# Showing papers in "Inventiones Mathematicae in 2011"

••

TL;DR: In this article, a general quadratic relation between these two dimensions was derived, which they view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819-826, 1988) relation from conformal field theory.

Abstract: Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫
D
∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on D is the weak limit as e→0 of the measures $$\varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz,$$
where dz is Lebesgue measure on D and h
e
(z) denotes the mean value of h on the circle of radius e centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819–826, 1988) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

461 citations

••

TL;DR: In this article, it was shown that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N − ε for some ε > 0.

Abstract: Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we prove that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N
−ζ for some ζ>0. The proof is based on an estimate of the entropy flow of the Dyson Brownian motion w.r.t. a “pseudo equilibrium measure”. As an application of this estimate, we prove that the eigenvalue spacing statistics in the bulk of the spectrum for N×N symmetric Wigner ensemble is the same as that of the Gaussian Orthogonal Ensemble (GOE) in the limit N→∞. The assumptions on the probability distribution of the matrix elements of the Wigner ensemble are a subexponential decay and some minor restriction on the support.

289 citations

••

TL;DR: In this paper, the authors consider the problem of global in time existence and uniqueness of solutions of the 3D infinite depth full water wave problem, in the setting that the interface tends to the horizontal plane, the velocity and acceleration on the interface tend to zero at spatial infinity.

Abstract: We consider the problem of global in time existence and uniqueness of solutions of the 3-D infinite depth full water wave problem, in the setting that the interface tends to the horizontal plane, the velocity and acceleration on the interface tend to zero at spatial infinity. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial interface that is sufficiently small in its steepness and velocity, we show that there exists a unique smooth solution of the full water wave problem for all time, and the solution decays at the rate 1/t.

259 citations

••

TL;DR: In this article, the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is shown to be equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Marino, and Vafa of local Calabi-Yau 3-folds are proven to be correct.

Abstract: We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Marino, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.

146 citations

••

TL;DR: In this article, the authors study cocycles of discrete countable groups with values in ε 2 G and the ring of affiliated operators UG and obtain strong results about the existence of free subgroups and subgroup structure.

Abstract: In this article we study cocycles of discrete countable groups with values in � 2 G and the ring of affiliated operators UG. We clarify properties of the first cohomology of a group G with coefficients in � 2 G and answer several questions from De Cornulier et al. (Transform. Groups 13(1):125-147, 2008). Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first � 2 -Betti num- ber. We give numerous applications and examples of groups which satisfy our assumptions.

134 citations

••

TL;DR: In this article, a more general approach to sofic entropy which produces both measure and topological dynamical invariants was developed, and the variational principle was established in this context.

Abstract: Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C∗-algebra.

132 citations

••

TL;DR: In this article, the authors show uniform boundedness on the exterior for solutions to the wave equation for stationary axisymmetric black hole exterior spacetimes with parameters a and M such that the Killing fields span the null generator of the event horizon, and that the energy flux is positive definite and does not degenerate at the horizon, i.e. it agrees with the energy as measured by a local observer.

Abstract: We consider Kerr spacetimes with parameters a and M such that |a|≪M, Kerr-Newman spacetimes with parameters |Q|≪M, |a|≪M, and more generally, stationary axisymmetric black hole exterior spacetimes $(\mathcal{M},g)$
which are sufficiently close to a Schwarzschild metric with parameter M>0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation □
g
ψ=0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Σ
τ
, then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions ψ, a suitable energy flux through Σ
τ
is bounded by C times the initial energy flux through Σ0. This energy flux is positive definite and does not degenerate at the horizon, i.e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of ψ, in terms of a higher order initial energy on Σ0. Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.

131 citations

••

TL;DR: In this article, the authors studied the supersingular locus of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ in the case that p is inert.

