Ph.D. Recipients and their Thesis Abstracts

Analysis

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

Random Walks on Solvable Groups

Abstract: We study a number of questions about random walks on solvable groups. For random walks on nilpotent groups, we determine which subgroups are recurrent, and for a random walk on the Heisenberg group, we study the number of distinct visited cosets at time n.

The bulk of the examples considered are about the behavior of random walks away from their starting point in groups of exponential growth. In particular, we examine the rate of escape of some inward biased random walks, as well as some unbiased walks that have an intermediate escape rate. We also compute asymptotics for transition probabilities on some semi-direct products, both at the origin and at more general points.

Using Spider Theory to Explore Parameter Spaces

Abstract: For a fixed integer $d\ge2$, consider the family of polynomials $P_{d,\lambda}(z)=\lambda(1+z/d)^d$, where $\lambda$ is a complex parameter. In this work, we study the location of parameters $\lambda$ for which $P_{d,\lambda}$ has an attracting cycle of a given length, multiplier, and combinatorial type.

Two main tools are used in determining an algorithm for finding these parameters: the well-established theories of external rays in the dynamical and parameter planes and Teichmüller theory. External rays are used to specify hyperbolic components in parameter space of the polynomials and study the combinatorics of the attracting cycle. A properly normalized space of univalent mappings is then employed to determine a linearizing neighborhood of the attracting cycle.

Since the image of a univalent mapping completely determines the mapping, we visualize these maps concretely on the Riemann sphere; with discs for feet and curves as legs connected at infinity, these maps conjure a picture of fat-footed spiders. Isotopy classes of these spiders form a Teichmüller space, and the tools found in Teichmüller theory prove useful in understanding the Spider Space. By defining a contracting holomorphic mapping on this spider space, we can iterate this mapping to a fixed point in Teichmüller space which in turn determines the parameter we seek.

Finally, we extend the results about these polynomial families to the exponential family $E_\lambda(z)=\lambda e^z$. Here, we are able to constructively prove the existence and location of hyperbolic components in the parameter space of $E_\lambda$.

Teichmüller Theory and Holomorphic Motions

Abstract: The subject of holomorphic motions over the open unit disc shows some interesting connections between classical complex analysis and problems on moduli. It has also found many applications in complex dynamics. In this dissertation, we study holomorphic motions over more general parameter spaces. For a closed subset E of the Riemann sphere we study its Teichmüller space, which is shown to be a universal parameter space for holomorphic motions of that set over a simply connected complex Banach manifold. This universal property has several important consequences; one of them is a generalization of the "Harmonic \lambda-lemma." This is the best general theorem about extending holomorphic motions that is known at present. We also study some other applications.

Holomorphic Functions and Heat Kernel Measure on an Infinite Dimensional Complex Orthogonal Group

Abstract: The heat kernel measure \mu_t is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, {\Cal H} L^2(SO_{HS},\mu_t), is one of two spaces of holomorphic functions we consider. The second space, {\Cal H} L^2(SO(\infty)), consists of functions which are holomorphic on an analog of the Cameron-Martin subspace for the group. It is proved that there is an isometry from the first space to the second one.

The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from {\Cal H} L^2(SO(\infty)) into the Hilbert space associated with a Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups.

All the results of this paper are formulated for one concrete group, the Hilbert-Schmidt complex orthogonal group, though our methods can be applied in more general situations.

Perturbation Theorems for Supercontractive Semigroups

Abstract: Let \mu be a probability measure on a Riemannian manifold. It is known that if the semigroup e^{–t\nabla^*\nabla} is hypercontractive, then any function g for which |\nabla g|_{\infty}<=1 will satisfy a Herbst inequality, \int \exp(\alpha g^2) d\mu < \infty, for small \alpha > 0. If the semigroup is supercontractive, then the above inequality will hold for all \alpha > 0. For any \alpha > 0 for which Z= \int \exp(\alpha g^2) d\mu < \infty, we define a measure \mu_g by d\mu_g= Z^{–1} \exp(\alpha g^2) d\mu. We show that if \mu is hyper- or supercontractive, then so is \mu_g. Moreover, under standard conditions on logarithmic Sobolev inequalities which yield ultracontractivity of the semigroup, Gross and Rothaus have shown that Z= \int \exp(\alpha g^2 |\log|g||^c) d\mu < \infty for some constants \alpha, c. We in addition show that the perturbed measure d\mu_g= Z^{–1}\exp(\alpha g^2|\log|g||^c) d\mu is ultracontractive.

