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 Annals of Mathematics   [SJR: 10.358]   [H-I: 78]   [0 followers]  Follow        Subscription journal  (Not entitled to full-text)    ISSN (Print) 0003-486X    Published by Princeton University Press  [1 journal]
• Dynamical spectral rigidity among $\mathbb{Z}_2$-symmetric strictly convex
domains close to a circle (with an appendix co-authored with Hamed Hezari)

• Authors: mak
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 which preserve the length of all periodic orbits of the associated … Continue reading →
PubDate: Thu, 16 Mar 2017 20:37:56 +000

• Wilkie’s conjecture for restricted elementary functions
• Authors: mak
Abstract: We consider the structure $\mathbb{R}^\mathrm{RE}$ obtained from $(\mathbb{R}, PubDate: Tue, 28 Feb 2017 15:36:48 +000 • Geometric monodromy – semisimplicity and maximality • Authors: mak Abstract: Let$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$. We prove that the tensor invariants … Continue reading → PubDate: Wed, 22 Feb 2017 18:39:16 +000 • Chern’s conjecture for special affine manifolds • Authors: mak Abstract: An affine manifold$X$in the sense of differential geometry is a differentiable manifold admitting an atlas of charts with value in an affine space, with locally constant affine change of coordinates. Equivalently, it is a manifold whose tangent bundle … Continue reading → PubDate: Tue, 14 Feb 2017 21:11:33 +000 • Tame distillation and desingularization by$p$-alterations • Authors: mak Abstract: We strengthen Gabber’s$l’$-alteration theorem by avoiding all prmes invertible on a scheme. In particular, we prove that any scheme$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: Fri, 10 Feb 2017 18:51:37 +000 • Homological stability for moduli spaces of high dimensional manifolds. II • Authors: mak 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, 08 Feb 2017 22:22:37 +000 • The Apollonian structure of integer superharmonic matrices • Authors: mak 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 which determines the continuum scaling limit of the Abelian sandpile … Continue reading → PubDate: Wed, 08 Feb 2017 22:12:37 +000 • Global smooth and topological rigidity of hyperbolic lattice actions • 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 • 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 • On the growth of L^2-invariants for sequences of lattices in Lie groups • Authors: mak 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: Fri, 03 Feb 2017 17:10:25 +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 • Higher ramification and the local Langlands correspondence • Authors: mak 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: Wed, 25 Jan 2017 16:03:26 +000 • Positive scalar curvature on foliations • Authors: mak 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: Mon, 23 Jan 2017 21:44:38 +000 • The sphere packing problem in dimension 24 • Authors: mak 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: Tue, 17 Jan 2017 19:04:05 +000 • Cubic curves and totally geodesic subvarieties of moduli space • Authors: mak Abstract: In this paper we present the first example of a primitive, totally geodesic subvariety$F \subset \mathcal{M}_{g,n}$with$\dim(F)>1$. 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: Tue, 17 Jan 2017 17:37:19 +000 • The sphere packing problem in dimension 8 • Authors: mak 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: Wed, 21 Dec 2016 17:42:41 +000 • Harmonic quasi-isometric maps between rank one symmetric spaces • Authors: mak 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: Mon, 19 Dec 2016 20:24:21 +000 • Monochromatic sums and products in N • Authors: mak 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 non-linear patterns which can … Continue reading → PubDate: Sun, 04 Dec 2016 14:31:39 +000 • Erratum to `Isotopies of homeomorphisms of Riemann surfaces’ • Authors: annadmin Abstract: None PubDate: Fri, 02 Dec 2016 16:39:00 +000 • On large subsets of$\mathbb{F}_q^n$with no three-term arithmetic progression • Authors: annadmin Abstract: In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of$\mathbb{F}_q^n$with no three terms in arithmetic progression by$c^n$with$c < q$. For$q=3$, … Continue reading → PubDate: Fri, 02 Dec 2016 16:38:00 +000 • Progression-free sets in$\mathbb Z_4^n$are exponentially small • Authors: annadmin Abstract: We show that for an integer$n\ge 1$, any subset$A\subseteq \mathbb{Z}_4^n$free of three-term arithmetic progressions has size$|A|\le 4^{\gamma n}$, with an absolute constant$\gamma\approx 0.926$. PubDate: Fri, 02 Dec 2016 16:37:00 +000 • A sharp counterexample to local existence of low regularity solutions to Einstein equations in wave coordinates • Authors: annadmin Abstract: We give a sharp counterexample to local existence of low regularity solutions to Einstein equations in wave coordinates. We show that there are initial data in$H^2$satisfying the wave coordinate condition such that there is no solution in$H^2$… Continue reading → PubDate: Fri, 02 Dec 2016 16:36:00 +000 • Lyapunov exponents for random perturbations of some area-preserving maps including the standard map • Authors: annadmin Abstract: We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the standard map. Lower bounds for Lyapunov exponents of such … Continue reading → PubDate: Fri, 02 Dec 2016 16:35:00 +000 • Quasidiagonality of nuclear C$^{*}$-algebras • Authors: annadmin Abstract: We prove that faithful traces on separable and nuclear$\mathrm{C}^*$-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear$\mathrm{C}^*$-algebras of finite nuclear … Continue reading → PubDate: Fri, 02 Dec 2016 16:04:00 +000 • Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps • Authors: annadmin Abstract: In this paper we study the regularity of stationary and minimizing harmonic maps$f:B_2(p)\subseteq M\to N$between Riemannian manifolds. If$S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$is the$k^{\rm th}$-stratum of the singular set of$f$, … Continue reading → PubDate: Fri, 02 Dec 2016 16:03:00 +000 • New G$_2$-holonomy cones and exotic nearly Kähler structures on$S^6$and$S^3 \times S^3$• Authors: annadmin Abstract: There is a rich theory of so-called (strict) nearly Kähler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the$6$-sphere induced by octonionic multiplication. Nearly Kähler$6$-manifolds play a distinguished role both in the general structure theory and … Continue reading → PubDate: Fri, 02 Dec 2016 16:02:00 +000 • Cyclic surfaces and Hitchin components in rank 2 • Authors: annadmin Abstract: We prove that given a Hitchin representation in a split real rank 2 group$\mathsf{G}_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrisation of the Hitchin component by a … Continue reading → PubDate: Fri, 02 Dec 2016 16:01:00 +000 • Ramsey numbers of degenerate graphs • Authors: mak 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: Wed, 30 Nov 2016 12:48:44 +000 • Global well-posedness for the Yang-Mills equation in 4+1 dimensions. Small energy • Authors: mak 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: Thu, 27 Oct 2016 16:46:19 +000 • Editors’ Note on “The symplectic sum formula for Gromov-Witten invariants” • Authors: annadmin Abstract: None PubDate: Fri, 11 Mar 2016 00:00:00 +000 • On the structure of${\mathscr A}$-free measures and applications • Authors: annadmin Abstract: We establish a general structure theorem for the singular part of${\mathscr A}$-free Radon measures, where${\mathscr A}$is a linear PDE operator. By applying the theorem to suitably chosen differential operators${\mathscr A}$, we obtain a simple proof of … Continue reading → PubDate: Thu, 10 Mar 2016 00:00:00 +000 • Discrete Riesz transforms and sharp metric$X_p$inequalities • Authors: annadmin Abstract: For$p\in [2,\infty)$, the metric$X_p$inequality with sharp scaling parameter is proven here to hold true in$L_p$. The geometric consequences of this result include the following sharp statements about embeddings of$L_q$into$L_p$when$2\lt q\lt p\lt … Continue reading →
PubDate: Wed, 09 Mar 2016 00:00:00 +000