Abstract: We complete the study of the supersingular locus $\mathcal{M}^{\mathrm{ss}}$
in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ in the case that p is inert. This was started by the first author in Can. J. Math. 62, 668–720 (2010) where complete results were obtained for n=2,3. The supersingular locus $\mathcal{M}^{\mathrm{ss}}$
is uniformized by a formal scheme $\mathcal{N}$
which is a moduli space of so-called unitary p-divisible groups. It depends on the choice of a unitary isocrystal N. We define a stratification of $\mathcal{N}$
indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of N. We show that the combinatorial behavior of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the irreducible components of $\mathcal{N}_{\mathrm{red}}$
) are identified with (generalized) Deligne-Lusztig varieties. We show that the Bruhat-Tits stratification is a refinement of the Ekedahl-Oort stratification and also relate the Ekedahl-Oort strata to Deligne-Lusztig varieties. We deduce that $\mathcal{M}^{\mathrm{ss}}$
is locally a complete intersection, that its irreducible components and each Ekedahl-Oort stratum in every irreducible component is isomorphic to a Deligne-Lusztig variety, and give formulas for the number of irreducible components of every Ekedahl-Oort stratum of
$\mathcal{M}^{\mathrm{ss}}$
.

130 citations

••

TL;DR: In this paper, the nonlinear Schrodinger equation discretized on a hypercubic lattice has been shown to have a limit as λ→0 for t = λ ≥ 0 for t=λ ≥ 0, with τ fixed and |τ| sufficiently small.

Abstract: It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for the general case a proof of the kinetic limit remains open, we report on first progress. As wave equation we consider the nonlinear Schrodinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to the corresponding Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution ψ
t
(x) of the nonlinear Schrodinger equation yields then a stochastic process stationary in x∈ℤ
d
and t∈ℝ. If λ denotes the strength of the nonlinearity, we prove that the space-time covariance of ψ
t
(x) has a limit as λ→0 for t=λ
−2
τ, with τ fixed and |τ| sufficiently small. The limit agrees with the prediction from kinetic theory.

119 citations

••

TL;DR: In this paper, the 6n-dimensional phase space of the planetary (1+n)-body problem is shown to be foliated by symplectic leaves of dimension (6n−2) invariant for the planetary Hamiltonian, described by means of a new global set of Darboux coordinates related to a partial reduction of rotations.

Abstract: The 6n-dimensional phase space of the planetary (1+n)-body problem (after the classical reduction of the total linear momentum) is shown to be foliated by symplectic leaves of dimension (6n−2) invariant for the planetary Hamiltonian ${\mathcal{H}}$
. Such foliation is described by means of a new global set of Darboux coordinates related to a symplectic (partial) reduction of rotations. On each symplectic leaf ${\mathcal{H}}$
has the same form and it is shown to preserve classical symmetries. Further sets of Darboux coordinates may be introduced on the symplectic leaves so as to achieve a complete (total) reduction of rotations. Next, by explicit computations, it is shown that, in the reduced settings, certain degeneracies are removed. In particular, full torsion is checked both in the partially and totally reduced settings. As a consequence, a new direct proof of Arnold’s theorem (Arnold in Russ. Math. Surv. 18(6):85–191, 1963) on the stability of planetary system (both in the partially and in the totally reduced setting) is easily deduced, producing Diophantine Lagrangian invariant tori of dimension (3n−1) and (3n−2). Finally, elliptic lower dimensional tori bifurcating from the secular equilibrium are easily obtained.

106 citations

••

TL;DR: In this paper, the mod p Satake transform is used to define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over a finite extension of ℚp to be supersingular.

Abstract: Let F be a finite extension of ℚp. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over \(\overline{ \mathbb{F}}_{p}\) to be supersingular. We then give the classification of irreducible admissible smooth GLn(F)-representations over \(\overline{ \mathbb{F}}_{p}\) in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel–Livne for n=2. For general split reductive groups we obtain similar results under stronger hypotheses.

••

TL;DR: In this paper, the authors developed new solvability methods for divergence in real and complex, elliptic systems above Lipschitz graphs with L 2 boundary data, where the divergence coefficients may depend on all the coefficients.

Abstract: We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all ...

••

TL;DR: In this paper, the authors consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution.

Abstract: We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equations $$\partial_tu=\partial_x^2u+u(1-u)+\epsilon\sqrt{u(1-u)}\dot{W},$$
and $$\partial_tu=\partial_x^2u+u(1-u)+\epsilon\sqrt{u}\dot{W},$$
where $\dot{W}=\dot{W}(t,x)$
is a space-time white noise. We prove the Brunet-Derrida conjecture that the speed of traveling fronts for small e is $$2-\pi^2|{\log}\,\epsilon^2|^{-2}+O((\log|{\log}\,\epsilon|)|{\log}\,\epsilon|^{-3}).$$

••

TL;DR: In this paper, counterexamples to Min-Oo's conjecture in dimension n ≥ 3 were constructed. But the latter conjecture has not yet been verified in general relativity.

