Université de Bourgogne

Fri 6 Jul 2012, 3:00pm
SPECIAL
PIMS Seminars and PDF Colloquiums
PIMS, WMAX 110

Special Geometry Seminar: Polarized CRstructures

PIMS, WMAX 110
Fri 6 Jul 2012, 3:00pm4:00pm
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 (infinitedimensional) analytic set I is isomorphic to the product of a (finitedimensional) 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 CRstructure on a smooth manifold X^{diff}. It is natural to ask the same question in that context that is: describe the set I of CRstructures 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 CRstructures, which are a very special type of CRstructures; and give a description of the set I in the case where (E,J) is polarized.
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Wed 18 Jul 2012, 12:30pm
SPECIAL
Graduate Student Center, Room 203

Doctoral Exam

Graduate Student Center, Room 203
Wed 18 Jul 2012, 12:30pm3:00pm
Details
In the first part of this thesis, we study the existence and stability of multispot patterns on the surface of a sphere for a singularly perturbed Brusselator and Schnakenburg reactiondiffusion 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: selfreplication 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 differentialalgebraic 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 twocomponent reactiondiffusion systems that includes the wellknown Gierer Meinhardt activatorinhibitor model with saturation (GMS model), and a predatorprey model. For such onedimensional patterns, we study oscillatory instabilities of the pattern by way of a Hopf bifurcation and from a reduction to a limiting ODEPDE system. In addition, explicit thresholds are derived characterizing transverse instabilities of planar mesastripe patterns in two spatial dimensions. The results of our asymptotic theory as applied to the GMS and predatorprey systems are confirmed with full numerical results.
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Thu 19 Jul 2012, 9:00am
SPECIAL
Graduate Student Center, Room 203

Doctoral Exam

Graduate Student Center, Room 203
Thu 19 Jul 2012, 9:00am12:00pm
Details
The Schrödinger equation, an equation central to quantum mechanics, is a dispersive equation which means, very roughly speaking, that its solutions have a wavelike 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 2sphere. We obtain a global wellposedness result for this equation with radially symmetric initial data without any size restriction on the initial data.
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Mathematics

Mon 23 Jul 2012, 12:30pm
SPECIAL
Graduate Student Center, Room 200

Doctoral Exam

Graduate Student Center, Room 200
Mon 23 Jul 2012, 12:30pm3:00pm
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 nearestneighbor and spreadout models on the d dimensional integer lattice \mathbb{Z}^d, have edge set consisting of pairs \set{x,y} with \xy\_1=1 and \xy\_\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/dexpansion for the critical points of nearestneighbor 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 selfavoiding walks and percolation. To provide the leading terms in the expansions, we use a meanfield model, related to the GaltonWatson branching process with critical Poisson offspring distribution, and results obtained with the lace expansion. The leading terms are also calculated in the spreadout model. Then we develop expansions for the nearestneighbor 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 nearestneighbor lattice trees.
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University of Saskatchewan

Tue 24 Jul 2012, 4:00pm
Discrete Math Seminar / Probability Seminar
MATH 126

Lattice Models of Polymer Entanglements

MATH 126
Tue 24 Jul 2012, 4:00pm5:00pm
Abstract
With the goal of understanding polymer entanglements, for over 20 years there has been interest in questions about knotting and linking of selfavoiding polygons on the simple cubic lattice. Notably, in 1988 Sumners and Whittington
proved that all but exponentially few sufficiently long selfavoiding polygons are knotted. This proved the long standing FrischWassermanDelbruck 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.
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University of Georgia

Mon 30 Jul 2012, 3:00pm
SPECIAL
Topology and related seminars
WMAX 110 (PIMS)

Classifying thick subcategories of the stable category

WMAX 110 (PIMS)
Mon 30 Jul 2012, 3:00pm4:00pm
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.
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Tue 31 Jul 2012, 12:30pm
SPECIAL
Graduate Student Center, Room 200

Doctoral Exam

Graduate Student Center, Room 200
Tue 31 Jul 2012, 12:30pm3:00pm
Details
We study questions in three arithmetic settings, each of which carries aspects of randomlike behaviour.
In the setting of arithmetic functions, we establish mild conditions under which the tuple of multiplicative functions [f_{1}(n), f_{2}(n), ..., f_{d}(n)] densely approximates points in R^{d} for a positive proportion of n; we obtain a further generalization allowing these functions to be composed with various arithmetic progressions.
Secondly, we examine the eigenvalues of random integer matrices, showing that most matrices have no rational eigenvalues; we also identify the precise distributions of both real and rational eigenvalues in the 2 × 2 case.
Finally, we consider the set S(k) of numbers represented by the quadratic form x^{2}+ky^{2}, showing that it contains infinitely many strings of five consecutive integers under many choices of k; we also characterize exactly which numbers can appear as the difference of two consecutive values in S(k).
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Seminar Information Pages

Note for Attendees
Cookies and coffee will be available at 2:45 pm.