• Geometric invariants for real quadratic fields
Abstract: To an ideal class of a real quadratic field we associate a certain surface. This surface, which is a new geometric invariant, has the usual modular closed geodesic as its boundary. Furthermore, its area is determined by the length of … Continue reading →
PubDate: Tue, 08 Mar 2016 00:00:00 +000

• Finite time singularity for themodified SQG patch equation
Abstract: It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law that is one derivative less regular than in the Euler case, and the question … Continue reading →
PubDate: Mon, 07 Mar 2016 00:00:00 +000

• Errata of “Isoparametric hypersurfaces with $(g,m)=(6,2)$”
Abstract: We give a correction of the proof of the homogeneity of isoparametric hypersurfaces with $(g,m)=(6,2)$.
PubDate: Mon, 07 Mar 2016 00:00:00 +000

• Characters of odd degree
Abstract: We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal … Continue reading →
PubDate: Sun, 06 Mar 2016 00:00:00 +000

• Multiplicative functions in short intervals
Abstract: We introduce a general result relating “short averages” of a multiplicative function to “long averages” which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of $\mu(n)$ … Continue reading →
PubDate: Sun, 06 Mar 2016 00:00:00 +000

• Anabelian geometry with étale homotopy types
Abstract: Anabelian geometry with étale homotopy types generalizes in a natural way classical anabelian geometry with étale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field $k$ that … Continue reading →
PubDate: Sat, 05 Mar 2016 00:00:00 +000

• Classicité de formes modulaires surconvergentes
Abstract: Nous généralisons le critère de classicité de formes modulaires surconvergentes sur les courbes modulaires dû à Coleman à certaines variétés de Shimura PEL de type (A) et~(C) associées à des groupes réductifs non ramifiés sur $\mathbb{Q}_p$. Notre démonstration s’inspire de … Continue reading →
PubDate: Sat, 05 Mar 2016 00:00:00 +000

• Gromov-Hausdorff limits of Kähler manifolds and the finite generation
conjecture
Abstract: We study the uniformization conjecture of Yau by using the Gromov-Hausdorff convergence. As a consequence, we confirm Yau’s finite generation conjecture. More precisely, on a complete noncompact Kähler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions … Continue reading →
PubDate: Fri, 04 Mar 2016 00:00:00 +000

• Large gaps between consecutive prime numbers
Abstract: Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdős, we show that $G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$ where … Continue reading →
PubDate: Fri, 04 Mar 2016 00:00:00 +000

• On the phase transition in random simplicial complexes
Abstract: It is well known that the $G(n,p)$ model of random graphs undergoes a dramatic change around $p=\frac 1n$. It is here that the random graph, almost surely, contains cycles, and here it first acquires a giant (i.e., order $\Omega(n)$) connected … Continue reading →
PubDate: Thu, 03 Mar 2016 00:00:00 +000

• Large gaps between primes
Abstract: We show that there exist pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. This answers a well-known question of Erdős.
PubDate: Thu, 03 Mar 2016 00:00:00 +000

• $p$-torsion monodromy representations of elliptic curves over geometric
function fields
Abstract: Given a complex quasiprojective curve $B$ and a nonisotrivial family $\mathcal{E}$ of elliptic curves over $B$, the $p$-torsion $\mathcal{E}[p]$ yields a monodromy representation $\rho_\mathcal{E}[p]:\pi_1(B)\rightarrow \mathrm{GL}_2(\mathbb{F}_p)$. We prove that if $\rho_{\mathcal E}[p]\cong \rho_{\mathcal E’}[p]$, then $\mathcal{E}$ and $\mathcal E’$ are isogenous, … Continue reading →
PubDate: Wed, 02 Mar 2016 00:00:00 +000

• Decay for solutions of the wave equation on Kerr exterior spacetimes III:
The full subextremalcase $|a| < M$
Abstract: This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I–II: the cases $|a| \ll M$ or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and … Continue reading →
PubDate: Wed, 02 Mar 2016 00:00:00 +000

• The clique density theorem
Abstract: Turán’s theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geqslant 2$, every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually adjacent vertices. … Continue reading →
PubDate: Tue, 01 Mar 2016 00:00:00 +000

• Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture
over function fields
Abstract: We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let $\ell > 2$ be prime and $A$ a finite abelian $\ell$-group. Then there exists … Continue reading →
PubDate: Tue, 01 Mar 2016 00:00:00 +000

• Proof of the main conjecture in Vinogradov’s Mean Value Theorem for
degrees higher than three
Abstract: We prove the main conjecture in Vinogradov’s Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves.
PubDate: Sun, 07 Feb 2016 00:00:00 +000

• Rigid inner forms of real and $p$-adic groups
Abstract: We define a new cohomology set $H^1(u \to W,Z \to G)$ for an affine algebraic group $G$ and a finite central subgroup $Z$, both defined over a local field of characteristic zero, which is an enlargement of the usual first … Continue reading →
PubDate: Sat, 06 Feb 2016 00:00:00 +000

• An integrable deformation of an ellipse of small eccentricity is an
ellipse
Abstract: The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we show that a version of this conjecture is true for … Continue reading →
PubDate: Fri, 05 Feb 2016 00:00:00 +000

