Homological methods for hypergeometric families

We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems  arising from a integer matrix  and a parameter . To do so we introduce an Euler-Koszul functor for hypergeometric families over  , whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter is rank-jumping for if and only if lies in the Zariski closure of the set of -graded degrees  where the local cohomology of the semigroup ring supported at its maximal graded ideal  is nonzero. Consequently, has no rank-jumps over  if and only if is Cohen-Macaulay of dimension .

     

Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi

In 1978 De Giorgi formulated the following conjecture. Let be a solution of in all of such that and 0$"> in . Is it true that all level sets of are hyperplanes, at least if ? Equivalently, does depend only on one variable? When , this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for . The question, however, remains open for . The results for and 3 apply also to the equation for a large class of nonlinearities .

     

On a lattice problem of H. Steinhaus

It is shown that there is a subset of such that each isometric copy of (the lattice points in the plane) meets in exactly one point. This provides a positive answer to a problem of H. Steinhaus.

     

Représentations -adiques et normes universelles I. Le cas cristallin

Let be a crystalline -adic representation of the absolute Galois group of an finite unramified extension of , and let be a lattice of stable by . We prove the following result: Let be the maximal sub-representation of with Hodge-Tate weights strictly positive and . Then, the projective limit of is equal up to torsion to the projective limit of . So its rank over the Iwasawa algebra is .

     

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.

     

Duality of Hardy and BMO spaces associated with operators with heat kernel bounds

Let be the infinitesimal generator of an analytic semigroup on with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space by means of an area integral function associated with the operator . By using a variant of the maximal function associated with the semigroup , a space of functions of BMO type was defined by Duong and Yan and it generalizes the classical BMO space. In this paper, we show that if has a bounded holomorphic functional calculus on , then the dual space of is where is the adjoint operator of . We then obtain a characterization of the space in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces of BMO when is a second-order elliptic operator of divergence form and when is a Schrödinger operator, and study the inclusion between the classical BMO space and spaces associated with operators.

     

Integral motives and special values of zeta functions

For each field , we define a category of rationally decomposed mixed motives with -coefficients. When is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers to certain groups.

     

Conformal restriction: The chordal case

We characterize and describe all random subsets of a given simply connected planar domain (the upper half-plane , say) which satisfy the ``conformal restriction' property, i.e., connects two fixed boundary points ( and , say) and the law of conditioned to remain in a simply connected open subset of is identical to that of , where is a conformal map from onto with and . The construction of this family relies on the stochastic Loewner evolution processes with parameter and on their distortion under conformal maps. We show in particular that SLE is the only random simple curve satisfying conformal restriction and we relate it to the outer boundaries of planar Brownian motion and SLE.     

boundedness of discrete singular Radon transforms

We prove that if is a Calderón-Zygmund kernel and is a polynomial of degree with real coefficients, then the discrete singular Radon transform operator

extends to a bounded operator on , . This gives a positive answer to an earlier conjecture of E. M. Stein and S. Wainger.

     

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 .

     

Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of

We give the first examples of infinite sets of primes such that Hilbert's Tenth Problem over has a negative answer. In fact, we can take to be a density 1 set of primes. We show also that for some such there is a punctured elliptic curve over such that the topological closure of in has infinitely many connected components.

     

Topology of symplectomorphism groups of rational ruled surfaces

Let be either or the one point blow-up of . In both cases carries a family of symplectic forms , where -1$"> determines the cohomology class . This paper calculates the rational (co)homology of the group of symplectomorphisms of as well as the rational homotopy type of its classifying space . It turns out that each group contains a finite collection , of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups ``asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as . However, for each fixed there is essentially one nonvanishing product that gives rise to a ``jumping generator" in and to a single relation in the rational cohomology ring . An analog of this generator was also seen by Kronheimer in his study of families of symplectic forms on -manifolds using Seiberg-Witten theory. Our methods involve a close study of the space of -compatible almost complex structures on .

     

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.

     

The spectra of nonnegative integer matrices via formal power series

We characterize the possible nonzero spectra of primitive integer matrices (the integer case of Boyle and Handelman's Spectral Conjecture). Characterizations of nonzero spectra of nonnegative matrices over and follow from this result. For the proof of the main theorem we use polynomial matrices to reduce the problem of realizing a candidate spectrum to factoring the polynomial as a product where the 's are polynomials in satisfying some technical conditions and is a formal power series in . To obtain the factorization, we present a hierarchy of estimates on coefficients of power series of the form to ensure nonpositivity in nonzero degree terms.

     

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.

     

On the image of the -adic Abel-Jacobi map for a variety over the algebraic closure of a finite field

Let be a smooth projective variety of dimension at most 4 defined over the algebraic closure of a finite field of characteristic . It is shown that the Tate conjecture implies the surjectivity of the -adic Abel-Jacobi map, , for all and almost all . For a special class of threefolds the surjectivity of is proved without assuming any conjectures.

     

Presentations of finite simple groups: A quantitative approach

There is a constant such that all nonabelian finite simple groups of rank over , with the possible exception of the Ree groups , have presentations with at most generators and relations and total length at most . As a corollary, we deduce a conjecture of Holt: there is a constant such that for every finite simple group , every prime and every irreducible -module .

     

A solution to the L space problem

In this paper I will construct a non-separable hereditarily Lindelöf space (L space) without any additional axiomatic assumptions. The constructed space is a subspace of where is the unit circle. It is shown to have a number of properties which may be of additional interest. For instance it is shown that the closure in of any uncountable subset of contains a canonical copy of .

I will also show that there is a function such that if are uncountable and , then there are in and respectively with . Previously it was unknown whether such a function existed even if was replaced by . Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality .

The results all stem from the analysis of oscillations of coherent sequences of finite-to-one functions. I expect that the methods presented will have other applications as well.

     

Cayley groups

The classical Cayley map, , is a birational isomorphism between the special orthogonal group SO and its Lie algebra , which is SO-equivariant with respect to the conjugating and adjoint actions, respectively. We ask whether or not maps with these properties can be constructed for other algebraic groups. We show that the answer is usually ``no", with a few exceptions. In particular, we show that a Cayley map for the group SL exists if and only if , answering an old question of LUNA.

     

Orbit equivalence for Cantor minimal -systems

We show that every minimal, free action of the group on the Cantor set is orbit equivalent to an AF-relation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AF-relations, -actions and -actions.