Abelian monopoles and complex geometry

Three remarks concerning the constructive properties of smooth 4-manifolds with nontrivial Seiberg-Witten invariants are presented. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 420–428, March, 1999.     

The Positivity of Intersection Multiplicities and Symbolic Powers of Prime Ideals

Serre's nonnegativity conjecture for intersection multiplicities has recently been proven by O. Gabber. In this paper we investigate Serre's positivity conjecture using the methods which he developed. We show in particular that the positivity conjecture has implications for properties of symbolic powers of prime ideals in regular local rings.     

Slice filtration on motives and the Hodge conjecture (with an appendix by J. Ayoub)

We clarify the expected properties of the slice filtration on triangulated motives from the point of view of the generalized Hodge conjecture. In the appendix, J. Ayoub proves unconditionally that the slice filtration does not respect geometric motives. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A sufficient condition for the bicolorability of a hypergraph

In this note we prove a long-standing conjecture of Sterboul [P. Duchet, Hypergraphs, in: R. Graham, M. Grötschel, L. Lovász (Eds.), Handbook of Combinatorics, 1995, pp. 381-432 (Chapter 7)], which states that a hypergraph is bicolorable provided it does not contain a specific kind of odd cycle. This is currently the strongest result of its kind, improving on results by Berge [Graphs and Hypergraphs, North-Holland, American Elsevier, Amsterdam, 1973] and Fournier and Las Vergnas [Une classe d’hypergraphes bichromatiques II, Discrete Math. 7 (1974) 99-106; A class of bichromatic hypergraphs, Ann. Discrete Math. 21, in: C. Berge, V. Chvátal (Eds.), Topics on Perfect Graphs, 1984, pp. 21-27].     

The Gold Partition Conjecture for 6-Thin Posets

We consider the Gold Partition Conjecture (GPC) that implies the 1/3–2/3 Conjecture. We prove the GPC in the case where every element of the poset is incomparable with at most six others. The proof involves the extensive use of computers. This paper contains results obtained using computer resources of the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM), University of Warsaw.     

Skew partitions in perfect graphs

We discuss some new and old results about skew partitions in perfect graphs.     

An illustrated theory of hyperbolic virtual polytopes

The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.      

Points of Bounded Height on Equivariant Compactifications of Vector Groups, I

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.     

On a cycle partition problem

Let G be any graph and let c(G) denote the circumference of G. We conjecture that for every pair c₁,c₂ of positive integers satisfying c₁+c₂=c(G), the vertex set of G admits a partition into two sets V₁ and V₂, such that V_i induces a graph of circumference at most c_i, i=1,2. We establish various results in support of the conjecture; e.g. it is observed that planar graphs, claw-free graphs, certain important classes of perfect graphs, and graphs without too many intersecting long cycles, satisfy the conjecture.This work is inspired by a well-known, long-standing, analogous conjecture involving paths.     

Cn中有界域的Bergman核函数的零点问题

多复变数空间Cn中有界域的Bergman核函数的零点问题集中表现为陆启铿猜想.陆启铿猜想是波兰数学家M.Skwarczynski对陆启铿1966年的一篇文章中关于Bergman核函数的零点问题而命名的,至今已经40年了.该猜想已写入了多复变函数论的多本专著,引起很多数学家的兴趣而研究之,已经成为多复变函数论中的一个活跃的研究方向.本文简述了陆启铿猜想的最初含意,综述了迄今为止关于有界域的Bergman核函数有无零点的各种研究成果以及所用的思想和方法.特别对近来出现的陆启铿猜想的新研究领域进行了较详细的阐述并在最后提出了关于陆启铿猜想的6个Open Problems,希望国内的年轻数学家对陆启铿猜想感到兴趣而研究之.