Short Time Behavior of Hermite Functions on Compact Lie Groups

Abstract: Let p_t(x) be the (Gaussian) heat kernel on R^n at time t. The classical Hermite polynomials at time t may be defined by a Rodriguez formula, given by H_\alpha(–x, t) p_t(x) = \alpha p_t(x), where \alpha is a constant coefficient differential operator on R^n. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new class of functions on a general compact Lie group, G. In analogy with the R^n case, these "Hermite functions" on G are obtained by the same formula, wherein p_t(x) is now the heat kernel on the group, –x is replaced by x^{–1}, and \alpha is a right invariant differential operator. Let {\frak g} be the Lie algebra of G. Composing a Hermite function on G with the exponential map produces a family of functions on {\frak g}. We prove that these functions, scaled appropriately in t, approach the classical Hermite polynomials at time 1 as t tends to 0, both uniformly on compact subsets of {\frak g} and in L^p({\frak g}, d\mu), where 1 <= p< \infty, and d\mu is a Gaussian measure on {\frak g}. Similar theorems are established when G is replaced by G/K, where K is some closed, connected subgroup of G.

Spectral Analysis on Infinite Sierpinski Gaskets

Abstract: We study the spectral properties of the Laplacian on infinite Sierpinski gaskets. We prove that the Laplacian with the Neumann boundary condition has pure point spectrum. Moreover, the set of eigenfunctions with compact support is complete. The same is true if the infinite Sierpinski gasket has no boundary, but is false for the Laplacian with the Dirichlet boundary condition. In all these cases we describe the spectrum of the Laplacian and all the eigenfunctions with compact support.

To obtain these results, first we prove certain new formulae for eigenprojectors of the Laplacian on finite Sierpinski pre-gaskets. Then we prove that the spectrum of the discrete Laplacian on a Sierpinski lattice is pure point, and the eigenfunctions are localized.

The Heat Kernel Weighted Hodge Laplacian on Noncompact Manifolds

Abstract: On a compact orientable Riemannian manifold, the Hodge Laplacian \Delta has compact resolvent, therefore a spectral gap, and the dimension of the space {\Cal H}^p = ker \Delta^p of harmonic p-forms is a topological invariant. By contrast, on complete noncompact Riemannian manifolds \Delta is known to have various pathologies, among them the absence of a spectral gap and either a "too large" or "too small" space {\Cal H}^p.

We use a heat kernel measure d\mu to determine the size of the space of square-integrable forms and in constructing the appropriate Laplacian \Delta_\mu. We recover in the noncompact case certain results of Hodge's theory of \Delta in the compact case. If the Ricci curvature of a noncompact connected Riemannian manifold M is bounded below, then this "heat kernel weighted Laplacian" \Delta_\mu acts on functions on M in precisely the manner we would wish, that is, it has a spectral gap and a one-dimensional kernel. We prove that the kernel of \Delta_\mu on n-forms is zero-dimensional on M, as we expect from topology, if the Ricci curvature is nonnegative. On Euclidean space, there is a complete Hodge theory for \Delta_\mu. Weighted Laplacians have a duality analogous to Poincaré duality on noncompact manifolds, and a Künneth formula. We show that heat kernel-like measures give desirable spectral properties (compact resolvent) in certain general cases. In particular, we use measures with Gaussian decay to justify the statement: "every topologically tame manifold has a strong Hodge decomposition."

Combinatorics and Holomorphic Dynamics: Captures, Matings and Newton's Method

Abstract: This thesis studies three interrelated topics of one dimensional holomorphic dynamics: captures, matings and Newton's method. The main ingredients are Thurston's characterization theorem of critically finite rational maps and Yoccoz's puzzle arguments. The latter works in many cases in which critical finiteness is not assumed. Both employ combinatorics heavily and essentially.

