Mathematics

Sparse Optimization Algorithms and Applications

Speaker: 
Stephen Wright
Date: 
Mon, Apr 4, 2011
Location: 
PIMS, University of British Columbia
Conference: 
IAM-PIMS-MITACS Distinguished Colloquium Series
Abstract: 
In many applications of optimization, an exact solution is less useful than a simple, well structured approximate solution. An example is found in compressed sensing, where we prefer a sparse signal (e.g. containing few frequencies) that matches the observations well to a more complex signal that matches the observations even more closely. The need for simple, approximate solutions has a profound effect on the way that optimization problems are formulated and solved. Regularization terms can be introduced into the formulation to induce the desired structure, but such terms are often non-smooth and thus may complicate the algorithms. On the other hand, an algorithm that is too slow for finding exact solutions may become competitive and even superior when we need only an approximate solution. In this talk we outline the range of applications of sparse optimization, then sketch some techniques for formulating and solving such problems, with a particular focus on applications such as compressed sensing and data analysis.

As Geometry is Lost - What Connections are Lost? What Reasoning is Lost? What Students are Lost? Does it Matter?

Speaker: 
Walter Whitley
Date: 
Fri, Apr 29, 2011
Location: 
SFU Harbour Center
Location: 
PIMS, Simon Fraser University
Conference: 
Changing the Culture 2011
Abstract: 
In a North American curriculum preoccupied with getting to calculus, we witness an erosion of geometric content and practice in high school. What remains is often detached from "making sense of the world", and from reasoning (beyond axiomatic work in University). We see the essential role of geometry in science, engineering, computer graphics and in solving core problems in applications put aside when revising math curriculum. A second feature is that most graduates with mathematics degrees are not aware of these rich connections for geometry. We will present some samples of: what we know about early childhood geometry.; and then of the critical role of geometry and geometric reasoning in work in multiple fields outside of mathematics. With a perspective from "modern geometry", we note the critical role of transformations, symmetries and invariance in many fields, including mathematics beyond geometry. With these bookends of school mathematics in mind, we consider some key issues in schools, such as which students are lost when the bridge of geometry is not there to carry them through (caught in endless algebra) and possible connections other subjects. We also consider the loss within these other disciplines. We will present some sample investigations and reasoning which can be supported by a broader more inclusive set of practices and which pays attention to geometric features and reasoning in various contexts. In particular, we illustrate the use of dynamic geometry investigations, hands on investigations and reflections, and making connections to deeper parts of the rest of mathematics and science.

Changing the Culture of Homework

Speaker: 
Justin Grey
Speaker: 
Jamie Mulholand
Date: 
Fri, Apr 29, 2011
Location: 
SFU Harbour Center
Location: 
PIMS, Simon Fraser University
Conference: 
Changing the Culture 2011
Abstract: 
Who do your students think their homework is for? Does attaching credit to homework promote student understanding, or encourage students to find answers by whatever means necessary? Are they focused on calculating the answer, or seeing the big picture? Is their homework grade a true reflection of their own understanding of the material, or does it better reflect the understanding of their "support network"? In this workshop we will describe our efforts to improve student feedback and to promote good study skills in first and second year mathematics classes.

Raising the Floor and Lifting the Ceiling: Math For All

Speaker: 
Sharon Friesen
Date: 
Fri, Apr 29, 2011
Location: 
SFU Harbour Center
Location: 
PIMS, Simon Fraser University
Conference: 
Changing the Culture 2011
Abstract: 
"Math. The bane of my existence for as many years as I can count. I cannot relate it to my life or become interested in what I'm learning. I find it boring and cannot find any way to apply myself to it since I rarely understand it." (high school student) Today, mathematics education faces two major challenges: raising the floor by expanding achievement for all, and lifting the ceiling of achievement to better prepare future leaders in mathematics, as well as in science, engineering, and technology. At first glance, these appear to be mutually exclusive: But are they? Is it possible to design learning that engages the vast majority of students in higher mathematics learning? In this presentation, I will present the findings and discuss the implications from a research study that explored the ways to teach mathematics that both raised the floor and lifted the ceiling.

