An algorithm for detecting abelian monopoles inSU 2-valued lattice gauge-Higgs systems

We present an algorithm which calculates the monopole number of anSU ₂-valued lattice gauge field, together with a lattice Higgs field, on a simplicial lattice of dimension ≧3. The calculation is gauge invariant. The expected value of the monopole density (for a fixed Higgs field) does not depend on the Higgs field. Partially supported by NSF grants DMS 8607168 and DMS 8907753 Partially supported by PSC-CUNY and by NSF grant DMS 8805485     

Several Complex Variables and the Distribution of Resonances in Potential Scattering

We study resonances associated to Schrödinger operators with compactly supported potentials on ℝ^d, d≥3, odd. We consider potentials depending holomorphically on a parameter For certain such families, for all z except those in a pluripolar set, the associated resonance–counting function has order of growth d.Partially supported by NSF grant DMS 0088922.     

Local Critical Perturbations of Unimodal Maps

We introduce a new complete metric in the space of unimodal C ²-maps of the interval, with two maps close if they are close in the C ²-metric and differ only on a small interval containing their critical points. We identify all structurally stable maps in the sense of this metric. They are maps for which either (1) the trajectory of the critical point is attracted to a topologically attracting (at least from one side) periodic orbit, but never falls into this orbit, or (2) the critical point is mapped by some iterate to the interior of an interval consisting entirely of periodic points of the same (minimal) period. We verify the generalized Fatou conjecture for and show that structurally stable maps form an open dense subset of . Partially supported by NSF grant DMS 0456748. Partially supported by NSF grant DMS 0456526.     

Using integrability to produce chaos: Billiards with positive entropy

A new open class of convex 2 dimensional planar billiards with positive Lyapunov exponent almost everywhere is constructed. We introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive Lyapunov exponents. We prove that under smallC ⁶ perturbations, focusing arcs remain focusing and thereby show that perturbations of the Bunimovich stadium billiard have positive Lyapunov exponents.Partially supported by NSF grant DMS 8806067     

Localization for random Schrödinger operators with correlated potentials

We prove localization at high disorder or low energy for lattice Schrödinger operators with random potentials whose values at different lattice sites are correlated over large distances. The class of admissible random potentials for our multiscale analysis includes potentials with a stationary Gaussian distribution whose covariance functionC(x,y) decays as |x–y|^–, where >0 can be arbitrarily small, and potentials whose probability distribution is a completely analytical Gibbs measure. The result for Gaussian potentials depends on a multivariable form of Nelson's best possible hypercontractive estimate.Partially supported by the NSF under grant PHY8515288Partially supported by the NSF under grant DMS8905627     

Caustics for inner and outer billiards

With a plane closed convex curve,T, we associate two area preserving twist maps: the (classical) inner billiard inT and the outer billiard in the exterior ofT. The invariant circles of these twist maps correspond to certain plane curves: the inner and the outer caustics ofT. We investigate how the shape ofT determines the possible location of caustics, establish the existence of open regions which are free of caustics, and estimate fro below the size of these regions in terms of the geometry ofT.Partially supported by NSF.Partially supported by NSF Grant DMS 9017995.     

Mass Under the Ricci Flow

In this paper, we study the change of the ADM mass of an ALE space along the Ricci flow. Thus we first show that the ALE property is preserved under the Ricci flow. Then, we show that the mass is invariant under the flow in dimension three (similar results hold in higher dimension with more assumptions). A consequence of this result is the following. Let (M, g) be an ALE manifold of dimension n = 3. If m(g) ≠ 0, then the Ricci flow starting at g can not have Euclidean space as its (uniform) limit. Partially supported by NSF and NSFC. The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20060003002.     

