Abstract

Let X=(X^{diff}, J) be a compact complex manifold. Kodaira and Spencer developed from 1957 a theory to address the problem: describe the set I of complex structures on X^{diff} close to J. The crowning piece of this theory is Kuranishi's Theorem (1962) which states that the (infinite-dimensional) analytic set I is isomorphic to the product of a (finite-dimensional) analytic set K by a vector space. Moreover, every structure of I is isomorphic to one of K, so that K contains all classes of  isomorphisms of complex structures close to J.

Let (E,J) be a CR-structure on a smooth manifold X^{diff}. It is natural to ask the same question in that context that is: describe the set I of CR-structures close to (E,J). However, this is out of reach for a general (E,J). 

In this talk, I will introduce the notion of polarized CR-structures, which are a very special type of CR-structures; and give a description of the set I in the case where (E,J) is polarized.

Details

In the first part of this thesis, we study the existence and stability of multi-spot patterns on the surface of a sphere for a singularly perturbed Brusselator and Schnakenburg reaction-diffusion model. The method of matched asymptotic expansions, tailored to problems with logarithmic gauge functions, is used to construct both symmetric and asymmetric spot patterns. There are three distinct types of instabilities of these patterns that are analyzed: self-replication instabilities, amplitude oscillations of the spots, and competition instabilities. By using a combination of spectral theory for nonlocal eigenvalue problems together with numerical computations, parameter thresholds for these three different classes of instabilities are obtained. For the Brusselator model, our results point towards the existence of cycles of creation and destruction of spots, and possibly to chaotic dynamics. For the Schnakenburg model, a differential-algebraic ODE system for the motion of the spots on the surface of the sphere is derived.

In the second part of the thesis, we study the existence and stability of mesa solutions in one spatial dimension and the corresponding planar mesa stripe patterns in two spatial dimensions. An asymptotic analysis is used in the limit of a large diffusivity ratio to construct mesa patterns in one spatial dimension for a general class of two-component reaction-diffusion systems that includes the well-known Gierer Meinhardt activator-inhibitor model with saturation (GMS model), and a predator-prey model. For such one-dimensional patterns, we study oscillatory instabilities of the pattern by way of a Hopf bifurcation and from a reduction to a limiting ODE-PDE system. In addition, explicit thresholds are derived characterizing transverse instabilities of planar mesa-stripe patterns in two spatial dimensions. The results of our asymptotic theory as applied to the GMS and predator-prey systems are confirmed with full numerical results.

Details

The Schrödinger equation, an equation central to quantum mechanics, is a dispersive equation which means, very roughly speaking, that its solutions have a wave-like nature, and spread out over time. We will consider global behaviour of solutions of two nonlinear variations of the Schrödinger equation.

In particular, we consider the nonlinear magnetic Schrödinger equation. We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in H^{^1}, then the solution of the above equation decomposes uniquely into a standing wave part, which converges as t goes to infinity, and a dispersive part, which scatters.
    
We also consider the Schrödinger map equation into the 2-sphere. We obtain a global well-posedness result for this equation with radially symmetric initial data without any size restriction on the initial data.

Details

We study lattice trees and lattice animals in high dimensions. Lattice trees and animals are interesting combinatorial objects used to model branched polymers in polymer science. They are also of interest in combinatorics and in the study of critical phenomena in statistical physics.

The nearest-neighbor and spread-out models on the d dimensional integer lattice \mathbb{Z}^d, have edge set consisting of pairs \set{x,y} with \|x-y\|_1=1 and \|x-y\|_\infty\leq L with L\geq1 fixed, respectively. On either graph, a \emph{lattice animal} is a finite connected subgraph, and a \emph{lattice tree} is an animal without cycles. Let t_n and a_n be the number of lattice trees and animals with n bonds that contain the origin, respectively. Standard subadditivity arguments provide the existence of the \emph{growth constants} \tau=\lim_{n\to\infty}t_n^{1/n} and \alpha=\lim_{n\to\infty}a_n^{1/n}. We are interested in the \emph{critical points}
of these models, which are the reciprocals of the corresponding growth constants.

We rigorously calculate the first three terms of a 1/d--expansion for the critical points of nearest-neighbor lattice trees and animals. The proof follows an inductive argument similar to the one used in \cite{HS95} and \cite{HS06}, to obtain analogous results for the critical points of self-avoiding walks and percolation. To provide the leading terms in the expansions, we use a mean-field model, related to the Galton-Watson branching process with critical Poisson offspring distribution, and results obtained with the lace expansion. The leading terms are also calculated in the spread-out model. Then we develop expansions for the nearest-neighbor generating functions and, together with the lace expansion, obtain the first and second correction terms.

Our result gives a rigorous proof for previous work on the subject \cite{GP00}, \cite{Harr82,PG95}. Given the algorithmic nature of the proof, it can be extended, with sufficient labor, to compute higher degree terms. It may provide the starting point for proving the existence of an asymptotic expansion with rational coefficients, for the critical point of nearest-neighbor lattice trees.

Abstract

With the goal of understanding polymer entanglements, for over 20 years there has been interest in questions about knotting and linking of self-avoiding polygons on the simple cubic lattice. Notably, in 1988 Sumners and Whittington
proved that all but exponentially few sufficiently long self-avoiding polygons are knotted. This proved the long standing Frisch-Wasserman-Delbruck conjecture that sufficiently long ring polymers will be knotted with high probability.
Since then there has been progress both theoretically and numerically using lattice polygon models to investigate polymer entanglements. Much of this progress has been motivated by questions arising from the study of
DNA topology. For lattice models, these questions lie at the interface between statistical mechanics, enumerative combinatorics, topology, graph theory and 
applied probability/Monte Carlo methods. I will review progress made and highlight new results and open problems, focusing on models of enzyme action on DNA and polymers in confined geometries.

Abstract

This is joint work with Srikanth Iyengar on results that connects work of Mike Hopkins in homotopy theory and commutative algebra with a theorem of Dave Benson, Jeremy Rickard and myself on group representations. I will spend most of the lecture talking about what the words mean and why we are interested in the results.

Details