Authors:
mak
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: Thu, 15 Jun 2017 23:17:06 +000

Authors:
mak
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: Mon, 12 Jun 2017 19:22:28 +000

Authors:
mak
Abstract: The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that the $i^{\mathrm{th}}$ Betti number $\beta_i(M)$ of a non-zero 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 follow … Continue reading → PubDate: Tue, 06 Jun 2017 14:18:12 +000

Authors:
mak
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 … Continue reading → PubDate: Tue, 02 May 2017 18:16:55 +000

Authors:
mak
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, 26 Apr 2017 14:34:21 +000

Authors:
mak
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: Tue, 11 Apr 2017 18:18:35 +000

Authors:
mak
Abstract: We describe the de Rham complex of the étale coverings of Drinfeld’s $p$-adic upper half-plane for ${\rm GL}_2(\mathbf{Q}_p)$. Conjectured by Breuil and Strauch, this description gives a geometric realization of the $p$-adic local Langlands correspondence for certain two-dimensional de Rham … Continue reading → PubDate: Fri, 07 Apr 2017 15:12:20 +000

Authors:
mak
Abstract: We prove non-trivial 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: Thu, 06 Apr 2017 20:59:05 +000

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

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

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

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

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

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

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

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

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

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

Authors:
mak
Abstract: In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose $\Gamma$ is a lattice in semisimple Lie group, all of whose factors have rank $2$ or higher. Let $\alpha$ be a smooth $\Gamma$-action on a … Continue reading → PubDate: Mon, 06 Feb 2017 15:01:12 +000

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

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

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

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

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

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

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

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

Authors:
annadmin
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

Authors:
annadmin
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

Authors:
annadmin
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

Authors:
annadmin
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

Authors:
annadmin
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