Regularity of the free boundary for the porous medium equation

We study the regularity of the free boundary for solutions of the porous medium equation , , on , with initial data nonnegative and compactly supported. We show that, under certain assumptions on the initial data , the pressure will be smooth up to the interface , when , for some . As a consequence, the free-boundary is smooth.

     

Pythagoras numbers of fields

A field of characteristic is said to have finite Pythagoras number if there exists an integer such that each nonzero sum of squares in can be written as a sum of squares, in which case the Pythagoras number of is defined to be the least such integer. As a consequence of Pfister's results on the level of fields, of a nonformally real field is always of the form or , and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form , , and can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer there exists a formally real field with . As a refinement, we will show that if and are integers such that , then there exists a uniquely ordered field with and (resp. ), where (resp. ) denotes the supremum of the dimensions of anisotropic forms over which are torsion in the Witt ring of (resp. which are indefinite with respect to each ordering on ).

     

The enumerative geometry of surfaces and modular forms

Let be a surface, and let be a holomorphic curve in representing a primitive homology class. We count the number of curves of geometric genus with nodes passing through generic points in in the linear system for any and satisfying .

When , this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Göttsche has generalized their conjecture to arbitrary in terms of quasi-modular forms. We prove these formulas using Gromov-Witten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus constrained to points are also given in terms of quasi-modular forms.     

-regularity, -fibrant Hochschild homology, and a conjecture of Vorst

In this paper we prove that for an affine scheme essentially of finite type over a field and of dimension , -regularity implies regularity, assuming that the characteristic of is zero. This verifies a conjecture of Vorst.

     

Compatibility of local and global Langlands correspondences

We prove the compatibility of local and global Langlands correspondences for , which was proved up to semisimplification in M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Studies 151, Princeton Univ. Press, Princeton-Oxford, 2001. More precisely, for the -dimensional -adic representation of the Galois group of an imaginary CM-field attached to a conjugate self-dual regular algebraic cuspidal automorphic representation of , which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of to the decomposition group of a place of not dividing corresponds to by the local Langlands correspondence. If is square integrable for some finite place we deduce that is irreducible. We also obtain conditional results in the case .

     

The essentially tame local Langlands correspondence, I

Let be a non-Archimedean local field (of characteristic or ) with finite residue field of characteristic . An irreducible smooth representation of the Weil group of is called essentially tame if its restriction to wild inertia is a sum of characters. The set of isomorphism classes of irreducible, essentially tame representations of dimension is denoted . The Langlands correspondence induces a bijection of with a certain set of irreducible supercuspidal representations of . We consider the set of isomorphism classes of certain pairs , called ``admissible', consisting of a tamely ramified field extension of degree and a quasicharacter of . There is an obvious bijection of with . Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of with , generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of with . We show that one obtains the Langlands correspondence by composing the map with a permutation of of the form , where is a tamely ramified character of depending on . This answers a question of Moy (1986). We calculate the character in the case where is totally ramified of odd degree.

     

Sieving by large integers and covering systems of congruences

An old question of Erdos asks if there exists, for each number , a finite set of integers greater than and residue classes for whose union is . We prove that if is bounded for such a covering of the integers, then the least member of is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number , the complement in of any union of residue classes , for distinct , has density at least for sufficiently large. Here is a positive number depending only on . Either of these new results implies another conjecture of Erdos and Graham, that if is a finite set of moduli greater than , with a choice for residue classes for which covers , then the largest member of cannot be . We further obtain stronger forms of these results and establish other information, including an improvement of a related theorem of Haight.

     

Mating Siegel quadratic polynomials

Let be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers and . Using a new degree Blaschke product model for the dynamics of and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers and .

     

Independence of of monodromy groups

Let be a smooth curve over a finite field of characteristic , let be a number field, and let be an -compatible system of lisse sheaves on the curve . For each place of not lying over , the -component of the system is a lisse -sheaf on , whose associated arithmetic monodromy group is an algebraic group over the local field . We use Serre's theory of Frobenius tori and Lafforgue's proof of Deligne's conjecture to show that when the -compatible system is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is ``independent of '. More precisely, after replacing by a finite extension, there exists a connected split reductive algebraic group over the number field such that for every place of not lying over , the identity component of the arithmetic monodromy group of is isomorphic to the group with coefficients extended to the local field .

     

The McKay correspondence as an equivalence of derived categories

Let be a finite group of automorphisms of a nonsingular three-dimensional complex variety , whose canonical bundle is locally trivial as a -sheaf. We prove that the Hilbert scheme parametrising -clusters in is a crepant resolution of and that there is a derived equivalence (Fourier-Mukai transform) between coherent sheaves on and coherent -sheaves on . This identifies the K theory of with the equivariant K theory of , and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.

