1.   The Laplacian on Spaces with Cone-Like Singularities, MIT Thesis, 1990.

This work focusses on problems concerning metric degeneration.  More precisely, let (M,g) be a Riemannian manifold, H an embedded hypersurface. By deforming the metric g in a neighborhood of H, we introduce a one parameter family of metrics on M which shrink H to a point. We develop the necessary machinery to analyze the corresponding change in the Hodge cohomology.

2.  On the functional logdet and related flows on the space of closed embedded curves on S² - (with D. Burghelea, L. Friedlander, and T. Kappeler),  Jour. Func. Anal. (1994) vol 120, pp 440-466.

Fix a metric on the two sphere. Given an oriented simple closed curve on the sphere, the curve partitions the sphere into two topological disks (the orientation allows for a labelling of each disk). The metric on the sphere induces Dirichlet problems on each disk. The difference of the logarithms of the zeta regularized determinant for the Dirichlet problem on each disk defines a map from the space of all oriented simple closed curves on the sphere to the real numbers. We prove that this map is a Morse-Bott function. We show that the map defines a gradient-like flow which deforms a given curve to a conformal circle.

3.  Analytic and Reidemeister torsion for representations in finite type Hilbert modules - (with D. Burghelea, L. Friedlander, T. Kappeler), GAFA (1996) vol 6, pp 751-859.

In this paper we considers unitary representations of the fundamental group on a Hilbert A-module, where A is a finite von Neumann algebra, and  we prove a corresponding L²version of the Cheeger-Müller theorem.  We prove the theorem for manifolds which are determinant class.  The analysis is complicated by the fact that, in general, zero is  in the spectrum of the Laplacian and hence the L² zeta function is not a priori well defined.

4.  The extended generalized lambda distribution system for fitting distributions to data: history, completion (theory and tables) and applications - the final word on moment fits  (with Z. Karian and E. Dudewics), Comm. Stat.: Simulation and Computation, (1996) vol 25, pp 611-642.

We characterize properties of the extended generalized lambda distribution.

5.  Brownian functionals on hypersurfaces in Rⁿ  (with K. Kinateder), Proc. A.M.S. (1997) vol 125, pp1815-1822.

Let $X_t$ be Brownian motion in Euclidean space, let $P_x$ be the measure charging Brownian paths beginning at $x$ and let $E_x$ denote the corresponding expectation.  Given a smoothly bounded domain with compact closure, $D,$in Euclidean space, let $\tau$ be the first exit time of $X_t$ from $D.$   Define a function, $A: \{D: D smoothly bounded domains with compact closure\} \to R$ according the rule $A(D) = \frac{1}{volume(D)} \int_D E_x[\tau] x$ where $dx$ is Lebesgue measure.  Using the natural Frechet manifold topology on the set of all such domains, we characterize the critical points of the smooth map $A$ obtained by restricting to domains of a fixed volume.    The characterization is given in terms of an overdetermined boundary value problem first studied by Serrin.  These boundary value problems have solutions if and only if the underlying domain is a ball of the appropriate volume.

6.  Hypersurfaces in Rⁿ  and the variance of exit times for Brownian motion (with K. Kinateder) Proc. A.M.S. (1997) vol 125, pp 2453-2462.

We extend the analysis of paper 5 above and study the variance of the exit time of Brownian motion from smoothly bounded domains in Euclidean space. Once again, for fixed volume critical are characterized by an overdetermined boundary value problem.

7.  Symmetry problems arising in probability (with S. Fromm) Proc. A.M.S. (1997) vol 125, pp 3293-3298.

As in papers 5 and 6 above, we study the exit time of Brownian motion from smoothly bounded Euclidean domains. Fixing volume, critical points for the first moment of the exit time and the variance of the exit time are characterized by overdetermined boundary value problems. These problems were first studied by Serrin. We prove that the overdetermined problems have a solution if and only if the underlying domain is a ball of the appropriate volume.
 

8.  Exit time moments, boundary value problems, and the geometry of domains in Euclidean space  (with K. Kinateder, D. Miller) Prob. Th. and Rel. (1998) vol 111, pp 469-487.

We study problems inspired by papers 5,6, and 7 above. More precisely, given a diffusion on Euclidean space whose infinitesimal generator is of divergence form, we study exit time moments for smoothly bounded domains in Euclidean space. We establish variational formulae for such moments when domains are smoothly perturbed. We classify critical points under the assumption that the domains are of fixed volume via overdetermined boundary value problems and we prove that these problems have solutions if and only if the domains are balls of the appropriate volume.

9.  Variational principles for average exit time moments for diffusions in Euclidean space  (with K. Kinateder) Proc. A.M.S. (1999) vol 127, pp 2767--2772

Building on papers 5-8, we characterize exit time moments for diffusions admitting an infinitesimal generator of divergence form from smoothly bounded domains in Euclidean space using natural variational quotients.

10.  Isoperimetric conditions, Poisson problems and diffusions in Riemannian manifolds,  Potential Analysis (2002) vol 16, pp 115-138.