We prove a conjecture on captures for quadratic polynomials. The result has a significant impact on the study of parameter spaces for many quadratic rational map families. For example, it is used in the study of matings involving Yoccoz polynomials in this thesis, although we mainly relied on Yoccoz's puzzle argument to do the proof.

Newton's method has been a favorite application topic for holomorphic dynamics. We study the Newton maps for a class of higher degree polynomials which includes all polynomials of form z^d+az+b (d>=3 and a,b are arbitrary parameters).

Internal Addresses in the Mandelbrot Set and Irreducibility of Polynomials

Abstract: For a complex parameter c, consider the quadratic polynomial p_c= z^2 + c. Its periodic points of period n solve the polynomial equation P_n(z)= p_c^{\circ n}(z) – z. Dividing out periodic points of lower period, we arrive at polynomials Q_n(z) which are recursively defined by P_n(z)= \prod_{k| n} Q_k(z). The first main result is the following: the polynomials Q_n are irreducible over C(c). We even determine its Galois group and show that it consists of all permutations of periodic points of exact period n which commute with the dynamics of the iteration.

The principal tool, and second main result, is a combinatorial description of the Mandelbrot set using internal addresses. Moreover, they give the Mandelbrot set an interpretation as the parameter space of kneading sequences, and they answer the question: How can you tell where a rational external ray lands at the Mandelbrot set, without having Adrien Douady at your side?

Both the irreducibility result and the internal addresses are generalized to the polynomials z^d + c, and applications to the exponential family \lambda e^z are briefly discussed. In addition, we give a new, much simplified proof of a classical result of Douady and Hubbard: every external ray of the Mandelbrot set at a rational angle lands at a well-defined point c, such that there is a precisely described relation between the angle of the ray and the dynamics of the polynomial z^2 + c.

Finiteness And Compactness For The Family Of Isospectral Riemannian Manifolds

Abstract: Riemannian manifolds are said to be isospectral if they have the same Laplacian spectrum by counting multiplicity. The purpose of this thesis is to study to what extent the Laplacian spectrum of a manifold M determines the manifold up to finitely many topological types, or up to a compact family of metrics. This research involves the studies of heat invariants, a-priori estimates for Ricci curvature equations, the properties of the space of isospectral Riemannian manifolds endowed with Gromov-Hausdorff or Lipschitz topology, and interpolation and embedding techniques, etc. Several finiteness and compactness results have been obtained with either geometric conditions or purely spectral conditions. One of the main theorems is as follows. Given constants p<= n and q> p/2. Let C^p_I(M) be the p-isoperimetric constant defined by

C^p_I(M)= inf_S (Area(S)) / ([min(Vol(A), Vol(B))]^{1–1/p}}),

for n<= p< \infty, where S runs over all hypersurfaces which divide M into two parts A and B. And let I(p,q) denote the space which consists of all closed, isospectral n-dimensional Riemannian manifolds with sectional curvatures uniformly bounded in L^q-norm and p-isoperimetric constants C^p_I(M) uniformly bounded away from zero. Then, I(p,q) contains only finitely many diffeomorphism types and it is compact in the C^\infty-topology, by which we mean that, on a fixed manifold in a given diffeomorphism class of I(p,q), the set of all pullback metrics of Riemannian metrics in the diffeomorphism class is compact with respect to the C^\infty-topology.

Quasisymmetric Conjugacy of Degree N Critical Circle Map

Abstract: Circle diffeomorphism has been studied since the time of Poincaré. Through the works of Arnold, Herman and Yoccoz, one has a very deep understanding of circle diffeomorphism. However the techniques employed by the above people fail if the map is merely a circle homeomorphism and has critical points. To overcome this difficulty, van Strien and independently \'Swiatek introduced a new technique called distortion of cross-ratio. People have since tried to analyze this technique and create in the process a new principle to study one dimensional real dynamics.

