We study two notions of generalized matroid representations motivated by algorithmic information theory and cryptographic secret sharing. The first (entropic representability) involves discrete random variables, while the second (almost multilinear representability) deals with approximate subspace arrangements. In both cases, we prove that determining whether an input matroid has such a representation is undecidable. Consequently, the conditional independence implication problem is also undecidable, providing an independent answer to a question posed by Geiger and Pearl, recently resolved by Cheuk Ting Li. These problems are also closely related to characterizing achievable rates in network coding and constructing secret sharing schemes. For example, another corollary of our work is that deciding whether an access structure admits an ideal secret sharing scheme is undecidable. Our approach reduces undecidable problems from group theory to matroid representation problems. Specifically, we reduce the uniform word problem for finite groups to entropic representability and the word problem for sofic groups to almost multilinear representability. A key part of this reduction involves modifying group presentations into forms where linear representations are generic in an appropriate sense when restricted to the generating set.
On a class of dynamical space-times which are asymptotic as t-OO to a stationary space-time containing a horizon 3e0, we show the existence of a unique null hypersurface 3e which is asymptotic to 3e0. This is a special case of a general unstable manifold theorem for perturbations of flows which translate in time and have a normal sink at an invariant manifold in space. Examples of horizons 3e0 to which our result applies include event horizons of subextremal Kerr and Kerr-Newman black holes as well as event and cosmological horizons of subextremal Kerr-Newman-de Sitter black holes. In the Kerr(-Newman) case, we show that 3e is equal to the boundary of the black hole region of the dynamical space-time.
We prove that area-minimizing submanifolds are not generically smooth, settling a conjecture of White that asks the generic smoothness of area-minimizing submanifolds. We furthermore establish a lower bound on the Hausdorff dimension of the singular sets of area-minimizing submanifolds with respect to open sets of Riemannian metrics. The lower bound is max{d-5, d-c}, where d denotes the dimension of the submanifold and c denotes the codimension.
We provide a proof for the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) Feynman rule conjecture, which applies to all genus Gromov-Witten potentials of the quintic threefold. This is achieved by extracting an equivalent A-model Feynman rule from the N-Mixed-Spin-P field (NMSP) theory. For genus greater than 1, the Feynman rule effectively computes the genus g Gromov-Witten potential using 3g-3 initial data. As a result, the Yamaguchi-Yau equations and the genus 1 and 2 mirror formulas are confirmed. A notable discovery is that the propagators in the BCOV theory can be interpreted in the A-model as enumerative counts of chains of rational curves in the NMSP master space outside the quintic threefold. In this way these propagators afford direct visibility of BCOV Feynman structures in the moduli of NMSP fields. Furthermore, we demonstrate that the set of all propagators capable of yielding Feynman rules forms a torsor under a "gauge" group action, which includes all BCOV's "gauge"freedom of propagators.
We generalize the Hernandez-Jimbo category O of representations of Borel subalgebras of quantum affine algebras to the case of quantum loop algebras for arbitrary Kac-Moody g (as well as related algebras, such as quantum toroidal gl_1). Moreover, we give explicit realizations of all simple modules, and devise tools for the computation of q-characters that are new even for g of finite type. Our techniques allow us to generalize classic results of Frenkel-Hernandez, Frenkel-Mukhin, Hernandez-Jimbo and Hernandez-Leclerc, as well as prove conjectures of Feigin-Jimbo-Miwa-Mukhin and Mukhin-Young.
We prove that a variety of examples of minimal complex surfaces admit exotic diffeomorphisms, providing the first known instances of exotic diffeomorphisms of irreducible 4-manifolds. We also give sufficient conditions for the boundary Dehn twist on a spin 4-manifold with S^3 boundary to be non-trivial in the relative mapping class group. This gives many new examples of non-trivial boundary Dehn twists.
We prove that, for every smooth Jordan curve y C C and for every set Q C C of six concyclic points, there exists a nonconstant quadratic polynomial p E C[z] such that p(Q) C y. The proof relies on a theorem of Fukaya and Irie. We also prove that if Q is the union of the vertex sets of two concyclic regular n-gons, there exists a nonconstant polynomial p E C[z] of degree at most n-1 such that p(Q) C y. The proof is based on a computation in Floer homology. These results support a conjecture about which point sets Q C C admit a polynomial inscription of a given degree into every smooth Jordan curve y.
We study a random configuration of N soliton solutions psi(N)(x, t; lambda/ of the cubic focusing nonlinear Schr & ouml;dinger (fNLS) equation in one space dimension. The N soliton solutions are parametrized by 2N complex numbers (lambda, c), where lambda is an element of C-+(N) are the eigenvalues of the Zakharov-Shabat linear operator, and c is an element of C-N\{0} are the norming constants of the corresponding eigenfunctions. The randomness is obtained by choosing the complex eigenvalues to be independent and identically distributed random variables sampled from a probability distribution with compact support in the complex plane. The corresponding norming constants are interpolated by a smooth function of the eigenvalues. Then we consider the expectation of the random measure associated to this random spectral data. Such expectation uniquely identifies, via the Zakharov-Shabat inverse spectral problem, a solution psi(infinity)(x, t) of the fNLS equation. This solution can be interpreted as a soliton gas solution. We prove a law of large numbers and a central limit theorem for the differences psi(N)(x,t; lambda) -psi(infinity)(x, t) and vertical bar psi(N) (x,t; lambda)vertical bar(2) - vertical bar psi(infinity)(x, t)vertical bar(2) when (x, t) are in a compact set of R x R+; we additionally compute the correlation functions.