Given a Riemannian manifold $M$ with a diffusion admitting an infinitesimal generator of divergence form, we study exit time moments from smoothly bounded domains in $M$ of fixed volume, characterizing critical domains via ovedetermined boundary value problems. We develop the PDE required to prove that in the case of constant curvature, these boundary value problems have solutions if and only if the domain is a ball of the appropriate volume. We use rearrangement arguments to establish, for each moment, and analog of the Faber-Krahn theorem. Other comparison results follow along similar lines.

11.  An Ito formula for domain-valued processes driven by stochastic flows  (with K. Kinateder) Prob. Th. and Rel. (2002) vol 124, pp 73-99.

Given a stochastic flow on Euclidean space generated as the solution of a stochastic differential equation, we study the domain valued process obtained by allowing the flow to act on the space of smoothly bounded domains with compact closure in Euclidean space. We describe the evolution of geometric invariants associated to an initial domain in terms of the driving fields of the SDE by developing an Ito type formula.

12.   Diffusions on graphs, Poisson problems, and spectral geometry  (with R. Meyers) Trans. AMS (2002) vol 347, pp 5111-5136.

In this paper we study discrete potential theory and random walks on locally finite graphs. We establish many relationships between certain natural random walks on finite subgraphs and the spectrum of the discrete Laplacian. In particular, we prove that under certain natural assumptions on the underlying graphs, the collection of average exit time moments determines the heat content asy mptotics of the domain, as well as a "large" portion of the Dirichlet spectrum.

13.   Spectral geometry, the Polya conjecture and diffusions  Proceedings of the 34th Florida MAA meeting http://www/spcollege.edu/central/maa/proceedings/2001 (electronic).

This paper is a survey of many of the natural connections between spectral geometry, differential geometry, and exit time moments for domains in Riemannian manifolds. It is intended for a general mathematical audience.

14.   Redefining spinors in Lorentz violating QED   (with D. Colladay) Jour.  Math. Physics (2002) vol 43, pp 3554-3564.

In this paper we consider certain Lorentz violating extensions of quantum electrodynamics.  We analyze the corresponding spin fields and conserved currents.  We find that a number of paramters that apriori violate Lorentz invariance can be removed from the Lagrangian via an appropriate redefinition of spin field components.  We show that the conserved currents can be defined using a modified action of the complex extension of the Lorentz group, which in turn implies a natural correspondence between certain Lorentz-violating theories and standard QED.

15.   Dirichlet spectrum and heat content  (with R. Meyers) Jour. Funct. Analysis (2003) vol 200, pp 150-159.

In this paper we us Brownian motion to study the relationship between the Dirichlet spectrum of smoothly bounded domains in Riemannian manifolds and heat content.  We attach to each domain a sequence of invariants (the L¹ -norm of the exit time moments) and we prove that our invariants determine that part of the spectrum corresponding to eigenspaces which are not orthogonal to constant functions.  We prove that our invariants determine the heat content and that for generic smoothly bounded domains in Euclidean space, our invariants determine the Dirichlet spectrum.

16.   Isospectral polygons, planar graphs and heat content (with R. Meyers)  Proc. AMS (to appear).

Given a pair of planar isospectral, nonisometric  polygons constructed as a quotient of the plane by a finite group, we construct an associated pair of planar isospectral, nonisometric  weighted graphs.  To each such graph we associate a heat content.  We prove that the coefficients in the small time asymptotic expansion of the heat content distinguish our isospectral pairs.

17.   Diffusions, exit time moments, and Weierstrass theorems  (with V. de la Pena)  Proc. AMS (to appear)
Let X_t be a diffusion on the real line with infinitesimal generator L= 1/2(a(x)d/dx)² + b(x) d/dx where a(x) and b(x) are smooth, a(x) is positive and the ratio of a(x) and b(x) is a constant.  Given a compact interval on the line.  We prove a Weierstrass theorem for the exit time moments of X_t   and their corresponding naturally weighted  first derivatives.  We provide an algorithm which produces uniform approximations of arbitrary continuous functions  by exit time moments.

18.   Moment problems and the causal set approach to quantum gravity  (with A. Ash)  Jour. Math. Physics (2003) vol 44, pp 1666-1678.

In this paper we study a collection of discrete Markov chains related to the causal set approach to quantum gravity.  Such chains are characterized by properties of their transition probabilities: they satisfy a general covariance principle, a causality condition and a renormalizability condition.  The corresponding dynamics are completely determined by a sequence of nonnegative coupling constants.  For generic chains, we use techniques from the classical moment problem to give a complete description of an such sequence of coupling constants:  the coupling constants must be given by the moments of a probability distribution function on the positive real line.

19.   Recent Results in geometric analysis involving probability In: Recent Advances
in Applied Probability, ed. H.  Gzyl, Kluwer,  Dodrecht (to appear).

This paper is a survey of recent results in geometric analysis which explicitly involve both the geometry of (finite dimensional) Riemannian manifolds and probability.  We include developments in spectral geometry, the study of isoperimetric phenomena, comparison geometry, minimal varieties, harmonic functions and Hodge theory.

20.   Percolation, causet evolution and random partial orders (with A. Ash) (in preparation).

21.   Dynamic instability and optimal design  (in preparation).