We propose a hybrid method for solving large sparse linear least squares problems. The method is iterative in nature, as it is based on preconditioned LSQR. However, the preconditioner comes from an orthogonal factorization of a submatrix of the original matrix associated with the least squares problem; the construction of the preconditioner is based on well-known techniques in sparse direct methods. In this talk, we will discuss the quality of the preconditioner and the choices of the submatrix.

Consider a nonlinear Schr\"{o}dinger equation in $\mathbb{R}^3$ with a short-range potential. The linear Hamiltonian is assumed to have three or more eigenvalues satisfying some resonance conditions. We study the asymptotic behavior at time infinity of solutions with small initial data in $H^1 \cap L^1(\mathbb{R}^3)$. The results include the case that all of the eigenvalues are simple and also the case that the second eigenvalues are degenerate. These are joint works with Stephen Gustafson, Kenji Nakanishi and Tai-Peng Tsai.

This talk will be about the collection of theories that started from a
lecture by M. Kontsevich in 1995, and which are now known
by the name of "motivic integration". One of these theories
introduces integration-like techniques in order to prove results in
algebraic geometry (this was the original motivation behind it), and
the other one provides a geometric interpretation of p-adic integrals.
I will try to survey the main ideas behind the construction of motivic
measures, and mention some of the most spectacular applications.

Given a homogeneous Poisson point process it is well known that selecting each point independently with some fixed probability gives a homogeneous Poisson process of lower intensity. This is often referred to as thinning. Can thinning be achieved without additional randomization; that is, is it possible to choose a subset of the Poisson points as a deterministic function of the Poisson process so that the chosen points form a Poisson process of any given lower intensity?
On an infinite volume, it is always possible. Furthermore, on R^d , it is possible to define the deterministic function to be a translation-equivariant factor (that is, if a translation is applied to the original process, the chosen points are translated by the same vector). On a finite volume, the answer depends on both the intensities of the original and resulting Poisson processes. We will discuss joint work with Omer Angel, Alexander Holroyd, and Russell Lyons.

The last talk this term for UBC/UMC, the undergraduate mathematics colloquium, will be given by Eric Cytrynbaum.

Title: How cells get by without a compass and ruler

Abstract:

Cells need to be able to read their own geometries for various purposes. For example, cell division requires that a division plane through the "middle" of a cell be determined. Cell motility requires that various structures within the cell be properly aligned with the direction of "desired" motion. In this talk, I will discuss a few mechanisms by which cells carry out these space-sensing calculations.

This seminar continues with an overview of the topics in the just published (November 2009) Springer book "Applications of Symmetry Methods to Partial Differential Equations" by Bluman, Cheviakov and Anco. In Part IV, it will be shown how to find directly the multipliers and then the corresponding fluxes for conservation laws of PDE systems. This will be followed by a discussion of connections between symmetries and conservation laws.In particular, it will be shown how to use symmetries to find new conservation laws from known conservation laws.Furthermore relationships will be established between symmetries, solutions of adjoint equations and conservation laws.

Elliptic curves are very old mathematical objects but they are still element of current research. In my talk I will define the notion of an elliptic curve E motivated by some beautiful connection to meromorphic functions on complex tori. We will illustrate (but not prove) that an elliptic curve E carries an abelian group structure and will define its group of rational points, which leads to the famous Mordell-Weil Theorem. Time permits I will say something about the conjecture of Birch and Swinnerton-Dyer.

It is expected that a general K3 surface does not admit
self rational maps of degree > 1. I'll give a proof of this conjecture for
K3 surfaces of genus at least 4.

The outer automorphism group of a free group of finite rank shares many properties with linear groups and mapping class groups of surfaces. However the techniques for studying these three families of groups are generally quite different; problems which are difficult for one class may be easier and more intuitive for another. I will describe some algebraic, topological, and dynamical methods for studying such problems, thereby highlighting some of the interesting features which distinguish these groups.

Abstract: Fully irreducible outer automorphisms of a free group are analogous to loxodromic isometries of hyperbolic space, or to pseudo-Anosov elements of the mapping class group of a surface. We develop methods for constructing customized fully irreducible elements of a free group F of rank k. For example, there exists for any matrix A in GL(k,Z) a non-geometric fully irreducible element inducing the action of A on the non-abelian free group of rank k. This is an analogue of a well-known theorem for the mapping class group. This is joint work with Matt Clay.

