Causal geometry of Einstein-Vacuum spacetimes with finite curvature flux

One of the central difficulties of settling the L²-bounded curvature conjecture for the Einstein-Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of Enstein-Vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface. Mathematics Subject Classification (1991) 35J10     

On Planes of Order p2 in Which Every Quadrangle Generates a Subplane of Order p

After Gleason's result, in the late fifties the following conjecture appeared: if in a finite projective plane every quadrangle is contained in a unique Desarguesian proper subplane of order p, then the plane is Desarguesian (and its order is p ^d for some d). In this paper we prove the conjecture in the case when the plane is of order p ² and p is a prime.     

The geometry of the asymptotics of polynomial maps

The aim of this paper is to develop a theory for the asymptotic behavior of polynomials and of polynomial maps overR and overC and to apply it to the Jacobian conjecture. This theory gives a unified frame for some results on polynomial maps that were not related before. A well known theorem of J. Hadamard gives a necessary and sufficient condition on a local diffeomorphismf: R ⁿ →R ⁿ to be a global diffeomorphism. In order to show thatf is a global diffeomorphism it suffices to exclude the existence of asymptotic values forf. The real Jacobian conjecture was shown to be false by S. Pinchuk. Our first application is to understand his construction within the general theory of asymptotic values of polynomial maps and prove that there is no such counterexample for the Jacobian conjecture overC. In a second application we reprove a theorem of Jeffrey Lang which gives an equivalent formulation of the Jacobian conjecture in terms of Newton polygons. This generalizes a result of Abhyankar. A third application is another equivalent formulation of the Jacobian conjecture in terms of finiteness of certain polynomial rings withinC[U, V]. The theory has a geometrical aspect: we define and develop the theory of etale exotic surfaces. The simplest such surface corresponds to Pinchuk's construction in the real case. In fact, we prove one more equivalent formulation of the Jacobian conjecture using etale exotic surfaces. We consider polynomial vector fields on etale exotic surfaces and explore their properties in relation to the Jacobian conjecture. In another application we give the structure of the real variety of the asymptotic values of a polynomial mapf: R ² →R ² .     

Remarks on the second neighborhood problem

The second neighborhood conjecture of Seymour asserts that for any orientation G = (V,E), there exists a vertex υ ∈ V so that |N⁺(υ)| ≤ |N⁺⁺(υ)|. The conjecture was resolved by Fisher for tournaments. In this article, we prove the second neighborhood conjecture for several additional classes of dense orientations. We also prove some approximation results, and reduce an asymptotic version of the conjecture to a finite case. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 208–220, 2007     

Involutions and linear systems on holomorphic symplectic manifolds

A K3 surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that similar statement holds for the generic couple (X, H) with X a deformation of (K3)^[n] and H an ample divisor of square 2 for Beauville’s quadratic form. If n = 2 then according to the conjecture X is a double cover of a singular) sextic 4-fold in It follows from the conjecture that a deformation of (K3)^[n] carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture. In doing so we bump into some interesting geometry: examples of two antisymplectic involutions generating an interesting dynamical system, a case Strange duality and what is probably an involution on the moduli space degree-2 quasi-polarized (X, H) where X is a deformation of (K3)^[2]. Received: June 2004 Revision: December 2004 Accepted: January 2005     

Direct sums of balanced functions,perfect nonlinear functions,and orthogonal cocycles

Determining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by Nyberg on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg's result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non‐cyclic abelian groups and use it to find all the orthogonal cocycles over Z ₂^t, 2 ≤ t ≤ 4. We conjecture that any orthogonal cocycle over Z ₂^t, t ≥ 2, must be multiplicative. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 173–181, 2008     

From the Mahler Conjecture to Gauss Linking Integrals

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product v(K)  = (Vol K)(Vol K°) of the volume of a symmetric convex body and its polar body K°. The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain is minimized when K is an ellipsoid. It implies the Mahler conjecture up to a factor of (π/4) ⁿ γ _n , where γ _n is a monotonic factor that begins at 4/π and converges to . This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of c ⁿ . The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K ^◇, with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere S ^n-1 and in hyperbolic space H ^n-1. Received: December 2006, Accepted: January 2007     

Proof of the local REM conjecture for number partitioning. II. Growing energy scales

