American Journal of Mathematics最新文献

abstract:

Let $lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne--Ferenczi--Mauduit--Rivat--S'ark"ozy asks to show that if $P(x)inmathbb{Z}[x]$, then the sequence $lambda(P(n))$ changes sign infinitely often, assuming only that $P(x)$ is not the square of another polynomial.

We show that the sequence $lambda(P(n))$ indeed changes sign infinitely often, provided that either (i) $P$ factorizes into linear factors over the rationals; or (ii) $P$ is a reducible cubic polynomial; or (iii) $P$ factorizes into a product of any number of quadratics of a certain type; or (iv) $P$ is any polynomial not belonging to an exceptional set of density zero.

Concerning (i), we prove more generally that the partial sums of $g(P(n))$ for $g$ a bounded multiplicative function exhibit nontrivial cancellation under necessary and sufficient conditions on $g$. This establishes a ``99% version'' of Elliott's conjecture for multiplicative functions taking values in the roots of unity of some order. Part (iv) also generalizes to the setting of $g(P(n))$ and provides a multiplicative function analogue of a recent result of Skorobogatov and Sofos on almost all polynomials attaining a prime value.

abstract:

We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $Sp(4,R)$-symmetric space. We describe a homeomorphism between the "Hitchin component" of wild $Sp(4,R)$-Higgs bundles over $CP^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $h^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $R^4$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $h^{2,2}$ associated to $Sp(4,R)$-Hitchin representations along rays of holomorphic quartic differentials.

abstract:

We study the Fourier transforms $widehat{mu}(xi)$ of non-atomic Gibbs measures $mu$ for uniformly expanding maps $T$ of bounded distortions on $[0,1]$ or Cantor sets with strong separation. When $T$ is totally non-linear, then $widehat{mu}(xi)to 0$ at a polynomial rate as $|xi|toinfty$.

abstract:

This article is concerned with the Fourier coefficients of cusp forms (not necessarily eigenforms) of half-integer weight lying in the plus space. We give a soft proof that there are infinitely many fundamental discriminants $D$ such that the Fourier coefficients evaluated at $|D|$ are non-zero. By adapting the resonance method, we also demonstrate that such Fourier coefficients must take quite large values.

abstract:

This is the first step in an attempt at a deformation theory for $G_{2}$-instantons with $1$-dimensional conic singularities. Under a set of model data, the linearization yields a Dirac operator $P$ on a certain bundle over $mathbb{S}^{5}$, called the textit{link operator}. As a dimension reduction, the link operator also arises from Hermitian Yang--Mills connections with isolated conic singularities on a Calabi--Yau $3$-fold.

Using the quaternion structure in the Sasakian geometry of $mathbb{S}^{5}$, we describe the set of all eigenvalues of $P$, denoted by $Spec P$. We show that $Spec P$ consists of finitely many integers induced by certain sheaf cohomologies on $mathbb{P}^{2}$, and infinitely many real numbers induced by the spectrum of the rough Laplacian on the pullback endomorphism bundle over $mathbb{S}^{5}$. The multiplicities and the form of an eigensection can be described fairly explicitly.

In particular, there is a relation between the spectrum on $mathbb{S}^{5}$ to certain sheaf cohomologies on~$mathbb{P}^{2}$.

Moreover, on a Calabi--Yau $3$-fold, the index of the linearized operator for admissible singular Hermitian Yang--Mills connections is also calculated, in terms of these sheaf cohomologies.

Using the representation theory of $SU(3)$ and the subgroup $S[U(1)times U(2)]$, we show an example in which $Spec P$ and the multiplicities can be completely determined.

abstract:

We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. Consequently, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semi-linear equation with low regularity data as on hyperbolic spaces.

abstract:

We prove a priori interior curvature estimates for hypersurfaces of prescribing scalar curvature equations in $mathbb{R}^{3}$. The method is motivated by the integral method of Warren and Yuan. The new observation here is that we construct a ``Lagrangian'' graph which is a submanifold of bounded mean curvature if the graph function of a hypersurface satisfies a scalar curvature equation.

abstract:

We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in ``very short intervals'' of the form $I(f):={f(x) + a : a infp}$ for $f(x)infp[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the M"{o}bius $mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for ``generic'' set of shifts. We can also exhibit examples for which there is no cancellation at all in M"{o}bius/Chowla type sums (in fact, it turns out that (square root) cancellation in M"{o}bius sums is {em equivalent} to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic ``primes are independent'' fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the M"{o}bius $mu$ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated). We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for $f(x)infq[x]$ and intervals of the form $f(x)+a$ for $ainfq$, where $p$ is fixed and $q=p^{l}$ grows.

abstract:

We discover a new characterization of plurisubharmonic functions in terms of $L^p$ extension from one point and Griffiths positivity of holomorphic vector bundles with singular Finsler metrics in terms of $L^p$ extensions. As applications, we give a stronger result or new proof of some well-known theorems on the Griffiths positivity of the holomorphic vector bundles and their direct image sheaves associated to certain holomorphic fibrations.

abstract:

In the early 1940s, P. A. Smith showed that if a finite $p$-group $G$ acts on a finite dimensional complex $X$ that is mod $p$ acyclic, then its space of fixed points, $X^G$, will also be mod $p$ acyclic.

In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $G$-space $X$ is a equivariant homotopy retract of the $p$-localization of a based finite $G$-C.W. complex, given $H<G$ and $n$, what is the smallest $r$ such that if $X^H$ is acyclic in the $(n+r)$th Morava $K$-theory, then $X^G$ must be acyclic in the $n$th Morava $K$-theory? Barthel et.~al. then answered this when $G$ is abelian, by finding general lower and upper bounds for these ``blue shift'' numbers which agree in the abelian case.

In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod $p$ homology, and thus applies to all finite dimensional $G$-spaces. This unlocks new techniques and applications in chromatic fixed point theory.

Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups.

Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for $K(n)_*$-homology disks and spheres.

Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava $K$-theories of all real Grassmanians.