     

Double affine Hecke algebras and 2-dimensional local fields

We give an interpretation of the double affine Hecke algebra of Cherednik as a (suitably regularized) algebra of double cosets of a group by a subgroup , extending the well-known interpretations of the finite and affine Hecke algebras. In this interpretation, consists of -points of a simple algebraic group, where is a 2-dimensional local field such as or , and is a certain analog of the Iwahori subgroup.

     

Criteria for -ampleness

In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a -ample divisor, where is an automorphism of a projective scheme . Many open questions regarding -ample divisors have remained.

We derive a relatively simple necessary and sufficient condition for a divisor on to be -ample. As a consequence, we show right and left -ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms yield a -ample divisor.

     

The strange duality conjecture for generic curves

Let be a smooth connected projective algebraic curve of genus . The strange duality conjecture connects non-abelian theta functions of rank and level and those of rank and level on (for and , respectively). In this paper we prove this conjecture for generic in the moduli space of curves of genus .

     

The homotopy theory of fusion systems

We define and characterize a class of -complete spaces which have many of the same properties as the -completions of classifying spaces of finite groups. For example, each such has a Sylow subgroup , maps for a -group are described via homomorphisms , and is isomorphic to a certain ring of ``stable elements' in . These spaces arise as the ``classifying spaces' of certain algebraic objects which we call ``-local finite groups'. Such an object consists of a system of fusion data in , as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.

     

Green currents for holomorphic automorphisms of compact Kähler manifolds

Let be a holomorphic automorphism of a compact Kähler manifold of dimension . We study the convex cones of positive closed -currents , which satisfy a functional relation

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and some regularity condition (PB, PC). Under appropriate assumptions on dynamical degrees we introduce closed finite dimensional cones, not reduced to zero, of such currents. In particular, when the topological entropy of is positive, then for some , there is a positive closed -current which satisfies the relation

Moreover, every quasi-p.s.h. function is integrable with respect to the trace measure of . When the dynamical degrees of are all distinct, we construct an invariant measure as an intersection of closed currents. We show that this measure is mixing and gives no mass to pluripolar sets and to sets of small Hausdorff dimension.

     

Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients

Let be a quiver without oriented cycles. For a dimension vector let be the set of representations of with dimension vector . The group acts on . In this paper we show that the ring of semi-invariants is spanned by special semi-invariants associated to representations of . From this we show that the set of weights appearing in is saturated. In the case of triple flag quiver this reduces to the results of Knutson and Tao on the saturation of the set of triples of partitions for which the Littlewood-Richardson coefficient is nonzero.

     

The size of the singular set in mean curvature flow of mean-convex sets

We prove that when a compact mean-convex subset of (or of an -dimensional riemannian manifold) moves by mean-curvature, the spacetime singular set has parabolic hausdorff dimension at most . Examples show that this is optimal. We also show that, as , the surface converges to a compact stable minimal hypersurface whose singular set has dimension at most . If , the convergence is everywhere smooth and hence after some time , the moving surface has no singularities

     

Intermediate subfactors with no extra structure

If are type II factors with and we show that restrictions on the standard invariants of the elementary inclusions , , and imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto and do not commute, then is or . In the former case is the fixed point algebra for an outer action of on and the angle is , and in the latter case the angle is and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.

     

Finite group extensions and the Atiyah conjecture

The Atiyah conjecture for a discrete group states that the -Betti numbers of a finite CW-complex with fundamental group are integers if is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of .

Here we establish conditions under which the Atiyah conjecture for a torsion-free group implies the Atiyah conjecture for every finite extension of . The most important requirement is that is isomorphic to the cohomology of the -adic completion of for every prime number . An additional assumption is necessary e.g. that the quotients of the lower central series or of the derived series are torsion-free.

We prove that these conditions are fulfilled for a certain class of groups, which contains in particular Artin's pure braid groups (and more generally fundamental groups of fiber-type arrangements), free groups, fundamental groups of orientable compact surfaces, certain knot and link groups, certain primitive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture, provided the group does.

As a consequence, if such an extension is torsion-free, then the group ring contains no non-trivial zero divisors, i.e. fulfills the zero-divisor conjecture.

In the course of the proof we prove that if these extensions are torsion-free, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin's full braid group, therefore answering question B6 on http://www.grouptheory.info.

Our methods also apply to the Baum-Connes conjecture. This is discussed by Thomas Schick in his preprint ``Finite group extensions and the Baum-Connes conjecture', where for example the Baum-Connes conjecture is proved for the full braid groups.

     

Plurisubharmonic domination

For a large class of separable Banach spaces we prove the following. Given a pseudoconvex open and that is locally bounded above, there is a plurisubharmonic such that . We also discuss applications of this result.