Conjecture

We reformulate a conjecture of Deligne on 1motives by using the integral weight ﬁltration of Gillet and Soul´ on cohomology, and prove it. This implies e the original conjecture up to isogeny. If the degree of cohomology is at most two, we can prove the conjecture for the Hodge realization without isogeny, and even for 1motives with torsion. j Let X be a complex algebraic variety. We denote by H(1) (X, Z) the maximal mixed Hodge structure of type {(0, 0), (0, 1), (1, 0), (1, 1)} contained in j j H j (X, Z). Let H(1) (X, Z)fr...
42p tuanloccuoi 04012013 22 5 Download

This paper should be regarded as a sequel to [7]. There it was shown that the geometric Langlands conjecture for GLn follows from a certain vanishing conjecture. The goal of the present paper is to prove this vanishing conjecture. Let X be a smooth projective curve over a ground ﬁeld k. Let E be an mdimensional local system on X, and let Bunm be the moduli stack of rank m vector bundles on X.
67p tuanloccuoi 04012013 20 5 Download

We give a proof of the NirenbergTreves conjecture: that local solvability of principaltype pseudodiﬀerential operators is equivalent to condition (Ψ). This condition rules out sign changes from − to + of the imaginary part of the principal symbol along the oriented bicharacteristics of the real part. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case).
41p noel_noel 17012013 25 5 Download

We classify the measures on SL(k, R)/ SL(k, Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood’s conjecture has Hausdorﬀ dimension zero. 1. Introduction 1.1. Number theory and dynamics. There is a long and rich tradition of applying dynamical methods to number theory.
49p noel_noel 17012013 19 5 Download

D. Mumford conjectured in [33] that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes κi of dimension 2i. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by BΓ∞ , where Γ∞ is the group of isotopy classes of automorphisms of a smooth oriented connected surface of “large” genus.
100p noel_noel 17012013 18 5 Download

To any two graphs G and H one can associate a cell complex Hom (G, H) by taking all graph multihomomorphisms from G to H as cells. In this paper we prove the Lov´sz conjecture which states that a if Hom (C2r+1 , G) is kconnected, then χ(G) ≥ k + 4, where r, k ∈ Z, r ≥ 1, k ≥ −1, and C2r+1 denotes the cycle with 2r +1 vertices. The proof requires analysis of the complexes Hom (C2r+1 , Kn ). For even n, the obstructions to graph colorings are provided by the presence of torsion...
44p noel_noel 17012013 32 5 Download

In this paper we will prove the CalabiYau conjectures for embedded surfaces (i.e., surfaces without selfintersection). In fact, we will prove considerably more. The heart of our argument is very general and should apply to a variety of situations, as will be more apparent once we describe the main steps of the proof later in the introduction.
34p dontetvui 17012013 18 7 Download

We prove an old conjecture of Erd˝s and Graham on sums of unit fractions: o There exists a constant b 0 such that if we rcolor the integers in [2, br ], then there exists a monochromatic set S such that n∈S 1/n = 1. 1. Introduction We will prove a result on unit fractions which has the following corollary. Corollary. There exists a constant b so that for every partition of the integers in [2, br ] into r classes, there is always one class containing a subset S with the property n∈S 1/n = 1....
13p tuanloccuoi 04012013 26 6 Download

We prove the strong Macdonald conjecture of Hanlon and Feigin for reductive groups G. In a geometric reformulation, we show that the Dolbeault cohomology H q (X; Ωp ) of the loop Grassmannian X is freely generated by de Rham’s forms on the disk coupled to the indecomposables of H • (BG). Equating the two Euler characteristics gives an identity, independently known to Macdonald [M], which generalises Ramanujan’s 1 ψ1 sum. For simply laced root systems at level 1, we also ﬁnd a ‘strong form’ of Bailey’s 4 ψ4 sum. ...
47p dontetvui 17012013 27 6 Download

A longstanding conjecture due to Michael Freedman asserts that the 4dimensional topological surgery conjecture fails for nonabelian free groups, or equivalently that a family of canonical examples of links (the generalized Borromean rings) are not A − B slice. A stronger version of the conjecture, that the Borromean rings are not even weakly A − B slice, where one drops the equivariant aspect of the problem, has been the main focus in the search for an obstruction to surgery.
21p dontetvui 17012013 38 6 Download

