Let (M, g) be a compact Riemannian spin manifold. The Atiyah-Singer index theorem yields a lower bound for the dimension of the kernel of the Dirac operator. We prove that this bound can be attained by changing the Riemannian metric g on an arbitrarily small open set.

2. Benguria, Rafael D.

et al.

Frank, Rupert L.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

It is shown that the sharp constant in the Hardy-Sobolev-Maz'ya inequality on the upper half space H-3 subset of R-3 is given by the Sobolev constant. This is achieved by a duality argument relating the problem to a Hardy-Littlewood-Sobolev type inequality whose sharp constant is determined as well.

3. Bridy, Andrew

et al.

Ingram, Patrick

Jones, Rafe

Juul, Jamie

Levy, Alon

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

Given a finite endomorphism phi of a variety X defined over the field of fractions K of a Dedekind domain, we study the extension K (phi(-infinity)(alpha)) := boolean OR(n >= 1) K (phi(-n) (alpha)) generated by the preimages of alpha under all iterates of phi. In particular when phi is post-critically finite, i.e., there exists a non-empty, Zariski-open W subset of X such that phi(-1) (W) subset of W and phi : W -> X is etale, we prove that K (phi(-infinity) (alpha)) is rami fied over only finitely many primes of K. This provides a large supply of in finite extensions with restricted rami fication, and generalizes results of Aitken-Hajir-Maire [1] in the case X = A(1) and Cullinan-Hajir, Jones-Manes [7, 13] in the case X = P-1. Moreover, we conjecture that this finite rami fication condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for X = P-1. The proof relies on Faltings' theorem and a local argument.

4. Costa, L.

et al.

Di Rocco, Sandra

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Miro-Roig, R. M.

Derived category of fibrations2011In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 18, no 3, p. 425-432Article in journal (Refereed)

Abstract [en]

In this paper we give a structure theorem for the derived category D(b)(X) of a Zariski locally trivial fibration X over Z with fiber F provided both F and Z have a full strongly exceptional collection of line bundles.

5. Friedlander, John B.

et al.

Kurlberg, Pär

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Shparlinski, Igor E.

PRODUCTS IN RESIDUE CLASSES2008In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 15, no 5-6, p. 1133-1147Article in journal (Refereed)

Abstract [en]

We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results "on average" over moduli m and residue classes modulo m and somewhat stronger results when the average is restricted to prime moduli m = p. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results.

6.

Iakovlev, Alexander S.

et al.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Strömberg, Jan-Olov

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

In this paper, we will provide the quantitative estimation for the dependence of a lower bound of the Hardy-Littlewood maximal function. This work was inspired by the paper [1] of Stein and Strömberg where general properties of the maximal function were studied. In that work, the increase with the dimension d of the constant Ad that appears in the weak type (1, 1) inequality for the maximal function was proved however no estimation were given. In a recent paper [2], J.M. Aldaz showed that the lowest constant Ad tends to infinity as the dimension d → ∞. In this paper, we improve the result of J.M. Aldaz providing quantitative estimation of Ad ≥ Cd1/4, where C is a constant independent of d.

In this paper, we show that for almost all primes p there is an integer solution xε [2,p-1] to the congruence xx ≡ x (mod p). The solutions can be interpretated as fixed points of the map x→xx (mod p), and we study numerically and discuss some unexpected properties of the dynamical system associated with this map.