Israel Journal of Mathematics最新文献

For a prime field k of characteristic p > 2, we construct the Bökstedt periodicity generator v ∈ THH₂(k) as an explicit class in the stabilization of K-theory with coefficients K(k, −), and we show directly that v is not nilpotent in THH(k). This gives an alternative proof of the “multiplicative” part of Bökstedt periodicity.

A Poisson point process of unit intensity is placed in the square [0, n]². An increasing path is a curve connecting (0, 0) with (n, n) which is non-decreasing in each coordinate. Its length is the number of points of the Poisson process which it passes through. Baik, Deift and Johansson proved that the maximal length of an increasing path has expectation 2n − n^1/3(c₁ + o(1)), variance n^2/3(c₂ + o(1)) for some c₁, c₂ > 0 and that it converges to the Tracy–Widom distribution after suitable scaling. Johansson further showed that all maximal paths have a displacement of ({n^{{2 over 3} + o(1)}}) from the diagonal with probability tending to one as n → ∞. Here we prove that the maximal length of an increasing path restricted to lie within a strip of width n^γ, (gamma < {2 over 3}), around the diagonal has expectation 2n − n^1−γ+o(1), variance ({n^{1 - {gamma over 2} + o(1)}}) and that it converges to the Gaussian distribution after suitable scaling.

The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem of Edmonds and Fulkerson. A packing for a family (({M_i}:i in Theta)) of matroids on the common edge set E is a system (({S_i}:i in Theta)) of pairwise disjoint subsets of E where S_i is panning in M_i. Similarly, a covering is a system (I_i: i ∈ Θ) with ({cup _{i in Theta}}{I_i} = E) where I_i is independent in M_i. The conjecture states that for every matroid family on E there is a partition (E = {E_p} sqcup {E_c}) such that (({M_i}upharpoonright{E_p}:i in Theta)) admits a packing and (({M_i}.{E_c}:i in Theta)) admits a covering. We prove the case where E is countable and each M_i is either finitary or cofinitary. To do so, we give a common generalisation of the singular matroid intersection theorem of Ghaderi and the countable case of the Matroid Intersection Conjecture by Nash-Williams by showing that the conjecture holds for countable matroids having only finitary and cofinitary components.

Abstract

We investigate the failure of Bézout’s Theorem for two symplectic surfaces in ℂP² (and more generally on an algebraic surface), by proving that every plane algebraic curve C can be perturbed in the ({{cal C}^infty }) -topology to an arbitrarily close smooth symplectic surface C_ϵ with the property that the cardinality #C_ϵ ∩ Z_d of the transversal intersection of C_ϵ with an algebraic plane curve Z_d of degree d, as a function of d, can grow arbitrarily fast. As a consequence we obtain that, although Bézout’s Theorem is true for pseudoholomorphic curves with respect to the same almost complex structure, it is “arbitrarly false” for pseudoholomorphic curves with respect to different (but arbitrarily close) almost-complex structures (we call this phenomenon “instability of Bézout’s Theorem”).

In this paper, we investigate linear relations among regularized motivic iterated integrals on ℙ¹ ∖ {0, 1, ∞} of depth two, which we call regularized motivic double zeta values. Some mysterious connections between motivic multiple zeta values and modular forms are known, e.g., Gangl–Kaneko–Zagier relation for the totally odd double zeta values and Ihara–Takao relation for the depth graded motivic Lie algebra. In this paper, we investigate so-called non-admissible cases and give many new Gangl–Kaneko–Zagier type and Ihara–Takao type relations for regularized motivic double zeta values. Specifically, we construct linear relations among a certain family of regularized motivic double zeta values from odd period polynomials of modular forms for the unique index two congruence subgroup of the full modular group. This gives the first non-trivial example of a construction of the relations among multiple zeta values (or their analogues) from modular forms for a congruence subgroup other than the SL₂(ℤ).

We give an alternative proof for discrete Brunn–Minkowski type inequalities, recently obtained by Halikias, Klartag and the author. This proof also implies somewhat stronger weighted versions of these inequalities. Our approach generalizes ideas of Gozlan, Roberto, Samson and Tetali from the theory of measure transportation and provides new displacement convexity of entropy type inequalities on the n-dimensional integer lattice.

Given a k-uniform hypergraph ℋ and sufficiently large m ≫ m₀(ℋ), we show that an m-element set I ⊆ V(ℋ), chosen uniformly at random, with probability 1 − e−^ω(m) is either not independent or is contained in an almost-independent set in ℋ which, crucially, can be constructed from carefully chosen o(m) vertices of I. As a corollary, this implies that if the largest almost-independent set in ℋ is of size o(v(ℋ)) then I itself is an independent set with probability e^−ω(m). More generally, I is very likely to inherit structural properties of almost-independent sets in ℋ.

The value m₀(ℋ) coincides with that for which Janson’s inequality gives that I is independent with probability at most ({e^{- Theta ({m_0})}}). On the one hand, our result is a significant strengthening of Janson’s inequality in the range m ≫ m₀. On the other hand, it can be seen as a probabilistic variant of hypergraph container theorems, developed by Balogh, Morris and Samotij and, independently, by Saxton and Thomason. While being strictly weaker than the original container theorems in the sense that it does not apply to all independent sets of size m, it is nonetheless sufficient for many applications and admits a short proof using probabilistic ideas.

We show that the principal algebraic actions of countably infinite groups associated to lopsided elements in the integral group ring satisfying some orderability condition are Bernoulli.

We prove an effective form of Hilbert’s irreducibility theorem for polynomials over a global field K. More precisely, we give effective bounds for the number of specializations (t in {{cal O}_K}) that do not preserve the irreducibility or the Galois group of a given irreducible polynomial F(T, Y) ∈ K[T, Y]. The bounds are explicit in the height and degree of the polynomial F(T, Y), and are optimal in terms of the size of the parameter (t in {{cal O}_K}). Our proofs deal with the function field and number field cases in a unified way.