Showing papers in "Journal of the American Mathematical Society in 2016"

Testing the manifold hypothesis

Abstract: The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for fitting a manifold to an unknown probability distribution supported in a separable Hilbert space, only using i.i.d samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution $P$ supported on the unit ball of a separable Hilbert space $H$. Let $G(d, V, \tau)$ be the set of submanifolds of the unit ball of $H$ whose volume is at most $V$ and reach (which is the supremum of all $r$ such that any point at a distance less than $r$ has a unique nearest point on the manifold) is at least $\tau$. Let $L(M, P)$ denote mean-squared distance of a random point from the probability distribution $P$ to $M$. We obtain an algorithm that tests the manifold hypothesis in the following sense. The algorithm takes i.i.d random samples from $P$ as input, and determines which of the following two is true (at least one must be): (a) There exists $M \in G(d, CV, \frac{\tau}{C})$ such that $L(M, P) \leq C \epsilon.$ (b) There exists no $M \in G(d, V/C, C\tau)$ such that $L(M, P) \leq \frac{\epsilon}{C}.$ The answer is correct with probability at least $1-\delta$.

Decoupling, exponential sums and the Riemann zeta function

Abstract: We establish a new decoupling inequality for curves in the spirit of [B-D1], [B-D2] which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in [H] In particular, this leads to an improved bound $|\zeta(\frac 12+it)|\ll t^{53/342+\varepsilon}$ for the zeta function on the critical line

Gröbner methods for representations of combinatorial categories

Abstract: Given a category C of a combinatorial nature, we study the following fundamental question: how does the combinatorial behavior of C affect the algebraic behavior of representations of C? We prove two general results. The first gives a combinatorial criterion for representations of C to admit a theory of Grobner bases. From this, we obtain a criterion for noetherianity of representations. The second gives a combinatorial criterion for a general “rationality” result for Hilbert series of representations of C. This criterion connects to the theory of formal languages, and makes essential use of results on the generating functions of languages, such as the transfer-matrix method and the Chomsky–Schutzenberger theorem. Our work is motivated by recent work in the literature on representations of various specific categories. Our general criteria recover many of the results on these categories that had been proved by ad hoc means, and often yield cleaner proofs and stronger statements. For example: we give a new, more robust, proof that FI-modules (originally introduced by Church–Ellenberg–Farb), and a family of natural generalizations, are noetherian; we give an easy proof of a generalization of the Lannes–Schwartz artinian conjecture from the study of generic representation theory of finite fields; we significantly improve the theory of ∆modules, introduced by Snowden in connection to syzygies of Segre embeddings; and we establish fundamental properties of twisted commutative algebras in positive characteristic.

Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics

Abstract: Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the original conjecture in the presence of continuous automorphisms. The construction hinges upon an alternative approach to properness that uses in an essential way the metric completion with respect to a Finsler metric and its quotients with respect to group actions. This approach also allows us to formulate and prove new optimal versions of Tian's conjecture in the setting of smooth and singular Kahler-Einstein metrics, with or without automorphisms, as well as for Kahler-Ricci solitons. Moreover, we reduce both Tian's original conjecture (in the absence of automorphisms) and our modification of it (in the presence of automorphisms) in the general case of constant scalar curvature metrics to a conjecture on regularity of minimizers of the K-energy in the Finsler metric completion. Finally, our results also resolve Tian's conjecture on the Moser-Trudinger inequality for Fano manifolds with Kahler-Einstein metrics.

The Monge-Ampère equation for (n − 1)-plurisubharmonic functions on a compact Kähler manifold

Abstract: A C function on C is called (n− 1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of any n − 1 eigenvalues of its complex Hessian is nonnegative. We show that the associated MongeAmpère equation can be solved on any compact Kähler manifold. As a consequence we prove the existence of solutions to an equation of FuWang-Wu, giving Calabi-Yau theorems for balanced, Gauduchon and strongly Gauduchon metrics on compact Kähler manifolds.

Schubert calculus and torsion explosion

Abstract: The author observes that certain numbers occurring in Schubert calculus for SL also occur as entries in intersection forms controlling decompositions of Soergel bimodules in higher rank. These numbers grow exponentially. This observation gives many counter-examples to the expected bounds in Lusztig’s conjecture on the characters of simple rational modules for SL over fields of positive characteristic. The examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups.

