Ann-dimensional Borg-Levinson theorem

We show that the potentialq is uniquely determined by the spectrum, and boundary values of the normal derivatives of the eigenfunctions of the Schrödinger operator –+q with Dirichlet boundary conditions on a bounded domain in ⁿ . This and related results can be viewed as a direct generalization of the theorem in the title, which states that the spectrum and the norming constants determine the potential in the one dimensional case.Supported by NSF grant DMS-8602033Supported by NSF grant DMS-8600797Supported by NSF grant DMS-8601118 and an Alfred P. Sloan Research Fellowship     

Random walk in random environment: A counterexample?

We describe a family of random walks in random environments which have exponentially decaying correlations, nearest neighbor transition probabilities which are bounded away from 0, and yet are subdiffusive in any dimensiond<.This author partially supported by NSF grant DMS 83-1080This author partially supported by NSF grant DMS-85-05020 and the Army Research Office through the Mathematical Sciences Institute at Cornell University     

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     

A Particular Bit of Universality: Scaling Limits of Some Dependent Percolation Models

We study families of dependent site percolation models on the triangular lattice and hexagonal lattice that arise by applying certain cellular automata to independent percolation configurations. We analyze the scaling limit of such models and show that the distance between macroscopic portions of cluster boundaries of any two percolation models within one of our families goes to zero almost surely in the scaling limit. It follows that each of these cellular automaton generated dependent percolation models has the same scaling limit (in the sense of Aizenman-Burchard [3]) as independent site percolation on .The work was conducted while this author was at Department of Physics, New York University, New York, NY 10003, USA. Research partially supported by the U.S. NSF under grants DMS-98-02310 and DMS-01-02587.Research partially supported by the U.S. NSF under grants DMS-98-03267 and DMS-01-04278.Research partially supported by FAPERJ grant E-26/151.905/2000 and CNPq.     

Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

Solutions of the Bethe ansatz equations associated to the XXX model of a simple Lie algebra come in families called the populations. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra The proof is based on the correspondence between the solutions of the Bethe ansatz equations and special difference operators which we call the discrete Miura opers. The notion of a discrete Miura oper is one of the main results of the paper.For a discrete Miura oper D, associated to a point of a population, we show that all solutions of the difference equation DY=0 are rational functions, and the solutions can be written explicitly in terms of points composing the population.Supported in part by NSF grant DMS-0140460Supported in part by NSF grant DMS-0244579     

A note on the Ising model in high dimensions

We consider thed-dimensional Ising model with a nearest neighbor ferromagnetic interactionJ(d)=1/4d. We show that asd the+phase (and the — phase) approaches a product measure with density given by the mean field approximation. In particular the spontaneous magnetization converges to its mean field value. A similar result holds for the unique Gibbs measure of the system subject to an external fieldh0.Part of this work was done while this author was visiting Rutgers University, supported by NSF grant DMR-86-12369 and Princeton University, support by NSF grant PHY-85-15288-A01Partially supported by a NSF grant to Cornell UniversityPartially supported by NSF grant DMR 86-12369Supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell and by a NSF grant to Cornell University. This author was supported by the NSF grant DMR-86-12369 while visiting Rutgers University (when this work was started). On leave from São Paulo University     

Tail estimates for one-dimensional random walk in random environment

Suppose that the integers are assigned i.i.d. random variables { _x } (taking values in the unit interval), which serve as an environment. This environment defines a random walk {X _k } (called a RWRE) which, when atx, moves one step to the right with probability _x , and one step to the left with probability 1- _x . Solomon (1975) determined the almost-sure asymptotic speed (=rate of escape) of a RWRE. For certain environment distributions where the drifts 2 _x -1 can take both positive and negative values, we show that the chance of the RWRE deviating below this speed has a polynomial rate of decay, and determine the exponent in this power law; for environments which allow only positive and zero drifts, we show that these large-deviation probabilities decay like exp(–Cn ^1/3). This differs sharply from the rates derived by Greven and den-Hollander (1994) for large deviation probabilities conditioned on the environment. As a by product we also provide precise tail and moment estimates for the total population size in a Branching Process with Random Environment.Partially supported by NSF DMS-9209712 and DMS-9403553 grants, by a US-ISRAEL BSF grant and by the S. and N. Grand research fund.Research partially supported by NSF grant # DMS-9404391 and a Junior Faculty Fellowship from the Regents of the University of California.Partially supported by NSF grant # DMS-9302709, by a US-Israel BSF grant and by the fund for promotion of research at the Technion.     

Chern-Simons theory with finite gauge group

We construct in detail a 2+1 dimensional gauge field theory with finite gauge group. In this case the path integral reduces to a finite sum, so there are no analytic problems with the quantization. The theory was originally introduced by Dijkgraaf and Witten without details. The point of working it out carefully is to focus on the algebraic structure, and particularly the construction of quantum Hilbert spaces on closed surfaces by cutting and pasting. This includes the Verlinde formula. The careful development may serve as a model for dealing with similar issues in more complicated cases.The first author is supported by NSF grant DMS-8805684, an Alfred P. Sloan Research Fellowship, a Presidential Young Investigators award, and by the O'Donnell Foundation. The second author is supported by NSF grant DMS-9207973     

A Genus-3 Topological Recursion Relation

In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A by-product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1-pointed genus-3 stable curves. Research of the first author was partially supported by NSF grant DMS-0204824 Research of the second author was partially supported by NSF grant DMS-0505835     

Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

Let H be a one-dimensional discrete Schrödinger operator. We prove that if _ess(H)[–2,2], then H–H₀ is compact and _ess(H)=[–2,2]. We also prove that if has at least one bound state, then the same is true for H₀+V. Further, if has infinitely many bound states, then so does H₀+V. Consequences include the fact that for decaying potential V with , H₀+V has infinitely many bound states; the signs of V are irrelevant. Higher-dimensional analogues are also discussed. Supported in part by NSF grant DMS-0227289On leave from Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801-2975, USASupported in part by NSF grant DMS-0140592     

The large time stability of sound waves

We demonstrate the existence of solutions to the full 3×3 system of compressible Euler equations in one space dimension, up to an arbitrary timeT>0, in the case when the initial data has arbitrarily large total variation, and sufficiently small supnorm. The result applies to periodic solutions of the Euler equations, a nonlinear model for sound wave propagation in gas dynamics. Our analysis establishes a growth rate for the total variation that depends on a new length scaled that we identify in the problem. This length scale plays no role in 2×2 systems, (or any system possessing a full set of Riemann coordinates), nor in the small total variation problem forn×n systems, the cases originally addressed by Glimm in 1965. Recent work by a number of authors has demonstrated that when the total variation is sufficiently large, solutions of 3×3 systems of conservation laws can in general blow up in finite time, (independent of the supnorm), due to amplifying instabilities created by the non-trivial Lie algebra of the vector fields that define the elementary waves. For the large total variation problem, there is an interaction between large scale effects that amplify and small scale effects that are stable, and we show that the length scale on which this interaction occurs isd. In the limitd, we recover Glimm's theorem, and we observe that there exist linearly degenerate systems within the class considered for which the growth rate we obtain is sharp.Supported in part by NSF Applied Mathematics grant numbers DMS-92-06631, DMS-95000694, in part by ONR, US Navy grant number N00014-94-1-0691, A Guggenheim fellowship, and by the Institute of Theoretical Dynamics, UC-Davis.Partially supported by DOE grant number DE-FG02-88ER25053 while at the Courant Institute, and by NSF grant number DMS-9201581 and DOE grant number DE-FG02-90ER25084.     

Induced modules for vertex operator algebras

For a vertex operator algebraV and a vertex operator subalgebraV which is invariant under an automorphismg ofV of finite order, we introduce ag-twisted induction functor from the category ofg-twistedV-modules to the category ofg-twistedV-modules. This functor satisfies the Frobenius reciprocity and transitivity. The results are illustrated withV being theg-invariants in simpleV orV beingg-rational.The first author was supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.The second author was supported by NSF grant DMS-9401389.     

Local Borel summability of Euclidean 311-1311-1311-1a simple proof via differential flow equations

It is shown how the differential flow equation (or, equivalently, the continous renormalization group) method can be employed to give an astonishingly easy proof of the local Borel summability of the renormalized perturbative Euclidean massive ₄ ⁴ .Supported by NSF grant # DMS-9100383     

Renormalization Group and the Ginzburg-Landau equation

We use Renormalization Group methods to prove detailed long time asymptotics for the solutions of the Ginzburg-Landau equations with initial data approaching, asx±, different spiraling stationary solutions. A universal pattern is formed, depending only on this asymptotics at spatial infinity.Supported by NSF grant DMS-8903041 and by EEC Grant SCI-CT91-0695TSTS     

Completely integrable gradient flows

In this paper we exhibit the Toda lattice equations in a double bracket form which shows they are gradient flow equations (on their isospectral set) on an adjoint orbit of a compact Lie group. Representations for the flows are given and a convexity result associated with a momentum map is proved. Some general properties of the double bracket equations are demonstrated, including a discussion of their invariant subspaces, and their function as a Lie algebraic sorter.Supported in part by NSF Grant DMS-90-02136, NSF PYI Grant DMS-9157556, and a Seed Grant from Ohio State UniversitySupported in part by AFOSR grant AFOSR-96-0197, by U.S. Army Research Office grant DAAL03-86-K-0171 and by NSF grant CDR-85-00108Supported in part by NSF Grant DMS-8922699     

Quantization ofSL(2,R) Chern-Simons theory

In this paper we consider the bosonic sector of the electroweak theory. It has been shown in the work of Ambjorn and Olesen that when the Higgs mass equals to the mass of theZ boson, the model in two dimensions subject to the 't Hooft periodic boundary condition may be reduced to a Bogomol'nyi system and that the solutions of the system are vortices in a dual superconductor. We shall prove using a constrained variational reformulation of the problem the existence of such vortices. Our conditions for the existence of solutions are necessary and sufficient when the vortex numberN=1,2.Research supported in part by NSF grant DMS-88-02858 and DOE grant DE-FG02-86ER250125     

On the Largest Eigenvalue of a Random Subgraph of the Hypercube

Let G be a random subgraph of the n-cube where each edge appears randomly and independently with probability p. We prove that the largest eigenvalue of the adjacency matrix of G is almost surely where (G) is the maximum degree of G and the o(1) term tends to zero as max(^1/2(G),np) tends to infinity.Research was supported in part by the NSF grant DMS-0103948.Research was supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey.     

A low temperature expansion for classicalN-vector models. I. A renormalization group flow

A class of low temperature lattice classical spin models with a symmetry groupO(N) is considered, including the classical Heisenberg model. In this paper a renormalization group approach in a small field approximation is formulated and studied, with a goal to prove the so-called spin wave picture displaying massless behavior of the models.The work has been partially supported by the NSF Grant DMS-9102639     

Projective embeddings of complex supermanifolds

We generalize the Kodaira Embedding Theorem and Chow's Theorem to the context of families of complex supermanifolds. In particular, we show that every family of super Riemann surfaces is a family of projective superalgebraic varieties.Research supported in part by NSF grant DMS-8704401Research supported in part by NSF grant DMS-4253943Research also supported in part by NSF grant DMS-4253943