Abstract: Consider a compact Riemannian manifold M of dimension n whose boundary ∂M is totally geodesic and is isometric to the standard sphere S
n−1. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least n(n−1), then M is isometric to the hemisphere $S_{+}^{n}$
equipped with its standard metric. This conjecture is inspired by the positive mass theorem in general relativity, and has been verified in many special cases. In this paper, we construct counterexamples to Min-Oo’s Conjecture in dimension n≥3.

••

TL;DR: In this paper, a super duality was established for the ortho-symplectic Lie superalgebras in a parabolic category O, which includes all finite dimensional irreducible modules, in terms of classical Kazhdan-Lusztig polynomials.

Abstract: We formulate and establish a super duality which connects parabolic categories O for the ortho-symplectic Lie superalgebras and classical Lie algebras of BCD types. This provides a complete and conceptual solution of the irreducible character problem for the ortho-symplectic Lie superalgebras in a parabolic category O, which includes all finite dimensional irreducible modules, in terms of classical Kazhdan-Lusztig polynomials.

••

TL;DR: In this paper, the authors describe the interaction of two solitons with nearly equal speeds for the quartic (gKdV) equation 0.1 and show that the collision is not perfectly elastic.

Abstract: This paper describes the interaction of two solitons with nearly equal speeds for the quartic (gKdV) equation 0.1
$$\partial_tu+\partial_x(\partial_x^2u+u^4)=0,\quad t,x\in \mathbb{R}.$$
We call soliton a solution of (0.1) of the form u(t,x)=Q
c
(x−ct−y
0), where c>0, y
0∈ℝ and $Q_{c}''+Q_{c}^{4}=cQ_{c}$
. Since (0.1) is not an integrable model, the general question of the collision of two given solitons $Q_{c_{1}}(x-c_{1}t)$
, $Q_{c_{2}}(x-c_{2}t)$
with c
1≠c
2 is an open problem. We focus on the special case where the two solitons have nearly equal speeds: let U(t) be the solution of (0.1) satisfying $$\lim_{t\to-\infty}\|{U}(t)-Q_{c_1^-}(.-c_1^-t)-Q_{c_2^-}(.-c_2^-t)\|_{H^1}=0,$$
for $\mu_{0}=(c_{2}^{-}-c_{1}^{-})/(c_{1}^{-}+c_{2}^{-})>0$
small. By constructing an approximate solution of (0.1), we prove that, for all time t∈ℝ, $$\begin{array}{l}\displaystyle{U}(t)={Q}_{c_1(t)}(x-y_1(t))+{Q}_{c_2(t)}(x-y_2(t))+{w}(t)\\[6pt]\displaystyle\quad\mbox{where }\|w(t)\|_{H^1}\leq|\ln\mu_0|\mu_0^2,\end{array}$$
with y
1(t)−y
2(t)>2|ln μ
0|+C, for some C∈ℝ. These estimates mean that the two solitons are preserved by the interaction and that for all time they are separated by a large distance, as in the case of the integrable KdV equation in this regime. However, unlike in the integrable case, we prove that the collision is not perfectly elastic, in the following sense, for some C>0, $$\lim_{t\to+\infty}c_1(t)>c_2^-\biggl(1+\frac{\mu_0^5}{C}\biggr),\quad \lim_{t\to+\infty}c_2(t)

••

TL;DR: In this paper, the Universal Coefficient Theorem was used to show that all unital simple AH-algebras with tracial rank at most one are isomorphic to ones with no dimension growth.