Kink dynamics in the $\phi ^4$ model: Asymptotic stability for odd perturbations in the energy space

Abstract: We consider a classical equation known as the $\phi^4$ model in one space dimension. The kink, defined by $H(x)=\tanh(x/{\sqrt{2}})$, is an explicit stationary solution of this model. From a result of Henry, Perez and Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability of the kink for odd perturbations in the energy space. The proof is based on Virial-type estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general Klein-Gordon equations with potential: the interactions of the so-called internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.

The distribution of sandpile groups of random graphs

Abstract: We determine the distribution of the sandpile group (a.k.a. Jacobian) of the Erd\H{o}s-R\'enyi random graph G(n,q) as n goes to infinity. Since any particular group appears with asymptotic probability 0 (as we show), it is natural ask for the asymptotic distribution of Sylow p-subgroups of sandpile groups. We prove the distributions of Sylow p-subgroups converge to specific distributions conjectured by Clancy, Leake, and Payne. These distributions are related to, but different from, the Cohen-Lenstra distribution. Our proof involves first finding the expected number of surjections from the sandpile group to any finite abelian group (the "moments" of a random variable valued in finite abelian groups). To achieve this, we show a universality result for the moments of cokernels of random symmetric integral matrices that is strong enough to handle dependence in the diagonal entries. We then show these moments determine a unique distribution despite their p^{k^2}-size growth.

Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients

Abstract: We study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles We study those biorthogonal ensembles for which the underlying biorthogonal family s

Higher order Fourier analysis of multiplicative functions and applications

Abstract: We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm of an arbitrary degree. The proof uses tools from higher order Fourier analysis and some soft number theoretic input that comes in the form of an orthogonality criterion of Katai. We use variants of this structure theorem to derive applications of number theoretic and combinatorial flavor: $(i)$ we give simple necessary and sufficient conditions for the Gowers norms (over $\mathbb{N}$) of a bounded multiplicative function to be zero, $(ii)$ generalizing a classical result of Daboussi and Delange we prove asymptotic orthogonality of multiplicative functions to "irrational" nilsequences, $(iii)$ we prove that for certain polynomials in two variables all "aperiodic" multiplicative functions satisfy Chowla's zero mean conjecture, $(iv)$ we give the first partition regularity results for homogeneous quadratic equations in three variables showing for example that on every partition of the integers into finitely many cells there exist distinct $x,y$ belonging to the same cell and $\lambda\in \mathbb{N}$ such that $16x^2+9y^2=\lambda^2$ and the same holds for the equation $x^2-xy+y^2=\lambda^2$.

Diagonal cycles and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-functions

Abstract: This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over $ \mathbb{Q}$ viewed over the fields cut out by certain self-dual Artin representations of dimension at most $ 4$. When the associated $ L$-function vanishes (to even order $ \ge 2$) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be linearly independent assuming the non-vanishing of a Garrett-Hida $ p$-adic $ L$-function at a point lying outside its range of classical interpolation. The key tool for both results is the study of certain $ p$-adic families of global Galois cohomology classes arising from Gross-Kudla-Schoen diagonal cycles in a tower of triple products of modular curves.

Gromov-Witten/pairs correspondence for the quintic 3-fold

Abstract: We use the Gromov-Witten/Pairs descendent correspondence for toric 3-folds and degeneration arguments to establish the GW/P correspondence for several compact Calabi-Yau 3-folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of the GW/P correspondence for descendents in relative geometries. Projective bundles over surfaces relative to a section play a special role. The GW/P correspondence for Calabi-Yau complete intersections provides a structure result for the Gromov-Witten invariants in a fixed curve class. After change of variables, the Gromov-Witten series is a rational function in the variable −q = eiu invariant under q ↔ q−1.

Lens rigidity for manifolds with hyperbolic trapped set

Abstract: For a Riemannian manifold (M, g) with strictly convex boundary ∂M , the lens data consists in the set of lengths of geodesics γ with endpoints on ∂M , together with their endpoints (x − , x +) ∈ ∂M × ∂M and tangent exit vectors (v − , v +) ∈ T x− M × T x+ M. We show deformation lens rigidity for manifolds with hyperbolic trapped set and no conjugate points, a class which contains all manifolds with negative curvature and strictly convex boundary, including those with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension 2, we prove that the set of endpoints and exit vectors of geodesics (ie. the scattering data) determines the Riemann surface up to conformal diffeomorphism.

On the Erdős-Szekeres convex polygon problem

Abstract: Let $ES(n)$ be the smallest integer such that any set of $ES(n)$ points in the plane in general position contains $n$ points in convex position. In their seminal 1935 paper, Erdos and Szekeres showed that $ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)}$. In 1960, they showed that $ES(n) \geq 2^{n-2} + 1$ and conjectured this to be optimal. In this paper, we nearly settle the Erdos-Szekeres conjecture by showing that $ES(n) =2^{n +o(n)}$.

Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations

Abstract: We prove that in a certain asymptotic regime, solutions of the Gross-Pitaevskii equation converge to solutions of the incompressible Euler equation, and solutions to the parabolic Ginzburg-Landau equations converge to solutions of a limiting equation which we identify. We work in the setting of the whole plane (and possibly the whole three-dimensional space in the Gross-Pitaevskii case), in the asymptotic limit where $\epsilon$, the characteristic lengthscale of the vortices, tends to $0$, and in a situation where the number of vortices $N$ blows up as $\epsilon \to 0$. The requirements are that $N$ should blow up faster than $|\log \epsilon|$ in the Gross-Pitaevskii case, and at most like $|\log \epsilon|$ in the parabolic case. Both results assume a well-prepared initial condition and regularity of the limiting initial data, and use the regularity of the solution to the limiting equation. In the case of the parabolic Ginzburg-Landau equation, the limiting mean-field dynamical law that we identify coincides with the one proposed by Chapman-Rubinstein-Schatzman and E in the regime $N\ll |\log \epsilon|$, but not if $N$ grows faster.

Geometric complexity theory V: Efficient algorithms for Noether normalization

Abstract: We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noether's Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry. In particular, we show that: (1) The categorical quotient for any finite dimensional representation $V$ of $SL_m$, with constant $m$, is explicit in characteristic zero. (2) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in the dimension of $V$. (3) The categorical quotient of the space of $r$-tuples of $m \times m$ matrices by the simultaneous conjugation action of $SL_m$ is explicit in any characteristic. (4) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in $m$ and $r$ in any characteristic $p$ not in $[2,\ m/2]$. (5) NNL for every explicit variety in zero or large enough characteristic can be solved deterministically in quasi-polynomial time, assuming the hardness hypothesis for the permanent in geometric complexity theory. The last result leads to a geometric complexity theory approach to put NNL for every explicit variety in P.

A classification of irreducible admissible mod p representations of p-adic reductive groups

Abstract: Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible C-representations of G = G(F), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GL_m, as further expanded by the first-named author, who treated split groups G. As in the split case, we first get a classification in terms of supersingular representations of Levi subgroups, and as a consequence show that supersingularity is the same as supercuspidality.

Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension

Abstract: Reference EPFL-ARTICLE-231100doi:10.1090/jams/871 URL: http://dx.doi.org/10.1090/jams/871 Record created on 2017-09-19, modified on 2017-10-02

Weak mixing directions in non-arithmetic Veech surfaces

Abstract: We show that the billiard in a regular polygon is weak mixing in almost every invariant surface, except in the trivial cases which give rise to lattices in the plane (triangle, square and hexagon). More generally, we study the problem of prevalence of weak mixing for the directional flow in an arbitrary non-arithmetic Veech surface, and show that the Hausdorff dimension of the set of non-weak mixing directions is not full. We also provide a necessary condition, verified for instance by the Veech surface corresponding to the billiard in the pentagon, for the set of non-weak mixing directions to have positive Hausdorff dimension.

The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions

Abstract: We prove the nonvanishing hypothesis at infinity for Rankin-Selberg convolutions for $\GL(n)\times \GL(n-1)$.

Non-density of small points on divisors on Abelian varieties and the Bogomolov conjecture

Abstract: The Bogomolov conjecture for a curve claims finiteness of algebraic points on the curve which are small with respect to the canonical height. Ullmo has established this conjecture over number fields, and Moriwaki generalized it to the assertion over finitely generated fields over $\mathbb{Q}$ with respect to arithmetic heights. As for the case of function fields with respect to the geometric heights, Cinkir has proved the conjecture over function fields of characteristic $0$ and of transcendence degree $1$. However, the conjecture has been open over other function fields. In this paper, we prove that the Bogomolov conjecture for curves holds over any function field. In fact, we show that any non-special closed subvariety of dimension $1$ in an abelian variety over function fields has only a finite number of small points. This result is a consequence of the investigation of non-density of small points of closed subvarieties of abelian varieties of codimension $1$. As a by-product, we remark that the geometric Bogomolov conjecture, which is a generalization of the Bogomolov conjecture for curves over function fields, holds for any abelian variety of dimension at most $3$. Combining this result with our previous works, we see that the geometric Bogomolov conjecture holds for all abelian varieties for which the difference between its nowhere degeneracy rank and the dimension of its trace is not greater than $3$.