On the other hand, much less is known about covering circle map of degree n> 1. Even though such a map is topologically semi-conjugate to the map z\to z^n, we may not have a topological conjugacy because the map may have attracting periodic orbits. Using the new techniques developed by Sullivan, de Melo and van Strien, this work gives sufficient conditions for a degree n critical covering circle map to be quasisymmetrically conjugate to the map z\to z^n.

As an application of this theorem to complex dynamical systems, I make use of Douady's and Ghy's construction which gives sufficient conditions for a quadratic polynomial to have a Siegel disk which is a quasi-disk. We consider a one parameter family of cubic polynomials with a superattractive fixed point at the origin, the family being parametrized by the other free critical point. I am able to give sufficient conditions for the immediate basin of the origin to be bounded by a quasi-circle.

Band-Limited Wavelets With Rotational Symmetry

Abstract: Given a lattice \Gamma \subset R^n, a linear transformation M: \Gamma\to\Gamma with det(M) = m> 1, and a lattice-preserving rotation \rho:\Gamma\to\Gamma of integral order N, we consider the problem of finding band-limited functions {\psi_1,...,\psi_r} \subset L^2(R^n) for which

{m^{j/2} \psi_i(\rho^k M^jx– \gamma) : i= 1,...,r; k= 0,...,N–1; j\in Z; \gamma\in\Gamma}

is an orthonormal basis of L^2(R^2). Generalizing a result of A. Bonami, J. Soria, and G. Weiss, we find necessary and sufficient conditions on \hat\psi_i for such a basis to occur. By working in the frequency domain, we are able to bypass the consideration of a multiresolution analysis, thereby enlarging the class of wavelets we are able to consider. Having done this, we then use these results to construct some smooth wavelet bases for L^2(R^2) which are {\it rotationally symmetric} in the sense that they are generated through {\it rotation} as well as translation and dilation.

In the second chapter, we investigate orthonormal bases of L^2(R^2) which are generated through rotation and dilation. That is, we consider the problem of finding functions {\Psi_i: i\in I} for which

{m^{j/2} \Psi_i(\rho^k M^j x) : i\in I; k= 0,...,N–1; j\in Z}

is an orthonormal basis for L^2(R^2). Adapting a result of P. Auscher, G. Weiss, and M. Wickerhauser, we construct projections P_W on L^2(R^2) which are similar in spirit to multiplication by \chi_W, where W is the annular wedge

W= {(x(r,\theta), y(r,\theta)) : r\in[r_1, r_2], \theta\in[\theta_1,\theta_2]},

but are "smoother." Using these projections, we decompose L^2(R^2) = \oplus_W {\Cal H}_W into a direct sum of mutually orthogonal subspaces which, like the subspaces L^2(W), are mapped into one another by rotation and dilation. We then construct a smooth, compactly supported basis for {\Cal H}_W= P_W(L^2(R^2)) which is similar to the local sine and cosine bases of Coifmann and Meyer. This yields a basis of the desired form.

The Bargmann-Segal "Coherent State" Transform for Compact Lie Groups

Abstract: Let K be an arbitrary compact, connected Lie group. We describe on K an analog of the Bargmann-Segal "coherent state" transform, and we prove that this generalized coherent state transform maps L^2(K) isometrically onto the space of holomorphic functions in L^2(G, \mu), where G is the complexification of K and where \mu is an appropriate heat kernel measure on G. The generalized coherent state transform is defined in terms of the heat kernel on the compact group K, and its analytic continuation to the complex group G. We also define a "K-averaged" version of the coherent state transform, and we prove a similar result for it.

Local Connectivity in a Family of Cubic Polynomials

Abstract: In this work we consider the one parameter family of cubic polynomials with a critical fixed point, the family being parameterized by the other free critical point. Because these polynomials have a single free critical point we are able to use many of the techniques developed to study quadratics (which also have a single free critical point). In particular we will be using some relatively recent constructions due to J. C. Yoccoz which he introduced in his work on the Mandelbrot set.

Our first results are in the dynamical plane. We are able to show that if the polynomial is not renormalizable then its Julia set is locally connected. Furthermore we show that the boundary of the immediate basin of attraction of the critical fixed point is locally connected in all cases.

