On the degree of cocycles with values in the group<Emphasis Type="Italic">SU</Emphasis>(2)

In this paper are presented some properties of smooth cocycles over irrational rotations on the circle with values in the groupSU(2). It is proved that the degree of anyC ²-cocycle (the notion of degree was introduced in [2]) belongs to 2πℕ (ℕ={0, 1, 2,...}). It is also shown that if the rotation satisfies a Diophantine condition, then everyC ^∞-cocycle with nonzero degree isC ^∞-cohomologous to a cocycle of the form

On the asymptotic completeness of the Volterra calculus

The Volterra calculus is a simple and powerful pseudodifferential tool for inverting parabolic equations which has found many applications in geometric analysis. An important property in the theory of pseudodifferential operators is asymptotic completeness, which allows the construction of parametrices modulo smoothing operators. In this paper, we present new and fairly elementary proofs of the asymptotic completeness of the Volterra calculus. The author was partially supported by the European RT NetworkGeometric Analysis HPCRN-CT-1999-00118.     

The lower density conjecture for harmonic measure

In this paper, we establish the lower density conjecture for harmonic measure in simply connected plane domains.     

The structure of strongly stationary systems

Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson introduced the notion of strong stationarity, showing that certain recurrence properties hold for arbitrary measure preserving systems if they are valid for strongly stationary ones. We construct some new examples and prove a structure theorem for strongly stationary systems. The building blocks are Bernoulli systems and rotations on nilmanifolds.     

Invariant sets and measures of nonexpansive group automorphisms

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link inS ³. Our main tool is a careful use of the Århus integral and the (now proven) “Wheels” and “Wheeling” conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered spaces, but does separate Seifert fibered spaces which are integral homology spheres.     

Word maps have large image

An elementw in the free group onr letters defines a mapf _w,G:G^v→G for each groupG. In this note, we show that wheneverw≠1 andG is a semisimple algebraic group,f _w,G is dominant. As an application, we show that for fixedw and Γ _i a sequence of pairwise non-isomorphic finite simple groups,

Realizing deformations of curves using Lubin-Tate formal groups

Letk be an algebraically closed field of characteristicp>0 andR be a suitable valuation ring of characteristic 0, dominating the Witt vectorsW(k). We show how Lubin-Tate formal groups can be used to lift those orderp ⁿ automorphisms ofk[[Z]] toR[[Z]], which occur as endomorphisms of a formal group overk of suitable height. We apply this result to prove the existence of smooth liftings of galois covers of smooth curves from characteristicp to characteristic 0, provided thep-part of the inertia groups acting on the completion of the local rings at the points of the cover overk arep-power cyclic and determined by an endomorphism of a suitable formal group overk. The author would like to express his thanks to the Max-Planck-Institut für Mathematik, Bonn, for its hospitality and support, where this research was done during a visit from June – September 2001.     

A generalization of the Contou-Carrère symbol

Using techniques of Algebraic Geometry, the aim of this paper is to give a generalized definition of the Contou-Carrère symbol as a morphism of schemes. In fact, from formal schemes and Heisenberg groups, we provide a new definition of the Contou-Carrère symbol and a generalization of it associated with a separable extension . Moreover, a reciprocity law is proved and classical explicit reciprocity laws are recovered from it. This work is partially supported by the DGESYC research contract n. BFM2003-00078 and Castilla y León Regional Government contract SA064/01.     

Singular Fano compactifications of (I)

In this paper we study the singular Fano compactifications of with small Gorenstein singularities and determine the structure of such a compactification in the case of the Fano index r 2.Mathematics Subject Classification (2000): 14J45 , 32J05     

On index one covers of two-dimensional purely log terminal singularities in positive characteristic

Let S be a normal surface over an algebraically closed field k and let be a standard boundary. We consider index 1 covers of the purely log terminal pair (S,). We prove that when S is smooth and char k=p3, then is canonical under some conditions. To prove this, we classify the boundary =(1–1/b_i)D_i which makes (S,) a purely log terminal pair, and then reduce equations defining singularity of to the normal forms of RDP. Unfortunately there are some counterexamples in p=2, and we classify them. These results give partial solutions to the index 1 cover conjecture in positive characteristic.Mathematical Subject Classification (2000): 14E20