Exotic smooth structures on small 4-manifolds

Let M be either or . We construct the first example of a simply-connected irreducible symplectic 4-manifold that is homeomorphic but not diffeomorphic to M. Dedicated to Ronald J. Stern on the occasion of his sixtieth birthday Mathematics Subject Classification (2000)  Primary 57R55; Secondary 57R17, 57M05     

Signed words and permutations, V; a sextuple distribution

We calculate the distribution of the sextuple statistic over the hyperoctahedral group B _n that involves the flag-excedance and flag-descent numbers “fexc” and “fdes,” the flag-major index “fmaj,” the positive and negative fixed point numbers “ ” and “ ” and the negative letter number “neg.” Several specializations are considered. In particular, the joint distribution for the pair is explicitly derived.      

Duality of chordal SLE

We derive some geometric properties of chordal SLE(κ;) processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE(κ;) process has the same distribution as the image of a chordal SLE(κ’;’) trace, where κ>4, κ’=16/κ, and the forces and ’ are suitably chosen. We find that for κ≥8, the boundary of a standard chordal SLE(κ) hull stopped on swallowing a fixed is the image of some SLE(16/κ;) trace started from x. Then we obtain a new proof of the fact that chordal SLE(κ) trace is not reversible for κ>8. We also prove that the reversal of SLE(4;) trace has the same distribution as the time-change of some SLE(4;’) trace for certain values of and ’.     

Realization of the mapping class group by homeomorphisms

In this paper, we show that the mapping class group of a closed surface can not be geometrically realized as a group of homeomorphisms of that surface. More precisely, let denote the standard projection of the group of homeomorphisms to the mapping class group of a closed surface M of genus g>5. We show that there is no homomorphism , such that is the identity. This answers a question by Thurston (see [11]). Mathematics Subject Classification (2000)  Primary 20H10, 37F30     

Criteria for the L p -dissipativity of systems of second order differential equations

Abstract  We give complete algebraic characterizations of the L^p-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form , where are m× m matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is L^p-dissipative if and only if

On the relation of Voevodsky’s algebraic cobordism to Quillen’s <Emphasis Type="Italic">K</Emphasis>-theory

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P¹-spectrum MGL of Voevodsky is considered as a commutative P¹-ring spectrum. Setting we regard the bigraded theory MGL ^p,q as just a graded theory. There is a unique ring morphism which sends the class [X]_MGL of a smooth projective k-variety X to the Euler characteristic of the structure sheaf . Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories

Hardy Spaces and bmo on Manifolds with Bounded Geometry

We develop the theory of the “local” Hardy space and John-Nirenberg space when M is a Riemannian manifold with bounded geometry, building on the classic work of Fefferman-Stein and subsequent material, particularly of Goldberg and Ionescu. Results include – duality, L ^p estimates on an appropriate variant of the sharp maximal function, and bmo-Sobolev spaces, and action of a natural class of pseudodifferential operators, including a natural class of functions of the Laplace operator, in a setting that unifies these results with results on L ^p -Sobolev spaces. We apply results on these topics to some interpolation theorems, motivated in part by the search for dispersive estimates for wave equations.      

From 1-homogeneous supremal functionals to difference quotients: relaxation and Γ-convergence

In this paper we consider positively 1-homogeneous supremal functionals of the type . We prove that the relaxation $\bar{F}$ is a difference quotient, that is where is a geodesic distance associated to F. Moreover we prove that the closure of the class of 1-homogeneous supremal functionals with respect to Γ-convergence is given exactly by the class of difference quotients associated to geodesic distances. This class strictly contains supremal functionals, as the class of geodesic distances strictly contains intrinsic distances. Mathematics Subject Classification (2000) 47J20, 58B20, 49J45     

Absolute graphs with prescribed endomorphism monoid

We consider endomorphism monoids of graphs. It is well-known that any monoid can be represented as the endomorphism monoid M of some graph Γ with countably many colors. We give a new proof of this theorem such that the isomorphism between the endomorphism monoid $\mathop{\rm End}\nolimits (\Gamma)We consider endomorphism monoids of graphs. It is well-known that any monoid can be represented as the endomorphism monoid M of some graph Γ with countably many colors. We give a new proof of this theorem such that the isomorphism between the endomorphism monoid and M is absolute, i.e. holds in any generic extension of the given universe of set theory. This is true if and only if |M|,|Γ| are smaller than the first Erdős cardinal (which is known to be strongly inaccessible). We will encode Shelah’s absolutely rigid family of trees (Isr. J. Math. 42(3), 177–226, 1982) into Γ. The main result will be used to construct fields with prescribed absolute endomorphism monoids, see G?bel and Pokutta (Shelah’s absolutely rigid trees and absolutely rigid fields, in preparation). This work was supported by the project No. I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research & Development and a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD).     