Invariants for smooth conjugacy of hyperbolic dynamical systems. IV

We show that if twoC transitive Anosov flows in a three-dimensional manifold are topologically conjugate and the Lyapunov exponents on corresponding periodic orbits agree, then the conjugating homeomorphism isC .Partially supported by NSF grant # DMS 85-04984     

On the spectral problem for anyons

We consider the spectral problem resulting from the Schrödinger equation for a quantum system ofn2 indistinguishable, spinless, hard-core particles on a domain in two dimensional Euclidian space. For particles obeying fractional statistics, and interacting via a repulsive hard core potential, we provide a rigorous framework for analysing the spectral problem with its multi-valued wave functions.Partially supported by the Mathematical Sciences Research Institute, Berkeley California, under NSF Grant # DMS 8505550Partially supported under NSF Grant no. DMR-9101542     

Absence of shocks in an initially dilute collisionless plasma

The Cauchy Problem for the relativistic Vlasov-Maxwell equations is studied in three space dimensions. It is assumed that the initial data satisfy the required constraints and have compact support. If in addition the data have sufficiently smallC ² norm, then a uniqueC ¹ solution to this system is shown to exist on all of spacetime.Research supported in part by NSF DMS 85-20662 and NSF DMS 84-20957     

Regularity Criteria for the Dissipative Quasi-Geostrophic Equations in Hölder Spaces

We study regularity criteria for weak solutions of the dissipative quasi-geostrophic equation (with dissipation (−Δ) ^γ/2, 0 < γ ≤ 1). We show in this paper that if , or with is a weak solution of the 2D quasi-geostrophic equation, then θ is a classical solution in . This result improves our previous result in [18]. Partially supported by a start-up funding from the Division of Applied Mathematics of Brown University and NSF grant number DMS 0800129. Partially supported by a start-up funding from the College of Natural Sciences of the University of Texas at Austin, NSF grant number DMS 0758247 and an Alfred P. Sloan Research Fellowship.     

Spin Gromov-Witten Invariants

We define and study r-spin Gromov-Witten invariants and r-spin quantum cohomology of a projective variety V, where r≥2 is an integer. The main element of the construction is the space of r-spin maps, the stable maps into a variety V from n-pointed algebraic curves of genus g with the additional data of an r-spin structure on the curve. We prove that is a Deligne-Mumford stack and use it to define the r-spin Gromov-Witten classes of V. We show that these classes yield a cohomological field theory (CohFT) which is isomorphic to the tensor product of the CohFT associated to the usual Gromov-Witten invariants of V and the r-spin CohFT. Restricting to genus zero, we obtain the notion of an r-spin quantum cohomology of V, whose Frobenius structure is isomorphic to the tensor product of the Frobenius manifolds corresponding to the quantum cohomology of V and the r^th Gelfand-Dickey hierarchy (or, equivalently, the A_r−1 singularity). We also prove a generalization of the descent property which, in particular, explains the appearance of the ψ classes in the definition of gravitational descendants.Research of the first author was partially supported by NSA grant number MDA904-99-1-0039Research of the second author was partially supported by NSF grant number DMS-9803427Research of the third author was partially supported by NSF grant DMS-0104397     

The computation of characteristic classes of lattice gauge fields

AGL(p,C)-valued lattice gauge fieldu on a simplicial complex determines a principalGL(p,C)-bundle if the plaquette products are sufficiently small with respect to the maximum distortion coefficient of the transporters. A representative cocyclec _q for theq ^th Chern class of can be computed on each 2q-simplex by takingc _q() to be the intersection number of a certain singular 2q-cubeM with a Schubert-type variety _q in the space of allp×p matrices. This reduces to the solution of polynomial equations with coefficients coming fromu and thus avoids numerical integration or cooling-type procedures. An application of this method is suggested for the computation of the topological charge of anSU(3)-valued lattice gauge field on a 4-complex.Partially supported by NSF grant DMS 8607168Partially supported by PSC-CUNY and by NSF grant DMS 8805485     

