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 Annals of Mathematics   [SJR: 10.358]   [H-I: 78]   [0 followers]  Follow        Subscription journal    ISSN (Print) 0003-486X    Published by Princeton University Press  [1 journal]
• The André-Oort conjecture for $\mathcal{A}_g$
• Authors: mak
Abstract: We give a proof of the André-Oort conjecture for $\mathcal{A}_g$ – the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven `averaged’ version of the Colmez conjecture yields lower bounds for Galois orbits of … Continue reading →
PubDate: Sun, 08 Oct 2017 15:12:16 +000

• Positivity for quantum cluster algebras
• Authors: mak
Abstract: Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew–symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures … Continue reading →
PubDate: Wed, 04 Oct 2017 14:15:31 +000

• Ricci flow with surgery in higher dimensions
• Authors: mak
Abstract: We present a new curvature condition which is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton’s neck-like curvature pinching estimate. Using this estimate, we are able … Continue reading →
PubDate: Tue, 03 Oct 2017 01:49:45 +000

• On the averaged Colmez conjecture
• Authors: mak
Abstract: The Colmez conjecture is a formula expressing the Faltings height of an abelian variety with complex multiplication in terms of some linear combination of logarithmic derivatives of Artin L-functions. The aim of this paper to prove an averaged version of … Continue reading →
PubDate: Tue, 03 Oct 2017 01:41:43 +000

• The Gopakumar-Vafa formula for symplectic manifolds
• Authors: mak
Abstract: The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau $3$-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa conjecture holds for any … Continue reading →
PubDate: Sat, 30 Sep 2017 01:51:25 +000

• Faltings heights of abelian varieties with complex multiplication
• Authors: mak
Abstract: Let $M$ be the Shimura variety associated with the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points … Continue reading →
PubDate: Thu, 28 Sep 2017 18:44:36 +000

• Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture
and of the lower bound in Yau’s conjecture
• Authors: mak
Abstract: Let $u$ be a harmonic function in the unit ball $B(0,1) \subset \mathbb{R}^n$, $n \geq 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that $H^{n-1}(\{u=0 \}\cap B) \geq … Continue reading → PubDate: Mon, 25 Sep 2017 19:03:14 +000 • Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure • Authors: mak Abstract: Let$\mathbb{M}$be a compact$C^\infty$-smooth Riemannian manifold of dimension$n$,$n\geq 3$, and let$\varphi_\lambda: \Delta_M \varphi_\lambda + \lambda \varphi_\lambda = 0$denote the Laplace eigenfunction on$\mathbb{M}$corresponding to the eigenvalue$\lambda$. We show that$$H^{n-1}(\{ \varphi_\lambda=0\}) \leq … Continue reading → PubDate: Mon, 25 Sep 2017 18:56:51 +000 • Erratum on “On the number of generators of ideals in polynomial rings” • Authors: annadmin Abstract: We explain a mistake that occurred in the proof of Murthy’s conjecture by the author. PubDate: Wed, 09 Aug 2017 16:35:00 +000 • Total Betti numbers of modules of finite projective dimension • Authors: annadmin Abstract: The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that the$i^{\rm th}$Betti number$\beta_i(M)$of a nonzero module$M$of finite length and finite projective dimension over a local ring$R$of dimension$d$should be at least${d \choose i}$. It would … Continue reading → PubDate: Wed, 09 Aug 2017 16:35:00 +000 • A sharp Schrödinger maximal estimate in$\mathbb {R}^2$• Authors: annadmin Abstract: We show that$\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$almost everywhere for all$f \in H^s (\mathbb{R}^2)$provided that$s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling. PubDate: Wed, 09 Aug 2017 16:35:00 +000 • Borel circle squaring • Authors: annadmin Abstract: We give a completely constructive solution to Tarski’s circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If$k \geq 1$and$A, B \subset \mathbb{R}^k$are bounded Borel sets with the … Continue reading → PubDate: Wed, 09 Aug 2017 16:35:00 +000 • The triviality of the 61-stem in the stable homotopy groups of spheres • Authors: annadmin Abstract: We prove that the 2-primary$\pi_{61}$is zero. As a consequence, the Kervaire invariant element$\theta_5$is contained in the strictly defined 4-fold Toda bracket$\langle 2, \theta_4, \theta_4, 2\rangle$.Our result has a geometric corollary: the 61-sphere has a unique … Continue reading → PubDate: Wed, 09 Aug 2017 16:35:00 +000 • Bilinear forms with Kloosterman sums and applications • Authors: annadmin Abstract: We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Pólya-Vinogradov range). We then derive applications to the second moment of cusp forms twisted … Continue reading → PubDate: Wed, 09 Aug 2017 16:35:00 +000 • Revêtements du demi-plan de Drinfeld et correspondance de Langlands$p$-adique • Authors: annadmin Abstract: Nous décrivons le complexe de de Rham des revêtements du demi-plan de Drinfeld pour${\rm GL}_2(\mathbf{Q}_p)$. Cette description, conjecturée par Breuil et Strauch, fournit une réalisation géométrique de la correspondance de Langlands locale$p$-adique pour certaines représentations de de Rham … Continue reading → PubDate: Wed, 09 Aug 2017 16:35:00 +000 • A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang Formation • Authors: mak Abstract: We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on$(0,\infty) \times \mathbb{T}^3$. The Kasner solutions model a spatially uniform scalar field evolving in … Continue reading → PubDate: Wed, 05 Jul 2017 16:02:58 +000 • Monochromatic sums and products in$\mathbb N$• Authors: annadmin Abstract: An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair$\{x+y,xy\}$. We answer this question affirmatively in a strong sense by exhibiting a large new class of nonlinear patterns that can … Continue reading → PubDate: Fri, 10 Mar 2017 00:00:00 +000 • Positive scalar curvature on foliations • Authors: annadmin Abstract: We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of … Continue reading → PubDate: Thu, 09 Mar 2017 00:00:00 +000 • The sphere packing problem in dimension$24$• Authors: annadmin Abstract: Building on Viazovska’s recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, … Continue reading → PubDate: Wed, 08 Mar 2017 00:00:00 +000 • The sphere packing problem in dimension 8 • Authors: annadmin Abstract: In this paper we prove that no packing of unit balls in Euclidean space$\mathbb R^8$has density greater than that of the$E_8$-lattice packing. PubDate: Tue, 07 Mar 2017 00:00:00 +000 • Cubic curves and totally geodesic subvarieties of moduli space • Authors: annadmin Abstract: In this paper we present the first example of a primitive, totally geodesic subvariety$F \subset \mathcal{M}_{g,n}$with$\mathrm{dim}(F)>1$. The variety we consider is a surface$F \subset \mathcal{M}_{1,3}$defined using the projective geometry of plane cubic curves. We also … Continue reading → PubDate: Mon, 06 Mar 2017 00:00:00 +000 • Higher ramification and the local Langlands correspondence • Authors: annadmin Abstract: Let$F$be a non-Archimedean locally compact field. We show that the local Langlands correspondence over$F$has a property generalizing the higher ramification theorem of local class field theory. If$\pi$is an irreducible cuspidal representation of a general … Continue reading → PubDate: Sun, 05 Mar 2017 00:00:00 +000 • Harmonic quasi-isometric maps between rank one symmetric spaces • Authors: annadmin Abstract: We prove that a quasi-isometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture. PubDate: Sat, 04 Mar 2017 00:00:00 +000 • Global smooth and topological rigidity of hyperbolic lattice actions • Authors: annadmin Abstract: In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices.Suppose$\Gamma$is a lattice in a semisimple Lie group, all of whose factors have rank$2$or higher. Let$\alpha$be a smooth$\Gamma$-action on a … Continue reading → PubDate: Fri, 03 Mar 2017 00:00:00 +000 • Global well-posedness for the Yang-Millsequation in$4+1$dimensions. Small energy • Authors: annadmin Abstract: We consider the hyperbolic Yang-Mills equation on the Minkowski space$\mathbb{R}^{4+1}$. Our main result asserts that this problem is globally well-posed for all initial data whose energy is sufficiently small. This solves a longstanding open problem. PubDate: Fri, 03 Mar 2017 00:00:00 +000 • Shtukas and the Taylor expansion of$L$-functions • Authors: annadmin Abstract: We define the Heegner–Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with$r$-modifications for an even integer$r$. We prove an identity between (1) the$r$-th central derivative of the quadratic base change$L$-function associated to … Continue reading → PubDate: Thu, 02 Mar 2017 00:00:00 +000 • Ramsey numbers of degenerate graphs • Authors: annadmin Abstract: A graph is$d$-degenerate if all its subgraphs have a vertex of degree at most$d$. We prove that there exists a constant$c$such that for all natural numbers$d$and$r$, every$d$-degenerate graph$H$of chromatic number … Continue reading → PubDate: Thu, 02 Mar 2017 00:00:00 +000 • On the generic part of the cohomology of compact unitary Shimura varieties • Authors: annadmin Abstract: The goal of this paper is to show that the cohomology of compact unitary Shimura varieties is concentrated in the middle degree and torsion-free, after localizing at a maximal ideal of the Hecke algebra satisfying a suitable genericity assumption. Along … Continue reading → PubDate: Wed, 01 Mar 2017 00:00:00 +000 • On the growth of$L^2$-invariants for sequences of lattices in Lie groups • Authors: annadmin Abstract: We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge–Wallach Theorem, of a theorem of Delorme and various other limit multiplicity … Continue reading → PubDate: Wed, 01 Mar 2017 00:00:00 +000 • Measurable circle squaring • Authors: annadmin Abstract: Laczkovich proved that if bounded subsets$A$and$B$of$\mathbb{R}^k$have the same nonzero Lebesgue measure and the upper box dimension of the boundary of each set is less than$k$, then there is a partition of$A$into … Continue reading → PubDate: Mon, 06 Feb 2017 00:00:00 +000 • Derived categories and Deligne-Lusztig varieties II • Authors: annadmin Abstract: This paper is a continuation and a completion of the work of the first and the third author on the Jordan decomposition. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in … Continue reading → PubDate: Sun, 05 Feb 2017 00:00:00 +000 • On the stability threshold for the 3D Couette flow in Sobolev regularity • Authors: annadmin Abstract: We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number \textbfRe. Our goal is to estimate how the stability threshold scales in$\textbf{Re}$: the largest the initial perturbation can … Continue reading → PubDate: Sat, 04 Feb 2017 00:00:00 +000 • Maximal representations of uniform complexhyperbolic lattices • Authors: annadmin Abstract: Let$\rho$be a maximal representation of a uniform lattice$\Gamma\subset\mathrm{SU}(n,1)$,$n\geq 2$, in a classical Lie group of Hermitian type$G$. We prove that necessarily$G=\mathrm{SU}(p,q)$with$p\geq qn$and there exists a holomorphic or antiholomorphic$\rho$-equivariant map from … Continue reading → PubDate: Fri, 03 Feb 2017 00:00:00 +000 • Affine Grassmannians and the geometric Satake in mixed characteristic • Authors: annadmin Abstract: We endow the set of lattices in$\mathbb{Q}_p^n$with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake equivalence in mixed characteristic. We also give an application of our theory … Continue reading → PubDate: Thu, 02 Feb 2017 00:00:00 +000 • Measure reducibility of countable Borel equivalence relations • Authors: annadmin Abstract: We show that every basis for the countable Borel equivalence relations strictly above$\mathbb{E}_0$under measure reducibility is uncountable, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push many known results concerning the abstract structure of the … Continue reading → PubDate: Wed, 01 Feb 2017 00:00:00 +000 • Editors’ Note on “Invariant differential operators and eigenspace representations on an affine symmetric space” • Authors: annadmin Abstract: None PubDate: Mon, 09 Jan 2017 00:00:00 +000 • Corrigendum to the paper “On the$\boldsymbol{K^2}$of degenerations of surfaces and the multiple point formula” • Authors: annadmin Abstract: We correct an error in the Multiple Point Formula (7.3) in the paper mentioned in the title. This correction propagates to formulas (7.5), (7.6), (7.23) and (8.18), and it affects minor results in Section 8, where few statements require an … Continue reading → PubDate: Sun, 08 Jan 2017 00:00:00 +000 • Dynamical spectral rigidity among$\mathbb{Z}_2$-symmetric strictly convex domains close to a circle (Appendix B coauthored with H. Hezari) • Authors: annadmin Abstract: We show that any sufficiently (finitely) smooth$\mathbb{Z}_2$-symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid; i.e., all deformations among domains in the same class that preserve the length of all periodic orbits of the associated … Continue reading → PubDate: Sat, 07 Jan 2017 00:00:00 +000 • Wilkie’s conjecture for restricted elementary functions • Authors: annadmin Abstract: We consider the structure$\mathbb{R}^{\mathrm{RE}}$obtained from$(\mathbb{R},
PubDate: Fri, 06 Jan 2017 00:00:00 +000