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Proof of the Razumov–Stroganov conjecture for some inﬁnite families of link patterns...
15p thulanh6 14092011 26 5 Download

In 1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Pad´ approximants e converges uniformly in compact subsets of the ball omitting poles of f . There is also apparently a cruder version of the conjecture due to Pad´ himself, going e back to the early twentieth century. We show here that for carefully chosen q on the unit circle, the RogersRamanujan continued fraction 1+ qz q 2 z q 3 z + + + ··· 1 1 1 provides a counterexample to the conjecture. We also highlight some...
44p tuanloccuoi 04012013 23 5 Download

We introduce a class of metric spaces which we call “bolic”. They include hyperbolic spaces, simply connected complete manifolds of nonpositive curvature, euclidean buildings, etc. We prove the Novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a “bolic”, weakly geodesic metric space of bounded geometry. 1. Introduction This work has grown out of an attempt to give a purely KKtheoretic proof of a result of A. Connes and H. Moscovici ([CM], [CGM]) that hyperbolic groups satisfy the Novikov conjecture. ...
43p tuanloccuoi 04012013 18 5 Download

Let G = GLn (K) where K is either R or C and let P = Pn (K) be the subgroup of matrices in GLn (K) consisting of matrices whose last row is (0, 0, . . . , 0, 1). Let π be an irreducible unitary representation of G. Gelfand and Neumark [GelNeu] proved that if K = C and π is in the GelfandNeumark series of irreducible unitary representations of G then the restriction of π to P remains irreducible. Kirillov [Kir] conjectured that this should be true for all irreducible unitary representations π of GLn...
47p tuanloccuoi 04012013 20 5 Download

We prove the ionization conjecture within the HartreeFock theory of atoms. More precisely, we prove that, if the nuclear charge is allowed to tend to inﬁnity, the maximal negative ionization charge and the ionization energy of atoms nevertheless remain bounded. Moreover, we show that in HartreeFock theory the radius of an atom (properly deﬁned) is bounded independently of its nuclear charge. Contents 1. Introduction and main results 2. Notational conventions and basic prerequisites 3. HartreeFock theory 4. ThomasFermi theory ...
69p tuanloccuoi 04012013 23 5 Download

We construct a proper C 2 smooth function on R4 such that its Hamiltonian ﬂow has no periodic orbits on at least one regular level set. This result can be viewed as a C 2 smooth counterexample to the Hamiltonian Seifert conjecture in dimension four. 1. Introduction The “Hamiltonian Seifert conjecture” is the question whether or not there exists a proper function on R2n whose Hamiltonian ﬂow has no periodic orbits on at least one regular level set.
25p tuanloccuoi 04012013 28 5 Download

At a prime of ordinary reduction, the Iwasawa “main conjecture” for elliptic curves relates a Selmer group to a padic Lfunction. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the padic Lfunction is wellbehaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case. Namely, Kobayashi’s conjecture relates modiﬁed Selmer groups, which he deﬁned, with modiﬁed padic Lfunctions deﬁned by the ﬁrst author.
19p tuanloccuoi 04012013 23 5 Download

We prove that the existence of an automorphism of ﬁnite order on a Qvariety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special values of the Γfunction. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of GrossDeligne [11, p. 205]1 .
29p tuanloccuoi 04012013 29 5 Download

This project would not have been possible without the generous support of many people. I would particularly like to thank Kerri Smith, Sam Ferguson, Sean McLaughlin, Jeff Lagarias, Gabor Fejes T´oth, Robert MacPherson, and the referees for their support of this project. A more comprehensive list of those who contributed to this project in various ways appears in [Hal06b].
122p noel_noel 17012013 22 5 Download

A Hausdorﬀ measure version of the DuﬃnSchaeﬀer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of Rk to Hausdorﬀ measure theoretic statements. In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorﬀ theory. This is rather surprising since the latter theory is viewed to be a subtle reﬁnement of the former. ...
23p noel_noel 17012013 27 5 Download