Derived Hilbert schemes

We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme with given Hilbert polynomial . This is a dg-manifold (smooth dg-scheme) which carries a natural family of commutative (up to homotopy) dg-algebras, which over the usual Hilbert scheme is given by truncations of the homogeneous coordinate rings of subschemes in . In particular, differs from , the derived Quot scheme constructed in our previous paper, which carries only a family of -modules over the coordinate algebra of .

As an application, we construct the derived version of the moduli stack of stable maps of algebraic curves to a given projective variety , thus realizing the original suggestion of M. Kontsevich.

     

Random polynomials having few or no real zeros

Consider a polynomial of large degree whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly real zeros with probability as through integers of the same parity as the fixed integer . In particular, the probability that a random polynomial of large even degree has no real zeros is . The finite, positive constant is characterized via the centered, stationary Gaussian process of correlation function . The value of depends neither on nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability one may specify also the approximate locations of the zeros on the real line. The constant is replaced by in case the i.i.d. coefficients have a nonzero mean.

     

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point the local real-analytic CR automorphisms of fixing can be parametrized real-analytically by their jets at . As a direct application, we derive a Lie group structure for the topological group . Furthermore, we also show that the order of the jet space in which the group embeds can be chosen to depend upper-semicontinuously on . As a first consequence, it follows that given any compact real-analytic hypersurface in , there exists an integer depending only on such that for every point germs at of CR diffeomorphisms mapping into another real-analytic hypersurface in are uniquely determined by their -jet at that point. Another consequence is the following boundary version of H. Cartan's uniqueness theorem: given any bounded domain with smooth real-analytic boundary, there exists an integer depending only on such that if is a proper holomorphic mapping extending smoothly up to near some point with the same -jet at with that of the identity mapping, then necessarily .

Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.

     

Locally analytic distributions and

In this paper we study continuous representations of locally -analytic groups in locally convex -vector spaces, where is a finite extension of and is a spherically complete nonarchimedean extension field of . The class of such representations includes both the smooth representations of Langlands theory and the finite dimensional algebraic representations of , along with interesting new objects such as the action of on global sections of equivariant vector bundles on -adic symmetric spaces. We introduce a restricted category of such representations that we call ``strongly admissible' and we show that, when is compact, our category is anti-equivalent to a subcategory of the category of modules over the locally analytic distribution algebra of . As an application we prove the topological irreducibility of generic members of the -adic principal series for . Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous -valued representations of locally -analytic groups.

     

Ideal membership in polynomial rings over the integers

We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials , where is an -tuple of indeterminates, are there such that ? We show that the degree of the polynomials can be bounded by where is the maximum total degree and the maximum height of the coefficients of . Some related questions, primarily concerning linear equations in , where is the ring of integers of a number field, are also treated.

     

Curvature and injectivity radius estimates for Einstein 4-manifolds

Let denote an Einstein -manifold with Einstein constant, , normalized to satisfy . For , a metric ball, we prove a uniform estimate for the pointwise norm of the curvature tensor on , under the assumption that the -norm of the curvature on is less than a small positive constant, which is independent of , and which in particular, does not depend on a lower bound on the volume of . In case , we prove a lower injectivity radius bound analogous to that which occurs in the theorem of Margulis, for compact manifolds with negative sectional curvature, . These estimates provide key tools in the study of singularity formation for -dimensional Einstein metrics. As one application among others, we give a natural compactification of the moduli space of Einstein metrics with negative Einstein constant on a given .

     

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.     

Analytic continuation of overconvergent eigenforms

Let be an overconvergent -adic eigenform of level , , with non-zero -eigenvalue. We show how may be analytically continued to a subset of containing, for example, all the supersingular locus. Using these results we extend the main theorem of our earlier work with R. Taylor to many ramified cases.

     

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.

     

On the geometric Langlands conjecture

Let be a smooth, complete, geometrically connected curve over a field of characteristic . The geometric Langlands conjecture states that to each irreducible rank local system on one can attach a perverse sheaf on the moduli stack of rank bundles on (irreducible on each connected component), which is a Hecke eigensheaf with respect to . In this paper we derive the geometric Langlands conjecture from a certain vanishing conjecture. Furthermore, using recent results of Lafforgue, we prove this vanishing conjecture, and hence the geometric Langlands conjecture, in the case when the ground field is finite.

     

The failure of rational dilation on a triply connected domain

For a bounded triply connected domain with boundary consisting of disjoint analytic curves there exists an operator on a complex Hilbert space so that the closure of is a spectral set for , but does not dilate to a normal operator with spectrum in , the boundary of . There is considerable overlap with the construction of an example on such a domain recently obtained by Agler, Harland and Raphael using numerical computations and work of Agler and Harland.

     

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 .

     

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.

     

The threshold for random -SAT is

Let be a random -CNF formula formed by selecting uniformly and independently out of all possible -clauses on variables. It is well known that if , then is unsatisfiable with probability that tends to 1 as . We prove that if , where , then is satisfiable with probability that tends to 1 as .

Our technique, in fact, yields an explicit lower bound for the random -SAT threshold for every . For our bounds improve all previously known such bounds.

     

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.     

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.

     

Homoclinic points of algebraic -actions

Let be an action of by continuous automorphisms of a compact abelian group . A point in is called homoclinic for if as . We study the set of homoclinic points for , which is a subgroup of . If is expansive, then is at most countable. Our main results are that if is expansive, then (1) is nontrivial if and only if has positive entropy and (2) is nontrivial and dense in if and only if has completely positive entropy. In many important cases is generated by a fundamental homoclinic point which can be computed explicitly using Fourier analysis. Homoclinic points for expansive actions must decay to zero exponentially fast, and we use this to establish strong specification properties for such actions. This provides an extensive class of examples of -actions to which Ruelle's thermodynamic formalism applies. The paper concludes with a series of examples which highlight the crucial role of expansiveness in our main results.

     

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 .

     

Singularities of pairs via jet schemes

Let be a smooth variety and a closed subscheme. We use motivic integration on the space of arcs of to characterize the fact that is log canonical or log terminal using the dimension of the jet schemes of . This gives a formula for the log canonical threshold of , which we use to prove a result of Demailly and Kollár on the semicontinuity of log canonical thresholds.