We continue our analysis of the number partitioning problem with n weights chosen i.i.d. from some fixed probability distribution with density ρ. In Part I of this work, we established the so‐called local REM conjecture of Bauke, Franz and Mertens. Namely, we showed that, as n → ∞, the suitably rescaled energy spectrum above some fixed scale α tends to a Poisson process with density one, and the partitions corresponding to these energies become asymptotically uncorrelated. In this part, we analyze the number partitioning problem for energy scales α_n that grow with n, and show that the local REM conjecture holds as long as n^‐1/4α_n → 0, and fails if α_n grows like κn^1/4 with κ > 0. We also consider the SK‐spin glass model, and show that it has an analogous threshold: the local REM conjecture holds for energies of order o(n), and fails if the energies grow like κn with κ > 0. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Further results on correlation functions of de Bruijn sequences

In this paper we develop some bounds for the correlation functions of sequences with period 2 ⁿ and specific linear complexities. These bounds are also applicable to the correlation functions of de Bruijn sequences with spann and specific linear complexities. It is interesting that a conjecture of Chan, Games and Key's for the case ofn=2 ^m can be proved easily by using the results developed here. Their conjecture asserts that there are no de Bruijn sequences with spann and linear complexity 2 ^n–1+n+1. The comjecture was proved by Games by a different method.Institute of Systems Science, Academic Sinica     

A unified approach to known and unknown cases of Berge's conjecture

Berge's elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing , there exists a collection C^k of k disjoint independent sets, where each dipath P∈P meets exactly min{|P|, k} of the independent sets in C. This conjecture extends Linial's conjecture, the Greene–Kleitman Theorem and Dilworth's Theorem for all digraphs. The conjecture is known to be true for acyclic digraphs. For general digraphs, it is known for k=1 by the Gallai–Milgram Theorem, for k?λ (where λis the number of vertices in the longest dipath in the graph), by the Gallai–Roy Theorem, and when the optimal path partition P contains only dipaths P with |P|?k. Recently, it was proved (Eur J Combin (2007)) for k=2. There was no proof that covers all the known cases of Berge's conjecture. In this article, we give an algorithmic proof of a stronger version of the conjecture for acyclic digraphs, using network flows, which covers all the known cases, except the case k=2, and the new, unknown case, of k=λ?1 for all digraphs. So far, there has been no proof that unified all these cases. This proof gives hope for finding a proof for all k.     

A family of crooked functions

It has been proved in Bierbrauer and Kyureghyan (Des. Codes Cryptogr. 46:269–301, 2008) that a binomial function aX ⁱ  + bX ^j can be crooked only if both exponents i, j have 2-weight  ≤2. In the present paper we give a brief construction for all known examples of crooked binomial functions. These consist of an infinite family and one sporadic example. The construction of the sporadic example uses the properties of an algebraic curve of genus 3. Computer experiments support the conjecture that each crooked binomial is equivalent either to a member of the family or to the sporadic example.      

Coniveau 2 Complete Intersections and Effective Cones

Griffiths computation of the Hodge filtration on the cohomology of a smooth hypersurface X of degree d in \mathbbPⁿ{\mathbb{P}^n} shows that it has coniveau ≥ c once n ≥ dc. The generalized Hodge conjecture (GHC) predicts that the cohomology of X is then supported on a closed algebraic subset of codimension at least c. This is essentially unknown for c ≥ 2. In the case where c = 2, we exhibit a geometric phenomenon in the variety of lines of X explaining the estimate for the coniveau, and show that (GHC) would be implied in this case by the following conjecture on effective cones of cycles of intermediate dimension: Very moving subvarieties have their class in the interior of the effective cone.     

On Sectional Genus of Quasi—Polarized Manifolds with Non-Negative Kodaira Dimension

Let X be a smooth projective variety over ? and L be a nef-big divisor on X. Then (X, L) is called a quasi - polarized manifold. Then we conjecture that g(L) ≥ q(X), where g(L) is the sectional genus of L and q(X) = dim H¹(O_x) is the irregularity of X. In general it is unknown that this conjecture is true or not even in the case of dim X = 2. For example, this conjecture is true if dim X = 2 and dim H≥(L) > 0. But it is unknown if dim X ≥ 3 and dim H⁰(L) > 0. In this paper, we consider a lower bound for g(L) if dim X = 2, dim H⁰(L) ≥ 2, and k(X) ≥ 0. We obtain a stronger result than the above conjecture if dim Bs|L| ≤ 0 by a new method which can be applied to higher dimensional cases. Next we apply this method to the case in which dim X = n ≥ 3 and we obtain a lower bound for g(L) if dim X = 3, dim H⁰(L) ≥ 2, and k(X) ≥ 0.     