By Mostow's rigidity theorem, geometric invariants of hyperbolic 3-manifolds are in fact topological invariants. On the other hand, it follows from the work of Thurston and Perelman that a 3-manifold is hyperbolic if and only if it satisfies some rather mild conditions. In light of these results, it is an interesting question to try to understand how topological conditions on a 3-manifold $M$ which admits a hyperbolic metric affect the geometry of the hyperbolic metric. This question is rather imprecise. In other words, it has many different incarnations. In this talk I will describe a few results on different concrete formulations of the question above.

John Mighton will speak on his work with JUMP Math.

John Mighton is a mathematician, author, playwright, and the founder of JUMP Math. He tirelessly volunteers his time and expertise at JUMP as the lead curriculum developer for the JUMP Math Student Workbooks and Teacher's Manuals. He also donates all proceeds from publications to JUMP.

Dr. Mighton completed a Ph.D. in mathematics at the University of Toronto and was awarded an NSERC fellowship for postdoctoral research in knot and graph theory. He is currently a Fellow of the Fields Institute for Research in Mathematical Sciences and has also taught mathematics at the University of Toronto.

Capture-recapture methods are widely used to monitor endangered wildlife populations. A requirement of simple capture-recapture models is that all individuals alive on one sampling occasion have the same probability of capture. While this assumption may be reasonable in small, isolated populations, there are many variables that might influence an individual's catchability and estimates of survival rates or the abundance will be biased if these differences are
ignored. However, covariates of the capture probability which vary both between individuals and over time, like body mass, present a challenge in the analysis of capture-recapture data because 1) their values can only be measured for the individuals captured on each sampling occasion and 2) the unknown values are not missing at random and cannot be ignored. I will present Bayesian methods to incorporate the effects of such covariates in the Cormack-Jolly-Seber and Jolly-Seber models -- the two most common models for open-population capture-recapture data.

My talk will begin with an introduction to Bayesian statistics, capture-recapture methods, and the problems associated with time-dependent covariates. I will then describe my method for including such covariates in the Cormack-Jolly-Seber model to estimate survival rates and how this method can be extended to the Jolly-Seber model to obtain estimates of abundance. I will illustrate my methods by application to data from the study of Soay sheep on the Isle of Hirta, Scotland, and conclude by discussing applications to more complicated models and comparisons with other approaches.

Abstract: Suppose that \Sigma is a hyperbolic surface of finite type and let \Map(\Sigma) be its mapping class group. It is due to Morita that the canonical homomorphism \Diff(\Sigma)\to\Map(\Sigma) does not split. A first goal of this talk is to give a very simple proof of this fact showing in fact that this also remains true when restricted to some rather small subgroups of the mapping class group. After having proved this, I will show that while \Map(\Sigma) admits a natural Lipschitz action on the unit tangent bundle T^1\Sigma of \Sigma, this action is not homotopic to any smooth action. This is partly joint work with Mladen Bestvina and Tom Church.

The talk will address the issues of numerical approximations of dynamical systems in presence of high oscillation. For the systems of highly oscillatory ordinary differential equations given in the vector form y' = A_w y + f, where A_w is a constant nonsingular matrix, ||A_w|| >> 1, \sigma(A_w) in iR, f is a smooth vectorvalued function and w is an oscillatory parameter, we show how an appropriate choice of quadrature rule improves the accuracy of numerical approximation as w -> \infty. We present a Filon-type method to solve highly oscillatory linear systems and WRF method, a special combination of the Filon-type method and the waveform methods, for nonlinear systems. The work is accompanied by numerical examples.

MATH ANNEX 1102 (relocated because of PIMS closure)

Mon 21 Dec 2009, 1:30pm-3:00pm

Abstract

In his influential works, A. Okounkov showed how to associate a convex body to a very ample G-line bundle L on a projective G-variety X such that it projects to the moment polytope of X and the push-forward of the Lebesgue measure on it gives the Duistermaat-Heckamnn measure for the correspoding Hamiltonian action. He used this to prove the log-concavity of multiplicities in this case. Motivated by his work, recently Lazarsfeld-Mustata and Kaveh-Khovanskii developed a general theory of Newton-Okounkov bodies (without presence of a G-action).

In this talk, I will go back to the case where X has a G-action. I discuss how to associate different convex bodies to a graded G-algebra which in particular encode information about the multiplicities of the G-action. Using this I will define the Duistermaat-Heckmann measure for a graded G-algebra and prove a Brunn-Minkowski inequality for it. Also I will prove a Fujita approximation type result (from the theory of line bundles) for this Duistermaat-Heckmann measure. This talk is based on a preprint in preparation joint with A. G. Khovanskii.

## Note for Attendees

Tea & cookies afterwards!