We resolve an open problem posed by Alexeev-Knutson on the projectivity of the moduli of branchvarieties in the equidimensional case. As an application, we construct projective moduli spaces of reduced equidimensional varieties equipped with ample linear series and subject to a semistability condition.
We show that, given a real or complex hyperbolic metric g0 on a closed manifold M of dimension n >= 3, there exists a neighborhood U of g(0) in the space of negatively curved metrics such that for any g is an element of U, the topological entropy and Liouville entropy of g coincide if and only if g and g0 are homothetic. This provides a partial answer to Katok's entropy rigidity conjecture. As a direct consequence of our theorem, we obtain a local rigidity result of the hyperbolic rank near complex hyperbolic metrics.
The goal of the paper is to show that the event horizons of the spacetimes constructed in , see also , in the proof of the nonlinear stability of slowly rotating Kerr spacetimes 𝒦(a_0,m_0), are necessarily smooth null hypersurfaces. Moreover we show that the result remains true for the entire range of |a_0|/m_0 for which stability can be established.
We propose a refinement of the random matrix model for a certain family of L-functions over F-q[u], using techniques that we hope will eventually apply to an arbitrary family of L-functions. This consists of a probability distribution on power series in q(-s) which combines properties of the characteristic polynomials of Haar-random unitary matrices and random Euler products over F-q[u]. The support of our distribution is contained in the intersection of the supports of the two original distributions. The expectations of low-degree polynomials in the coefficients of our series approximate the expectations of the same polynomials in the coefficients of random Euler products, while the expectations of high-degree polynomials approximate the expectations of the same polynomials in the coefficients of the characteristic polynomials of random matrices. Furthermore, the expectations of absolute powers of our series approximate the predictions of the Conrey-Farmer-Keating-Rubinstein-Snaith/Adrade-Keating recipe for the moments of our family of L-functions.
We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of L(infinity)L(2 )weak solutions with vorticity in (LLp)-L-infinity for some p > 1, surpassing for the first time the critical scaling of the standard convex integration technique. To achieve this, we introduce several new ideas, including: (i) a new family of building blocks built from the Lamb-Chaplygin dipole; (ii) a new method to cancel the error based on time averages and nonperiodic, spatially anisotropic perturbations.
We show that the operator Cf (x, y) := sup(v is an element of & Ropf;)|p.v. integral(& Ropf; )f(x - t, y - t(2))e(ivt3) dt/t is bounded on L-p(& Ropf;(2)) for every 1 < p < infinity. This gives an affirmative answer to a question of Pierce and Yung.
We extend Ratner's theorem on equidistribution of individual orbits of unipotent flows on finite volume homogeneous spaces of Lie groups to trajectories of noncontracting curves definable in a polynomially bounded o-minimal structure. To be precise, let phi:[0,infinity)-> SL(n,R) be a continuous curve whose coordinate functions are definable in a polynomially bounded o-minimal structure; for example, rational functions. Suppose that phi is noncontracting; that is, for any linearly independent vectors v1,& mldr;,vk in R-n, phi(t)& sdot;(v1 boolean AND & ctdot;boolean AND vk)negated right arrow 0 as t ->infinity. Then there exists a unique closed connected subgroup H-phi subset of SL(n,R) of smallest dimension such that H-phi is generated by unipotent one-parameter subgroups and such that phi(t)H-phi -> g0H phi in SL(n,R)/H-phi as t ->infinity, for some g0 is an element of SL(n,R). Let G subset of SL(n,R) be a closed subgroup, and let Gamma subset of G be a lattice. Suppose that phi([0,infinity))subset of G. Then H phi subset of G, and for any x is an element of G/Gamma, the trajectory {phi(t)x:t is an element of[0,T]} is equidistributed with respect to the measure g0 mu Lx as T ->infinity, where L subset of G is a closed subgroup such that & strns;& strns;& strns;& strns;& strns;& strns;& strns;& strns;& strns;& strns;H phi x=Lx and Lx admits a unique L-invariant probability measure, denoted by mu Lx. A crucial new ingredient is the proof that for any finite-dimensional representation V of SL(n,R), there exist T0>0, C>0, and alpha>0 such that for any v is an element of V, the map t parallel to phi(t)v parallel to is (C,alpha)-good on [T0,infinity)
Following & Lstrok;ojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. We prove & Lstrok;ojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions for mean curvature flows in & Ropf;nC1 with neck or nondegenerate cylindrical singularities. In particular, we prove the conjectures for all mean convex mean curvature flows of surfaces, including the cases when the arrival time functions are not C2. The results also apply to mean curvature flows starting from two-spheres or generic closed surfaces.
We provide a condition on a set of directions Omega subset of S-1 ensuring that the associated directional maximal operator M-Omega is unbounded on L-p(R-2) for every 1 <= p < infinity. The techniques of proof extend ideas of Bateman and Katz involving probabilistic construction of Kakeya-type sets using sticky maps and Bernoulli percolation.
We give a counterexample to the PIA (precise inversion of adjunction) conjecture for minimal log discrepancies. We also give a counterexample to the LSC conjecture for families.
We prove Koebe's conjecture and a version of Schramm's cofat uniformization theorem for domains Q C & Copf;O satisfying conditions involving quasitripods, that is, quasisymmetric images of the standard tripod. If the nonpoint complementary components of Q contain uniform quasitripods with large diameters and satisfy a packing condition, then there exists a conformal map f: Q-D onto a circle domain D. Moreover, f preserves the classes of point-components and nonpoint-components. The packing condition is satisfied if Q is cospread, that is, if the complementary components contain uniform quasitripods in all scales.
We prove that the singular set of a 2-valued Lipschitz graph that is stationary for the area is of codimension 1.