In the last two chapters we turn to the parameter plane and study the connectedness locus for this family of cubics. We show that it contains infinitely many embedded copies of the Mandelbrot set and is locally connected everywhere else.

The Regularity of Solutions to the Heat Equation Over Group-Valued Path Space

Abstract: In the group-valued path space W_G= {\gamma\in C[0,1]; G | \gamma(0) = e}, one has a natural Laplacian operator which is associated to the finite energy subgroup. The main purpose of this thesis is to solve the following heat equation on W_G:

and to prove some regularity properties of the solutions. As in the finite dimensional case, we construct a W_G-valued Brownian motion and use the convolution of the transition measures (heat kernels) of the W_G-valued Browning motion and the initial condition f to solve the equation. As an extension of those techniques in proving the regularity theorems for solutions of the heat equation, we also prove some regularity properties for the Brownian bridge measure on the based loop group.

On the Essential Self-Adjointness of Dirichlet Operators on Nonlinear Path Space

Abstract: There are two points of view in studying loop groups depending on the domain one chooses for the differential operators. If G is a compact Lie group and g is its Lie algebra, one can consider functions defined on

W_g= {b: [0, 1] \rightarrow g; b continuous, b(0) = 0}

or functions defined on

W_G= {g: [0, 1] \rightarrow G; g continuous, g(0) = e.}

Getzler and Malliavin have formulated their work on analysis over Loop Spaces in terms of functions on W_g. We choose to work with spaces of functions on W_G since they reflect topological properties of G. We call W_G nonlinear path space. We make substantial progress towards the solution of the problem of essential self-adjointness of the number operator over the nonlinear path space in the natural presumed core. Consider the Cameron-Martin space H over g and for h in H let \partial _h be the directional derivative on W_G. The adjoint operator \partial*_h with respect to the "Wiener measure" P on W_G is given by –\partial_h+ j_h. For each partition 0 < T_1 < ... < T_m of [0,1] define the subspace {\cal F}^m_h of L^2(W_G, P) by pulling back C^{\infty}_c(R x G^m)with respect to the map \theta_m(\gamma) = (j_h, \gamma(T_1), ..., \gamma(T_m)) on W_G. The completion \bar{{\cal F}}^m_h of {\cal F}^m_h in L^2(W_G, P) is isomorphic to L^2(R x G^m, \rho_m) where \rho_m is the density of the distribution law of \theta_m. We appeal to Malliavin calculus to prove smoothness of \rho_m and then use Wielen's technique to prove that the Dirichlet operator \partial*_h \partial _h|_{{\cal F}^m_h} as an operator on L^2(R x G^m, \rho_m) is essentially self-adjoint in {\cal F}^m_h. The essential self-adjointness of \partial*_h \partial _h in U {\cal F}^m_h is deduced easily due to the invariance of {\cal F}^m_h with respect to \partial*_h \partial _h.

Eigenvalue and Heat Kernel Estimates for the Weighted Laplacian on a Riemannian Manifold

Abstract: Let 0 < w be a smooth function on a complete Riemannian manifold (M^n, g), and let Lf = –\Delta f– <\nabla(log w), \nabla f> be the weighted Laplacian acting on L^2(w dV). Generalizing techniques used in the case of the standard Laplacian, we obtain upper and lower bounds for the first non-zero eigenvalue of L, if M^n is compact, and for the bottom of its spectrum, if M^n is non-compact. We also show that if R_w\equiv Ric – w^{–1} Hess w is bounded below by –nK(K>= 0), then the positive solutions of (L+ \partial_t)u= 0 satisfy a gradient estimate of the same form as that obtained by Li and Yau when L is the Laplacian. This is used to obtain a parabolic Harnack inequality which, in turn, yields upper and lower Gaussian estimates for the heat kernel of L. These estimates are then applied to study the L^p mapping properties of t\rightarrow exp(–tL)\mu for measures \mu which are \alpha-dimensional in a sense that generalizes the local uniform \alpha-dimensionality introduced by R. S. Strichartz.

Last modified: February 6, 2004