• Birational boundedness for holomorphic symplectic varieties, Zarhin’s
trick for $K3$ surfaces, and the Tate conjecture
Abstract: We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin’s trick for $K3$ surfaces by constructing big line bundles of low degree on certain moduli spaces of stable sheaves, and … Continue reading →
PubDate: Thu, 04 Feb 2016 00:00:00 +000

• Imaginary geometry III: reversibility of $\mathrm{SLE}_\kappa$ for $\kappa \in (4,8)$
Abstract: Suppose that $D \subseteq \mathbf{C}$ is a Jordan domain and $x,y \in \partial D$ are distinct. Fix $\kappa \in (4,8)$, and let $\eta$ be an $\mathrm{SLE}_\kappa$ process from $x$ to $y$ in $D$. We prove that the law of the … Continue reading →
PubDate: Wed, 03 Feb 2016 00:00:00 +000

• Combinatorial theorems in sparse random sets
Abstract: We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Turán’s theorem, Szemerédi’s theorem and Ramsey’s theorem, hold almost surely inside sparse random sets. For instance, we extend Turán’s theorem … Continue reading →
PubDate: Tue, 02 Feb 2016 00:00:00 +000

• Extremal results for random discrete structures
Abstract: We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemerédi’s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions, and we determine the threshold for Turán-type problems for random … Continue reading →
PubDate: Mon, 01 Feb 2016 00:00:00 +000

• Embedded self-similar shrinkers of genus $0$
Abstract: We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution of mean curvature flow in $\mathbb{R}^3$ with genus $0$. More generally, we show that the only properly embedded self-similar shrinkers in $\mathbb{R}^3$ with … Continue reading →
PubDate: Mon, 11 Jan 2016 17:06:00 +000

• Splitting mixed Hodge structures over affine invariant manifolds
Abstract: We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in … Continue reading →
PubDate: Mon, 11 Jan 2016 17:05:00 +000

• Perverse sheaves over real hyperplane arrangements
Abstract: Let $\mathcal{H}$ be an arrangement of real hyperplanes in $\mathbb{R}^n$. The complexification of $\mathcal{H}$ defines a natural stratification of $\mathbb{C}^n$. We denote by $\mathrm{Perv}(\mathbb{C}^n, \mathcal{H})$ the category of perverse sheaves on $\mathbb{C}^n$ smooth with respect to this stratification. We give … Continue reading →
PubDate: Mon, 11 Jan 2016 17:04:00 +000

• Regularity of area minimizing currents III: blow-up
Abstract: This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren’s partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis … Continue reading →
PubDate: Mon, 11 Jan 2016 17:03:00 +000

• Regularity of area minimizing currents II: center manifold
Abstract: This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely, the construction of a center manifold, … Continue reading →
PubDate: Mon, 11 Jan 2016 17:02:00 +000

• Global solutions of the Euler–Maxwell two-fluid system in 3D
Abstract: The fundamental “two-fluid” model for describing plasma dynamics is given by the Euler–Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. We prove global stability of a constant neutral background, in the sense … Continue reading →
PubDate: Mon, 11 Jan 2016 17:01:00 +000

• On the number of generators of ideals in polynomial rings
Abstract: For an ideal $I$ in a noetherian ring $R$, let $\mu(I)$ be the minimal number of generators of $I$. It is well known that there is a sequence of inequalities $\mu(I/I^2)\leq \mu(I)\leq \mu(I/I^2)+1$ that are strict in general. However, Murthy … Continue reading →
PubDate: Sun, 03 Jan 2016 00:00:00 +000

• A two-dimensional polynomial mapping with a wandering Fatou component
Abstract: We show that there exist polynomial endomorphisms of $\mathbb{C}^2$, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of $\mathbb{P}^2(\mathbb{C})$. We also find real examples with wandering domains in $\mathbb{R}^2$. The proof … Continue reading →
PubDate: Sat, 02 Jan 2016 00:00:00 +000

• On the nonexistence of elements of Kervaire invariant one
Abstract: We show that the Kervaire invariant one elements $\theta_{j}\in\pi_{2^{j+1}-2}S^{0}$ exist only for $j\le 6$. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions $2$, $6$, $14$, $30$, $62$, and possibly $126$. Except … Continue reading →
PubDate: Fri, 01 Jan 2016 00:00:00 +000

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