Dvoretzky Type Theorems for Multivariate Polynomials and Sections of Convex Bodies

In this paper we prove the Gromov–Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on \mathbbRⁿ{\mathbb{R}^{n}}, and improve bounds on the number n(d, k) in the analogous conjecture for odd degrees d (this case is known as the Birch theorem) and complex polynomials.     

On connectedness of sets in the real spectra of polynomial rings

Let R be a real closed field. The Pierce–Birkhoff conjecture says that any piecewise polynomial function f on R ⁿ can be obtained from the polynomial ring R[x ₁,..., x _n ] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points , the existence of connected sets in the real spectrum of R[x ₁,..., x _n ], satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce–Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce–Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce–Birkhoff conjecture in dimension greater than 2. Thirdly, we apply these ideas to give two proofs that the Connectedness conjecture (and hence also the Pierce–Birkhoff conjecture in the abstract formulation) holds locally at any pair of points , one of which is monomial. One of the proofs is elementary while the other consists in deducing this result as an immediate corollary of the main connectedness theorem of this paper.     

Constructive bounds and exact expectations for the random assignment problem

The random assignment problem is to choose a minimum‐cost perfect matching in a complete n×n bipartite graph, whose edge weights are chosen randomly from some distribution such as the exponential distribution with mean 1. In this case it is known that the expectation does not grow unboundedly with n, but approaches some limiting value c* between 1.51 and 2. The limit is conjectured to be π²/6, while a recent conjecture is that for finite n, the expected cost is ∑ 1/i². This paper contains two principal results. First, by defining and analyzing a constructive algorithm, we show that the limiting expectation is c*<1.94. Second, we extend the finite‐n conjecture to partial assignments on complete m×n bipartite graphs and prove it in some limited cases. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 113–144, 1999     

A periodic isoperimetric problem related to the unique games conjecture

We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the unique games conjecture, less a small error. Let n ≥ 2. Suppose a subset Ω of n‐dimensional Euclidean space satisfies ?Ω = Ω^c and Ω + v = Ω^c (up to measure zero sets) for every standard basis vector . For any and for any q ≥ 1, let and let . For any x ∈ ?Ω, let N(x) denote the exterior normal vector at x such that ‖N(x)‖₂ = 1. Let . Our main result shows that B has the smallest Gaussian surface area among all such subsets Ω, less a small error: In particular, Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 10^?9 would prove the endpoint case of the Khot‐Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture.     

Abelian constraints in inverse Galois theory

We show that if a finite group G is the Galois group of a Galois cover of over , then the orders p ⁿ of the abelianization of its p-Sylow subgroups are bounded in terms of their index m, of the branch point number r and the smallest prime of good reduction of the branch divisor. This is a new constraint for the regular inverse Galois problem: if p ⁿ is suitably large compared to r and m, the branch points must coalesce modulo small primes. We further conjecture that p ⁿ should be bounded only in terms of r and m. We use a connection with some rationality question on the torsion of abelian varieties. For example, our conjecture follows from the so-called torsion conjectures. Our approach also provides a new viewpoint on Fried’s Modular Tower program and a weak form of its main conjecture.     

Twisted Equivariant KK-Theory and the Baum–Connes Conjecture for Group Extensions

In this paper we study the stability of the Baum–Connes conjecture with coefficients undertaking group extensions. For this, it is necessary to extend Kasparov's equivariant KK-theory to an equivariant theory for twisted actions of groups on C ^*-algebras. As a consequence of our stability results, we are able to reduce the problem of whether closed subgroups of connected groups satisfy the Baum–Connes conjecture, with coefficients to the special case of center-free semi-simple Lie groups.     

Self-Reciprocal Irreducible Pentanomials Over \mathbb{F}_2

Joseph Yucas and Gary Mullen conjectured that there is no self-reciprocal irreducible pentanomial of degree n over if n is divisible by 6. In this note we prove this conjecture for the case n ≡ 0, and disprove the conjecture for the case n ≡ 6 (mod 12) AMS Classifications: 11T55