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

The G-Fredholm Property of the \bar{\partial} -Neumann Problem

Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G→M→X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if G acts by holomorphic transformations in M, then the Laplacian on M has the following properties: The kernel of □ restricted to the forms Λ ^p,q with q>0 is a closed, G-invariant subspace in L ²(M,Λ ^p,q ) of finite G-dimension. Secondly, we show that if q>0, then the image of □ contains a closed, G-invariant subspace of finite G-codimension in L ²(M,Λ ^p,q ). These two properties taken together amount to saying that □ is a G-Fredholm operator. It is a corollary of the first property mentioned that the reduced L ²-Dolbeault cohomology spaces of M are finite G-dimensional for q>0. The boundary Laplacian □ _b has similar properties.      

Affine Quotients of Supergroups

In this paper we consider sheaf quotients of affine superschemes by affine supergroups that act on them freely. The necessary and sufficient conditions for such quotients to be affine are given. If G is an affine supergroup and H is its normal supersubgroup, then we prove that a dur K-sheaf is again an affine supergroup. Additionally, if G is algebraic, then a K-sheaf is also an algebraic supergroup and it coincides with . In particular, any normal supersubgroup of an affine supergroup is faithfully exact. Supported by RFFI 07-01-00392 and by FAPESP (proc. 07/54834-9).     

Grothendieck’s comparison theorem and multivariable Fredholm theory

We use a variant of Grothendieck’s comparison theorem to show that, for a Fredholm tuple T ∈ L(X)ⁿ on a complex Banach space, there are isomorphisms . We conclude that a Fredholm tuple T ∈ L(X)ⁿ satisfies Bishop’s property (β) at z = 0 if and only if the vanishing conditions hold for . We apply these observations and results from commutative algebra to show that a graded tuple on a Hilbert space is Fredholm if and only if it satisfies Bishop’s property (β) at z = 0 and that, in this case, its cohomology groups can grow at most like k^p. Received: 14 January 2009     

The subexponentiality of products revisited

Following the work of Cline and Samorodnitsky (Stoch. Process. Their Appl. 49(1):75–98, 1994), we reexamine the subexponentiality of the product of two random variables, X and Y, which are independent and have distributions F and G, respectively. The main result is the following: If F belongs to the class [that is to say, F is subexponential and holds for some v>1] and G, with G(0–)=0 and G(0)<0, satisfies for each u>0, then the distribution of XY also belongs to the class .      

Schur–Weyl Reciprocity between the Quantum Superalgebra and the Iwahori–Hecke Algebra

In this paper, we establish Schur–Weyl reciprocity between the quantum general super Lie algebra and the Iwahori–Hecke algebra . We introduce the sign -permutation representation of on the tensor space of dimensional -graded -vector space . This action commutes with that of derived from the vector representation on . Those two subalgebras of satisfy Schur–Weyl reciprocity. As special cases, we obtain the super case (), and the quantum case (). Hence this result includes both the super case and the quantum case, and unifies those two important cases.Presented by A. Verschoren.     

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

On the Existence of Weak Solutions for a Degenerate and Singular Elliptic System in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

This paper deals with the existence of weak solutions to a class of degenerate and singular elliptic systems in ℝ ^N , N ≧2 of the form

Equivariant vector bundles on Drinfeld’s upper half space

Let be Drinfeld’s upper half space over a finite extension K of ℚ _p . We construct for every GL _d+1-equivariant vector bundle on ℙ ^d _K , a GL _d+1(K)-equivariant filtration by closed subspaces on the K-Fréchet . This gives rise by duality to a filtration by locally analytic GL _d+1(K)-representations on the strong dual . The graded pieces of this filtration are locally analytic induced representations from locally algebraic ones with respect to maximal parabolic subgroups. This paper generalizes the cases of the canonical bundle due to Schneider and Teitelbaum [ST1] and that of the structure sheaf by Pohlkamp [P].     

Representation theory of $\mathcal{W}$-algebras

We study the representation theory of the -algebra associated with a simple Lie algebra at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of -algebras. Mathematics Subject Classification (1991)  17B68, 81R10     

Geometry of the mapping class groups I: Boundary amenability

We construct a geometric model for the mapping class group of a non-exceptional oriented surface S of genus g with k punctures and use it to show that the action of on the compact metrizable Hausdorff space of complete geodesic laminations for S is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every subgroup of .