Abstract: Let C and A be two unital separable amenable simple C
∗-algebras with tracial rank at most one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ϕ
1,ϕ
2:C→A are two unital monomorphisms. We show that there is a continuous path of unitaries {u
t
:t∈[0,∞)} of A such that $$\lim_{t\to\infty}u_t^*\varphi_1(c)u_t=\varphi_2(c)\quad\mbox{for all }c\in C$$
if and only if [ϕ
1]=[ϕ
2] in $KK(C,A),\varphi_{1}^{\ddag}=\varphi_{2}^{\ddag},(\varphi_{1})_{T}=(\varphi _{2})_{T}$
and a rotation related map $\overline{R}_{\varphi_{1},\varphi_{2}}$
associated with ϕ
1 and ϕ
2 is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class ${\mathcal{A}}$
of unital separable simple amenable C
∗-algebras which is strictly larger than the class of separable C
∗-algebras with tracial rank zero or one. Tensor products of two C
∗-algebras in ${\mathcal{A}}$
are again in ${\mathcal{A}}$
. Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras for which the state space of K
0 is the same as the tracial state space and also some unital simple ASH-algebras whose K
0-group is ℤ and whose tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are ${\mathcal{Z}}$
-stable are isomorphic to ones with no dimension growth.

••

TL;DR: In this article, the authors studied the Jacobian determinant of a map g from a smooth bounded open subset of RN into RN (N ≥ 2) under a fairly weak assumption on g.

Abstract: This paper is devoted to the study of the Jacobian determinant of a map g from Ω, a smooth bounded open subset of RN , into RN (N ≥ 2). More generally, Ω could be a smooth bounded open subset of an N -dimensional manifold. Starting with the seminal work of C. B. Morrey [27], Y. Reshetnyak [32], and J. Ball [1], it has been known that one can define the distributional Jacobian determinant Det(∇g) under fairly weak assumption on g; in particular, it is defined for all maps g ∈ W 1, N2 N+1 (Ω) and also for all

••

TL;DR: In this paper, it was shown that if a countable group Γ contains a copy of F_2, then it admits uncountably many non orbit equivalent actions (see Section 2.1).

Abstract: We prove that if a countable group Γ contains a copy of F_2, then it
admits uncountably many non orbit equivalent actions

••

TL;DR: In this article, the authors prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group of one of the five Platonic polyhedra.

Abstract: We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group \(\mathcal{R}\) of one of the five Platonic polyhedra. The number N coincides with the order \(|\mathcal{R}|\) of \(\mathcal{R}\) and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action \(\mathcal{A}\) on a suitable subset \(\mathcal{K}\) of the H1T-periodic maps u:ℝ→ℝ3N. The set \({\mathcal {K}}\) is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group \(\mathcal{R}\). There exist infinitely many such cones \({\mathcal {K}}\), all with the property that \({\mathcal {A}}|_{{\mathcal {K}}}\) is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure.

••

TL;DR: In this article, the supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ is uniformized by a formal scheme and the length of the corresponding local ring is determined by using a variant of the theory of quasi-canonical liftings.

Abstract: The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ is uniformized by a formal scheme $\mathcal{N}$
. In the case when p is an inert prime, we define special cycles ${\mathcal{Z}}({\bold x})$
in $\mathcal{N}$
, associated to collections ${\bold x}$
of m ‘special homomorphisms’ with fundamental matrix T∈Herm
m
(O
k
). When m=n and T is nonsingular, we show that the cycle ${\mathcal{Z}}({\bold x})$
is either empty or is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient condition, in terms of T, for ${\mathcal{Z}}({\bold x})$
to be irreducible. When ${\mathcal{Z}}({\bold x})$
is zero dimensional—in which case it reduces to a single point—we determine the length of the corresponding local ring by using a variant of the theory of quasi-canonical liftings. We show that this length coincides with the derivative of a representation density for hermitian forms.

••

TL;DR: In this article, a reduction to simple groups is proposed to prove the Alperin Weight Conjecture in the representation theory of finite groups, and as an application they prove the conjecture in several cases.

Abstract: The Alperin Weight Conjecture was stated by J. Alperin in 1986 and constitutes one of the main problems in the representation theory of finite groups. In this paper, we provide a reduction to simple groups and as an application we prove the conjecture in several cases.

••

TL;DR: In this paper, it was shown that if the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map $f\colon\widehat{\mathbb{C}}\rightarrow \widehat{C}$�{C]-consuming the Jordan regions such that f(S petertodd i ) is a round circle for all i∈I.

