The number of Reidemeister moves needed for unknotting

There is a positive constant such that for any diagram representing the unknot, there is a sequence of at most Reidemeister moves that will convert it to a trivial knot diagram, where is the number of crossings in . A similar result holds for elementary moves on a polygonal knot embedded in the 1-skeleton of the interior of a compact, orientable, triangulated 3-manifold . There is a positive constant such that for each , if consists of tetrahedra and is unknotted, then there is a sequence of at most elementary moves in which transforms to a triangle contained inside one tetrahedron of . We obtain explicit values for and .

     

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 .

     

Maximal properties of the normalized Cauchy transform

We study the normalized Cauchy transform in the unit disk. Our goal is to find an analog of the classical theorem by M. Riesz for the case of arbitrary weights.

Let be a positive finite measure on the unit circle of the complex plane and . Denote by and the Cauchy integrals of the measures and , respectively. The normalized Cauchy transform is defined as . We prove that is bounded as an operator in for but is unbounded (in general) for 2$">. The associated maximal non-tangential operator is bounded for and has weak type but is unbounded for 2$">.

     

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.

     

Order -adic field

Let be an algebraically closed field of characteristic the ring of Witt vectors and a complete discrete valuation ring dominating and containing a primitive -th root of unity. Let denote a uniformizing parameter for We study order automorphisms of the formal power series ring which are defined by a series

The set of fixed points of is denoted by and we suppose that they are -rational and that for Let be the minimal semi-stable model of the -adic open disc over in which specializes to distinct smooth points. We study the differential data that can be associated to each irreducible component of the special fibre of Using this data we show that if , then the fixed points are equidistant, and that there are only a finite number of conjugacy classes of order automorphisms in which are not the identity

     

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.

     

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 .

     

Quantum groups, the loop Grassmannian, and the Springer resolution

We establish equivalences of the following three triangulated categories:

Here, is the derived category of the principal block of finite-dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra ; the category is defined in terms of coherent sheaves on the cotangent bundle on the (finite-dimensional) flag manifold for ( semisimple group with Lie algebra ), and the category is the derived category of perverse sheaves on the Grassmannian associated with the loop group , where is the Langlands dual group, smooth along the Schubert stratification.

The equivalence between and is an ``enhancement' of the known expression (due to Ginzburg and Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between and can be viewed as a ``categorification' of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a -adic reductive group, while the other is in terms of equivariant -theory of a complex (Steinberg) variety for the dual group.

The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following the Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on the representation theory of Kac-Moody algebras). It also gives way to proving Humphreys' conjectures on tilting -modules, as will be explained in a separate paper.     

Minkowski's conjecture, well-rounded lattices and topological dimension

Let be the diagonal subgroup, and identify with the space of unimodular lattices in . In this paper we show that the closure of any bounded orbit

meets the set of well-rounded lattices. This assertion implies Minkowski's conjecture for and yields bounds for the density of algebraic integers in totally real sextic fields.

The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of and .

     

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.

     

On a correspondence between cuspidal representations of

Let be an irreducible, automorphic, self-dual, cuspidal representation of , where is the adele ring of a number field . Assume that has a pole at and that . Given a nontrivial character of , we construct a nontrivial space of genuine and globally -generic cusp forms on -the metaplectic cover of . is invariant under right translations, and it contains all irreducible, automorphic, cuspidal (genuine) and -generic representations of , which lift (``functorially, with respect to ") to . We also present a local counterpart. Let be an irreducible, self-dual, supercuspidal representation of , where is a -adic field. Assume that has a pole at . Given a nontrivial character of , we construct an irreducible, supercuspidal (genuine) -generic representation of , such that has a pole at , and we prove that is the unique representation of satisfying these properties.

     

Regularity of a free boundary in parabolic potential theory

We study the regularity of the free boundary in a Stefan-type problem

with no sign assumptions on and the time derivative .

     

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 .

     

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.

     

On the equation and application to control of phases

The main result is the following. Let be a bounded Lipschitz domain in , . Then for every with , there exists a solution of the equation div in , satisfying in addition on and the estimate

where depends only on . However one cannot choose depending linearly on .

Our proof is constructive, but nonlinear--which is quite surprising for such an elementary linear PDE. When there is a simpler proof by duality--hence nonconstructive.

     

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.

(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .

(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.

     

Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case

We establish global wellposedness and scattering for the -critical defocusing NLS in 3D

assuming radial data , . In particular, it proves global existence of classical solutions in the radial case. The same result is obtained in 4D for the equation

     

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 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.