Embedding obstructions and -complexes

The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial -complex into for , and it was recently shown to be incomplete for . We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of -complexes in .

     

Contact structures, equivariant spin bordism, and periodic fundamental groups

Using contact surgery and equivariant bordism theory, we prove the existence of contact structures on all 5-dimensional spin manifolds with certain finite fundamental groups. Received September 13, 1999 / Revised version September 13, 2000 / Published online April 12, 2001     

Blowup for

In this note we consider the global regularity of smooth solutions to the vector-valued Cauchy problem

We show that if , the gradient-blowup phenomenon occurs in finite time for suitably chosen vanishing at infinity. We also present a simple example of the -blowup solutions for for any 0$">, if .

     

Oort's conjecture for

We prove the conjecture of Oort that a compact subvariety of the moduli space of principally polarized Abelian varieties of genus has codimension strictly greater than , in characteristic zero, for .

     

Restriction of stable bundles in characteristic

Let be an algebraically closed field of characteristic . Let be a nonsingular projective variety defined over and an ample line bundle on . We shall prove that there exists an explicit number such that if is a -stable vector bundle of rank at most three, then the restriction is -stable for all and all smooth irreducible divisors . This result has implications to the geometry of the moduli space of -stable bundles on a surface or a projective space.

     

Spectral synthesis for

Let be a locally compact group, the Fourier algebra of and the von Neumann algebra generated by the left regular representation of . We introduce the notion of -spectral set and -Ditkin set when is an -invariant linear subspace of , thus providing a unified approach to both spectral and Ditkin sets and their local variants. Among other things, we prove results on unions of -spectral sets and -Ditkin sets, and an injection theorem for -spectral sets.

     

Crystal bases for

We construct crystal bases for certain infinite dimensional representations of the -deformation of the Lie superalgebra .

     

and applications

We study an analogue of the classical theory of weights in without assuming that the underlying measure is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderón-Zygmund operators. We also consider commutators of those Calderón- Zygmund operators with bounded mean oscillation functions (), extending the main result from R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. Finally, we study self-improving properties of Poincaré-B.M.O. type inequalities within this context; more precisely, we show that if is a locally integrable function satisfying for all cubes , then it is possible to deduce a higher integrability result for , assuming a certain simple geometric condition on the functional .     

Brauer induction and equivariant stable homotopy

Brauer's induction theorem, published in 1951, asserts that every element of the complex representation ring R(G) of a finite group G is a linear combination of classes induced from 1-dimensional representations of subgroups of G. In 1987, Snaith formulated an explicit version of the induction theorem. Using the methods of equivariant fibrewise stable homotopy theory, specifically fixed-point theory, this note clarifies the relation between the explicit Brauer induction theorem due to Snaith, Boltje and Symonds and a topological splitting theorem established by Segal in 1973.     

The rank stable topology of instantons on

Let be the moduli space of based (anti-self-dual) instantons on of charge and rank . There is a natural inclusion . We show that the direct limit space is homotopy equivalent to . Let be a line in the complex projective plane and let be the blow-up at a point away from . can be alternatively described as the moduli space of rank holomorphic bundles on with and and with a fixed holomorphic trivialization on .

     

Real hypersurfaces of with non-negative Ricci curvature

We prove the non-existence of Levi flat compact real hypersurfaces without boundary in , with non-negative totally real Ricci curvature.

     

Explicit primality criteria for

Deterministic polynomial time primality criteria for have been known since the work of Lucas in 1876-1878. Little is known, however, about the existence of deterministic polynomial time primality tests for numbers of the more general form , where is any fixed prime. When (p-1)/2$"> we show that it is always possible to produce a Lucas-like deterministic test for the primality of which requires that only modular multiplications be performed modulo , as long as we can find a prime of the form such that is not divisible by . We also show that for all with such a can be found very readily, and that the most difficult case in which to find a appears, somewhat surprisingly, to be that for . Some explanation is provided as to why this case is so difficult.

     

On peak-interpolation manifolds for

Let be a bounded, weakly convex domain in , , having real-analytic boundary. is the algebra of all functions holomorphic in and continuous up to the boundary. A submanifold is said to be complex-tangential if lies in the maximal complex subspace of for each . We show that for real-analytic submanifolds , if is complex-tangential, then every compact subset of is a peak-interpolation set for .

     

On and discriminants

We modify the -conjecture for number fields in order to make the support (like the height) well-behaved under field extensions. We show further that the exponent 1$"> of the absolute value of the discriminant cannot be replaced by , and even that an arbitrarily large power of must be present.

     

Mirror symmetry and

We show that counting functions of covers of are equal to sums of integrals associated to certain `Feynman' graphs. This is an analogue of the mirror symmetry for elliptic curves.

     

The standard double bubble is the unique stable double bubble in

We prove that the only equilibrium double bubble in which is stable for fixed areas is the standard double bubble. This uniqueness result also holds for small stable double bubbles in surfaces, where it is new even for perimeter-minimizing double bubbles.

     

Generation theorems for Hille-Yosida operators

This paper introduces the concept of Hille-Yosida operators and studies several generation theorems. We show that if a once-integrated semigroup satisfies for all 0 a. e.$">, then is locally bounded on and exponentially bounded. In addition, some other interesting results are presented.

     

A computational approach for solving

We present a computational approach for finding all integral solutions of the equation for even values of . By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for assuming the Generalized Riemann Hypothesis, and for unconditionally.

     

A counterexample for approximable functions

Let be the unit disk. We show that for some relatively closed set there is a function that can be uniformly approximated on by functions of , but such that cannot be written as , with and uniformly continuous on . This answers a question of Stray.