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.     

The Mukai conjecture for log Fano manifolds

For a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D ₁) of Picard groups is injective for any irreducible component D ₁ ? D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).     

An algebraic construction of the ring of¶piecewise polynomial functions

In this note we will prove an algebraic characterization of the piecewiese polynomial functions of a real ℚ-algebra A. This characterization is related to the realm of investigations concerning the Pierce–Birkhoff conjecture. Received: 23 June 1997 / Revised version: 26 May 1998     

Points with maximal Birkhoff average oscillation

Let f: X → X be a continuous map with the specification property on a compact metric space X. We introduce the notion of the maximal Birkhoff average oscillation, which is the “worst” divergence point for Birkhoff average. By constructing a kind of dynamical Moran subset, we prove that the set of points having maximal Birkhoff average oscillation is residual if it is not empty. As applications, we present the corresponding results for the Birkhoff averages for continuous functions on a repeller and locally maximal hyperbolic set.     

Birkhoff integral for multi-valued functions

The aim of this paper is to study Birkhoff integrability for multi-valued maps , where (Ω,Σ,μ) is a complete finite measure space, X is a Banach space and cwk(X) is the family of all non-empty convex weakly compact subsets of X. It is shown that the Birkhoff integral of F can be computed as the limit for the Hausdorff distance in cwk(X) of a net of Riemann sums ∑_nμ(A_n)F(t_n). We link Birkhoff integrability with Debreu integrability, a notion introduced to replace sums associated to correspondences when studying certain models in Mathematical Economics. We show that each Debreu integrable multi-valued function is Birkhoff integrable and that each Birkhoff integrable multi-valued function is Pettis integrable. The three previous notions coincide for finite dimensional Banach spaces and they are different even for bounded multi-valued functions when X is infinite dimensional and X∗ is assumed to be separable. We show that when F takes values in the family of all non-empty convex norm compact sets of a separable Banach space X, then F is Pettis integrable if, and only if, F is Birkhoff integrable; in particular, these Pettis integrable F's can be seen as single-valued Pettis integrable functions with values in some other adequate Banach space. Incidentally, to handle some of the constructions needed we prove that if X is an Asplund Banach space, then cwk(X) is separable for the Hausdorff distance if, and only if, X is finite dimensional.     

Symmetry theorems for the overdetermined eigenvalue problems

The well-known Schiffer conjecture saying that for a smooth bounded domain Ω⊂Rn, if there exists a positive Neumann eigenvalue such that the corresponding Neumann eigenfunction u is constant on the boundary of Ω, then Ω is a ball. In this paper, we shall prove that the Schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding Neumann eigenfunction is constant on the boundary. We also prove a similar result to the Berenstein conjecture for the overdetermined Dirichlet eigenvalue problem.     

The Tate conjecture for K3 surfaces over finite fields

Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.     

The critical exponent for continuous conventional powers of doubly nonnegative matrices

We prove that there exists an exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative. We show that this critical exponent cannot be less than n-2 and we conjecture that it is always n-2 (as it is with Hadamard powering). We prove this conjecture when n<6 and in certain other special cases. We establish a quadratic bound for the critical exponent in general.     

Alternating sums over π-subgroups

Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p-defect is determined by local subgroups. In this paper we replace p by a set of primes π and prove a π-version of Dade's conjecture for π-separable groups. This extends the (known) p-solvable case of the original conjecture and relates to a π-version of Alperin's weight conjecture previously established by the authors.     

On the multigraded Hilbert and Poincaré-Betti series and the Golod property of monomial rings

In this paper we study the multigraded Hilbert and Poincaré-Betti series of A=S/a, where S is the ring of polynomials in n indeterminates divided by the monomial ideal a. There is a conjecture about the multigraded Poincaré-Betti series by Charalambous and Reeves which they proved in the case where the Taylor resolution is minimal. We introduce a conjecture about the minimal A-free resolution of the residue class field and show that this conjecture implies the conjecture of Charalambous and Reeves and, in addition, gives a formula for the Hilbert series. Using Algebraic Discrete Morse theory, we prove that the homology of the Koszul complex of A with respect to x₁,…,x_n is isomorphic to a graded commutative ring of polynomials over certain sets in the Taylor resolution divided by an ideal r of relations. This leads to a proof of our conjecture for some classes of algebras A. We also give an approach for the proof of our conjecture via Algebraic Discrete Morse theory in the general case.The conjecture implies that A is Golod if and only if the product (i.e. the first Massey operation) on the Koszul homology is trivial. Under the assumption of the conjecture we finally prove that a very simple purely combinatorial condition on the minimal monomial generating system of a implies Golodness for A.     