Recovery of singularities for formally determined inverse problems

In this paper we considered several formally determined problems in two dimensions. There are no global identifiability results for these problems. However, we can recover an important feature of these functions, namely their singularities. More precisely, we prove that one can determine the location and strength of singularities of anL compactly supported potential by knowing the associated scattering amplitude at a fixed energy. Also we prove that one can determine the location and strength of the singularities of the sound speed of a medium by making measurements just on the boundary of the medium.Partially supported by NSF grant DMS-9123742Partially supported by NSF grant DMS-9100178     

Potts Model on Infinite Graphs and the Limit of Chromatic Polynomials

 Given an infinite graph 𝔾 quasi-transitive and amenable with maximum degree Δ, we show that reduced ground state degeneracy per site W _r (𝔾, q) of the q-state antiferromagnetic Potts model at zero temperature on 𝔾 is analytic in the variable 1/q, whenever |2Δe ³ /q|<1. This result proves, in an even stronger formulation, a conjecture originally sketched in [12] and explicitly formulated in [16 and 19], based on which a sufficient condition for W _r (𝔾, q) to be analytic at 1/q=0 is that 𝔾 is a regular lattice. Received: 16 January 2002 / Accepted: 17 October 2002 Published online: 18 February 2003 RID="*" ID="*" Partially supported by CNPq (Brazil) RID="**" ID="**" Partially supported by CNR, G.N.F.M. (Italy) Communicated by H. Spohn     

Anomalies in the one-dimensional Anderson model at weak disorder

We show that at the special energiesE=2cosp/q, the invariant measure, the Lyapunov exponent, and the density of states can be extended to zero disorder as C functions in the disorder parameter. In particular, we obtain asymptotic series in the disorder parameter. This gives a rigorous proof of the existence of the anomalies originally discovered by Kappus and Wegner and studied by Derrida and Gardner and by Bovier and Klein.Partially supported by NSF grant DMS 87-02301     

Anderson localization for Bernoulli and other singular potentials

We prove exponential localization in the Anderson model under very weak assumptions on the potential distribution. In one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment. In particular this solves the longstanding problem of localization for Bernoulli potentials (i.e., potentials that take only two values). In dimensions greater than one we prove localization at high disorder for potentials with Hölder continuous distributions and for bounded potentials whose distribution is a convex combination of a Hölder continuous distribution with high disorder and an arbitrary distribution. These include potentials with singular distributions.We also show that for certain Bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component.Partially supported by NSF grant DMS 85-03695Partially supported by NSF grant DMS 83-01889Partially supported by G.N.F.M. C.N.R.     

Asymptotic neutrality of large ions

It is proved that a nucleus of chargeZ can bind at mostZ+O(Z ^a) electrons, witha=47/56.Partially supported by a NSF grant at Princeton UniversitySupported by a Sloan Foundation Dissertation Fellowship at Princeton University     

Kähler-Einstein metrics on complex surfaces withC 1>0

Various estimates of the lower bound of the holomorphic invariant (M), defined in [T], are given here by using branched coverings, potential estimates and Lelong numbers of positive,d-closed (1, 1) currents of certain type, etc. These estimates are then applied to produce Kähler-Einstein metrics on complex surfaces withC ₁>0, in particular, we prove that there are Kähler-Einstein structures withC ₁>0 on any manifold of differential type .Dedicated to Walter Thirring on his 60^th birthdayResearch supported in part by Alfred P. Sloan Fellowship for doctoral dissertationResearch supported in part by NSF grant # DMS 84-08447 and ONR contract # N-00014-85-K-0367     

The short distance behavior of (φ 4)3

We consider the ₃ ⁴ quantum field theory on a torus and study the short distance behavior. We reproduce the standard result that the singularities can be removed by a simple mass renormalization. For the resulting model we give anL _p bound on the short distance regularity of the correlation functions. To obtain these results we develop a systematic treatment of the generating functional for correlations using a renormalization group method incorporating background fields.Research supported by NSF Grant DMS 9102564Research supported by NSF Grant PHY9200278.Research supported by the Natural Sciences and Engineering Research Council of Canada.