Abstract: Let S
i
, i∈I, be a countable collection of Jordan curves in the extended complex plane $\widehat{\mathbb{C}}$
that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map $f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}$
such that f(S
i
) is a round circle for all i∈I. This implies that every Sierpinski carpet in $\widehat{\mathbb{C}}$
whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpinski carpet by a quasisymmetric map.

••

TL;DR: In this paper, the authors derived a global in time, local in space Carleman estimate for a parabolic operator P in the neighborhood of any point of the interface, where the "observation" region can be chosen independently of the sign of the jump of the coefficient at the considered point.

Abstract: In (0,T)×Ω, Ω open subset of ℝ
n
, n≥2, we consider a parabolic operator P=∂
t
−∇
x
δ(t,x)∇
x
, where the (scalar) coefficient δ(t,x) is piecewise smooth in space yet discontinuous across a smooth interface S. We prove a global in time, local in space Carleman estimate for P in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient δ at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface. In the high-frequency regime we use Calderon projectors. In the low-frequency regime we follow a more classical approach. Because of the parabolic nature of the problem we need to introduce Weyl-Hormander anisotropic metrics, symbol classes and pseudo-differential operators. Each frequency regime and the associated technique require a different calculus. A global in time and space Carleman estimate on (0,T)×M, M a manifold, is also derived from the local result.

••

TL;DR: In this paper, the authors give geometric tools to compare Nash and Sobolev inequalities for pieces of the relevant Markov operators that give useful bounds on rates of convergence for the Metropolis algorithm.

Abstract: This paper gives geometric tools: comparison, Nash and Sobolev inequalities for pieces of the relevant Markov operators, that give useful bounds on rates of convergence for the Metropolis algorithm. As an exam- ple, we treat the random placement of N hard discs in the unit square, the original application of the Metropolis algorithm.

••

TL;DR: In this paper, the authors give a topological characterization of rational maps with disconnected Julia sets, and extend Thurston's characterization of postcritically finite rational maps in place of iteration on Teichmuller space.

Abstract: We give a topological characterization of rational maps with disconnected Julia sets Our results extend Thurston’s characterization of postcritically finite rational maps In place of iteration on Teichmuller space, we use quasiconformal surgery and Thurston’s original result

••

TL;DR: In this article, the authors proved many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod p Hilbert modular forms, by making use of modularity lifting theorems and computations in p-adic Hodge theory.

Abstract: We prove many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod p Hilbert modular forms, by making use of modularity lifting theorems and computations in p-adic Hodge theory.

••

TL;DR: In this article, the linear and nonlinear instability of the line solitary water wave with respect to transverse perturbations was shown to be linear and nonsmooth, respectively.

Abstract: We prove the linear and nonlinear instability of the line solitary water waves with respect to transverse perturbations.

••

TL;DR: For a convex, real analytic, e-close to integrable Hamiltonian system with n ≥ 5 degrees of freedom, the authors constructed an orbit exhibiting Arnold diffusion with the diffusion time bounded by

Abstract: For a convex, real analytic, e-close to integrable Hamiltonian system with n≥5 degrees of freedom, we construct an orbit exhibiting Arnold diffusion with the diffusion time bounded by $\exp(C\epsilon^{-\frac{1}{2(n-2)}})$
. This upper bound of the diffusion time almost matches the lower bound of order $\exp(\epsilon ^{-\frac{1}{2(n-1)}})$
predicted by the Nekhoroshev-type stability results. Our method is based on the variational approach of Bessi and Mather, and includes a new construction on the space of frequencies.

••

TL;DR: In this article, a universal theory of refined Stark conjectures is presented, including a full proof in several important cases, and explain the connection to previous conjectures of Bloch and Kato, of Lichtenbaum and of Serre and Tate.

Abstract: We present a universal theory of refined Stark conjectures. We give evidence in support of these explicit and very general conjectures, including a full proof in several important cases, and explain the connection to previous conjectures of Bloch and Kato, of Lichtenbaum and of Serre and Tate. We also deduce a wide range of unconditional consequences of these results concerning the annihilation, as Galois modules, of ideal class groups by explicit elements constructed from the values of higher order derivatives of (non-abelian) Artin L-series.