• Geometric monodromy — semisimplicity and maximality
Abstract: Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants … Continue reading →
PubDate: Thu, 05 Jan 2017 00:00:00 +000

• Homological stability for moduli spaces of high dimensional manifolds. II
Abstract: We prove a homological stability theorem for moduli spaces of manifolds of dimension $2n$, for attaching handles of index at least $n$, after these manifolds have been stabilised by countably many copies of $S^n \times S^n$. Combined with previous work … Continue reading →
PubDate: Wed, 04 Jan 2017 00:00:00 +000

• Tame distillation and desingularization by $p$-alterations
Abstract: We strengthen Gabber’s $l’$-alteration theorem by avoiding all primes invertible on a scheme. In particular, we prove that any scheme $X$ of finite type over a quasi-excellent threefold can be desingularized by a $\mathrm{char}(X)$-alteration, i.e., an alteration whose order is … Continue reading →
PubDate: Tue, 03 Jan 2017 00:00:00 +000

• Chern’s conjecture for special affine manifolds
Abstract: An affine manifold $X$ in the sense of differential geometry is a differentiable manifold admitting an atlas of charts with value in an affine space, with locally constant affine change of coordinates. Equivalently, it is a manifold whose tangent bundle … Continue reading →
PubDate: Mon, 02 Jan 2017 00:00:00 +000

• The Apollonian structure of integer superharmonic matrices
Abstract: We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice $\mathbb{Z}^2$ has the structure of an Apollonian circle packing. This completely characterizes the PDE that determines the continuum scaling limit of the Abelian sandpile … Continue reading →
PubDate: Sun, 01 Jan 2017 00:00:00 +000

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