet boundary value problem for the critical and supercritical Lagrangian mean curvature equation with $C^0$ boundary data. We also provide a uniform gradient estimate for lower regularity phases that satisfy certain additional hypotheses.
We give the first examples of (non-amenable group) amenable actions on stably finite simple C*-algebras. More precisely, we give such actions for any countable group in an explicit way. The main ingredients of our construction are the full Fock space of the regular representation and a trace-scaling automorphism.
The admissible locus F(G, mu, b)a inside the flag variety F(G, mu), attached to a reductive group G with a minuscule cocharacter mu of G, is a p-adic analogue of the complex analytic period spaces. It has an algebraic approximation F(G, mu, b)wainside the flag variety, called the weakly admissible locus. On the flag variety F(G, mu), we have the Newton stratification which has the admissible locus as its unique open stratum. In this paper, we study the relation between the Newton strata and the weakly admissible locus. We show that F(G, mu, b)wais maximal (in the sense that it's a union of Newton strata) is equivalent to (G, mu) weakly fully HN-decomposable, it's also equivalent to the condition that the Newton stratification is finer than the Harder-Narasimhan stratification. These equivalent conditions are generalizations of the fully HN-decomposable condition and the weakly accessible condition. Moreover, we give a criterion to determine whether a Newton stratum is completely contained in the weakly admissible locus involving G-bundles as extensions of M-bundles over the Fargues-Fontaine curve, where M is a Levi subgroup of G. When G = GLn, we also give a combinatorial inductive criterion to determine whether a vector bundle over the Fargues-Fontaine curve is an extension of two given vector bundles.
We reveal a new and refined application of (a weaker statement than) the Iwasawa main conjecture for elliptic curves to the structure of Selmer groups of elliptic curves of arbitrary rank. For a large class of elliptic curves, we obtain the following arithmetic consequences. 1. Kato's Kolyvagin systems is non-trivial. It is the cyclotomic analogue of the Kolyvagin conjecture. 2. The structure of Selmer groups of elliptic curves over the rationals is completely determined in terms of certain modular symbols. It is a structural refinement of Birch and Swinnerton-Dyer conjecture. 3. The rank zero p-converse, the p-parity conjecture, and a new upper bound of the ranks of elliptic curves are obtained. 4. The conjecture of Kurihara on the semi-local description of mod p Selmer groups is confirmed. 5. An application of the p-adic Birch and Swinnerton-Dyer conjecture to the structure of Iwasawa modules is discussed.
. We show that any Riemannian metric conformal to the round metric on S n , for n ≥ 4, arises as a limit of a sequence of Riemannian metrics of positive scalar curvature on S n in the sense of uniform convergence of Riemannian distance. In particular, non-negativity of scalar curvature is not preserved under such limits.