Circuits through specified edges

We prove a theorem implying the conjecture of Woodall [14] that, given any k independent edges in a (k+1)-connected graph, there is a circuit containing all of them. This implies the truth of a conjecture of Berge [1, p.214] and provides strong evidence to a conjecture of Lovász [8].     

Kernels and perfectness in arc-local tournament digraphs

In this paper we give a characterization of kernel-perfect (and of critical kernel-imperfect) arc-local tournament digraphs. As a consequence, we prove that arc-local tournament digraphs satisfy a strenghtened form of the following interesting conjecture which constitutes a bridge between kernels and perfectness in digraphs, stated by C. Berge and P. Duchet in 1982: a graph G is perfect if and only if any normal orientation of G is kernel-perfect. We prove a variation of this conjecture for arc-local tournament orientable graphs. Also it is proved that normal arc-local tournament orientable graphs satisfy a stronger form of Berge's strong perfect graph conjecture.     

Dimensions of some affine Deligne-Lusztig varieties

This paper concerns the dimensions of certain affine Deligne-Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold. Rapoport conjectured a formula for the dimensions of the varieties Xμ(b) in the affine Grassmannian. We prove his conjecture for b in the split torus; we find that these varieties are equidimensional; and we reduce the general conjecture to the case of superbasic b. In the affine flag manifold, we prove a formula that reduces the dimension question for Xx(b) with b in the split torus to computations of dimensions of intersections of Iwahori orbits with orbits of the unipotent radical. Calculations using this formula allow us to verify a conjecture of Reuman in many new cases, and to make progress toward a generalization of his conjecture.     

Remarks on the critical graph conjecture

The vertex-critical graph conjecture (critical graph conjecture respectively) states that every vertex-critical (critical) graph has an odd number of vertices. In this note we prove that if G is a critical graph of even order, then G has at least three vertices of less-than-maximum valency. In addition, if G is a 3-critical multigraph of smallest even order, then G is simple and has no triangles.     

A geometric model for odd differential K-theory

Odd K-theory has the interesting property that it admits an infinite number of inequivalent differential refinements. In this paper we provide a bundle theoretic model for odd differential K-theory using the caloron correspondence and prove that this refinement is unique up to a unique natural isomorphism. We characterise the odd Chern character and its transgression form in terms of a connection and Higgs field and discuss some applications. Our model can be seen as the odd counterpart to the Simons–Sullivan construction of even differential K-theory. We use this model to prove a conjecture of Tradler–Wilson–Zeinalian [16], which states that the model developed there also defines the unique differential extension of odd K-theory.     

Proof of semialgebraic covering mapping cylinder conjecture with semialgebraic covering homotopy theorem

In this paper we prove the semialgebraic version of Palais' covering homotopy theorem, and use this to prove Bredon's covering mapping cylinder conjecture positively in the semialgebraic category. Bredon's conjecture was originally stated in the topological category, and a topological version of our semialgebraic proof of the conjecture answers the original topological conjecture for topological G-spaces over “simplicial” mapping cylinders.     

On the KŁR conjecture in random graphs

The K?R conjecture of Kohayakawa, ?uczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G _n,p , for sufficiently large p:= p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.     

Divisibility properties of values of partial zeta functions at non-positive integers

We conjecture a new bound on the exact denominators of the values at non-positive integers of imprimitive partial zeta functions associated with an Abelian extension of number fields. At s?=?0, this conjecture is closely connected to a conjecture of David Hayes. We prove the new conjecture assuming that the Coates–Sinnott conjecture holds for the extension.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

Profinite groups, profinite completions and a conjecture of Moore

Let R be any ring (with 1), Γ a group and RΓ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an RΓ-module, whose restriction to RH is projective.Moore's conjecture (J. Pure Appl. Algebra 7(1976)287): Assume for every nontrivial element x in Γ, at least one of the following two conditions holds:

(M1)

〈x〉∩H≠{e} (in particular this holds if Γ is torsion free)

(M2)

ord(x) is finite and invertible in R.

Then M is projective as an RΓ-module.More generally, the conjecture has been formulated for crossed products R*Γ and even for strongly graded rings R(Γ). We prove the conjecture for new families of groups, in particular for groups whose profinite completion is torsion free.The conjecture can be formulated for profinite modules M over complete groups rings [[RΓ]] where R is a profinite ring and Γ a profinite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.