Inventiones mathematicae最新文献

A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least (C^{2}), we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family create robust heterodimensional dynamics, i.e., a pair of non-trivial hyperbolic basic sets with different numbers of positive Lyapunov exponents, such that the unstable manifold of each of the sets intersects the stable manifold of the second set and these intersections persist for an open set of parameter values. We also give a solution to the so-called local stabilization problem of coindex-1 heterodimensional cycles in any regularity class (r=2,ldots ,infty ,omega ). The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.

We consider the distribution of the Galois groups (operatorname {Gal}(K^{operatorname{un}}/K)) of maximal unramified extensions as (K) ranges over (Gamma )-extensions of ℚ or ({{mathbb{F}}}_{q}(t)). We prove two properties of (operatorname {Gal}(K^{operatorname{un}}/K)) coming from number theory, which we use as motivation to build a probability distribution on profinite groups with these properties. In Part I, we build such a distribution as a limit of distributions on (n)-generated profinite groups. In Part II, we prove as (qrightarrow infty ), agreement of (operatorname {Gal}(K^{operatorname{un}}/K)) as (K) varies over totally real (Gamma )-extensions of ({{mathbb{F}}}_{q}(t)) with our distribution from Part I, in the moments that are relatively prime to (q(q-1)|Gamma |). In particular, we prove for every finite group (Gamma ), in the (qrightarrow infty ) limit, the prime-to-(q(q-1)|Gamma |)-moments of the distribution of class groups of totally real (Gamma )-extensions of ({{mathbb{F}}}_{q}(t)) agree with the prediction of the Cohen–Lenstra–Martinet heuristics.

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) (J)-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.

We show that the (J)-totally nonnegative flag variety has a cellular decomposition into totally positive (J)-Richardson varieties. Moreover, each totally positive (J)-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive (J)-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the (J)-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of (U^{-}) for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.

We develop new techniques to compute the weighted (L^{2})-cohomology of quasi-fibered boundary metrics (QFB-metrics). Combined with the decay of (L^{2})-harmonic forms obtained in a companion paper, this allows us to compute the reduced (L^{2})-cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of (n) points on (mathbb{C}^{2}), for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3.

We prove that if (u:mathbb{R}^{n}to mathbb{R}) is strongly convex, then for every (varepsilon >0) there is a strongly convex function (vin C^{2}(mathbb{R}^{n})) such that (|{uneq v}|<varepsilon ) and (Vert u-vVert _{infty}<varepsilon ).

This note corrects an error in our paper Anosov flows, growth rates on covers and group extensions of subshifts, Invent. Math. 223:445–483, 2021. This leaves our main results, Theorem 1.1, Corollary 1.2, Theorem 1.3 and Theorem 5.1, unchanged. We also fill a gap in the arguments presented in Sect. 9; this requires a small modification to the results in this section.

The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces (H^{s}) for (s>-tfrac{1}{2}). The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard’s explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.

Let (mathfrak {g}) be a symmetrisable Kac–Moody algebra and (V) an integrable (mathfrak {g})–module in category (mathcal {O}). We show that the monodromy of the (normally ordered) rational Casimir connection on (V) can be made equivariant with respect to the Weyl group (W) of (mathfrak {g}), and therefore defines an action of the braid group (mathcal {B}_{W}) on (V). We then prove that this action is canonically equivalent to the quantum Weyl group action of (mathcal {B}_{W}) on a quantum deformation of (V), that is an integrable, category (mathcal {O}) module (mathcal {V}) over the quantum group (U_{hbar }mathfrak {g}) such that (mathcal {V}/hbar mathcal {V}) is isomorphic to (V). This extends a result of the second author which is valid for (mathfrak {g}) semisimple.

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of (i)-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories (mathscr{C}_{{mathfrak{g}}}^{0}) and (mathscr{C}_{{mathfrak{g}}}^{-}) provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.

Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include (k)-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of (U_{q}(widehat{mathfrak{sl}}_{ell }))-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.