Multi Variable Operator Theory with Relations

Speaker: 
Ken Davidson
Date: 
Tue, May 24, 2011
Location: 
PIMS, University of Victoria
Conference: 
Canadian Operator Symposium 2011 (COSY)
Abstract: 
TBA

Min Protein Patter Formation

Speaker: 
William Carlquist
Date: 
Thu, Jul 14, 2011
Location: 
PIMS, University of Victoria
Conference: 
AMP Math Biology Workshop
Conference: 
IGTC Summit
Abstract: 
This talk was one of the IGTC Student Presentations.

Memory Induced Animal Movement Patterns

Speaker: 
Ulrike Schlaegel
Date: 
Thu, Jul 14, 2011
Location: 
PIMS, University of Victoria
Conference: 
AMP Math Biology Workshop
Conference: 
2011 IGTC Summit
Abstract: 
This talk was one of the IGTC Student Presentations.

On Fourth Order PDEs Modelling Electrostatic Micro-Electronical Systems

Speaker: 
Nassif Ghoussoub
Date: 
Wed, Jul 8, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 

Micro-ElectroMechanical Systems (MEMS) and Nano-ElectroMechanical Systems (NEMS) are now a well established sector of contemporary technology. A key component of such systems is the simple idealized electrostatic device consisting of a thin and deformable plate that is held fixed along its boundary $ \partial \Omega $, where $ \Omega $ is a bounded domain in $ \mathbf{R}^2. $ The plate, which lies below another parallel rigid grounded plate (say at level $ z=1 $) has its upper surface coated with a negligibly thin metallic conducting film, in such a way that if a voltage l is applied to the conducting film, it deflects towards the top plate, and if the applied voltage is increased beyond a certain critical value $ l^* $, it then proceeds to touch the grounded plate. The steady-state is then lost, and we have a snap-through at a finite time creating the so-called pull-in instability. A proposed model for the deflection is given by the evolution equation

$$\frac{\partial u}{\partial t} - \Delta u + d\Delta^2 u = \frac{\lambda f(x)}{(1-u^2)}\qquad\mbox{for}\qquad x\in\Omega, t\gt 0 $$
$$u(x,t) = d\frac{\partial u}{\partial t}(x,t) = 0 \qquad\mbox{for}\qquad x\in\partial\Omega, t\gt 0$$
$$u(x,0) = 0\qquad\mbox{for}\qquod x\in\Omega$$

Now unlike the model involving only the second order Laplacian (i.e., $ d = 0 $), very little is known about this equation. We shall explain how, besides the above practical considerations, the model is an extremely rich source of interesting mathematical phenomena.

Law of Large Number and Central Limit Theorem under Uncertainty, the related New Itô's Calculus and Applications to Risk Measures

Speaker: 
Shige Peng
Date: 
Thu, Jul 9, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 

Let $ S_n= \sum_{i=1}^n X_i $ where $ \{X_i\}_{i=1}^\infty $ is a sequence of independent and identically distributed (i.i.d.) of random variables with $ E[X_1]=m $. According to the classical law of large number (LLN), the sum $ S_n/n $ converges strongly to $ m $. Moreover, the well-known central limit theorem (CLT) tells us that, with $ m = 0 $ and $ s^2=E[X_1^2] $, for each bounded and continuous function $ j $ we have $ \lim_n E[j(S_n/\sqrt{n}))]=E[j(X)] $ with $ X \sim N(0, s^2) $.

These two fundamentally important results are widely used in probability, statistics, data analysis as well as in many practical situation such as financial pricing and risk controls. They provide a strong argument to explain why in practice normal distributions are so widely used. But a serious problem is that the i.i.d. condition is very difficult to be satisfied in practice for the most real-time processes for which the classical trials and samplings becomes impossible and the uncertainty of probabilities and/or distributions cannot be neglected.

In this talk we present a systematical generalization of the above LLN and CLT. Instead of fixing a probability measure P, we only assume that there exists a uncertain subset of probability measures $ \{P_q:q \in Q\} $. In this case a robust way to calculate the expectation of a financial loss $ X $ is its upper expectation: $ [\^\,(\mathbf{E})][X]=\sup_{q \in Q} E_q[X] $ where $ E_q $ is the expectation under the probability $ P_q $. The corresponding distribution uncertainty of $ X $ is given by $ F_q(x)=P_q(X \leq x) $, $ q \in Q $. Our main assumptions are:

  1. The distributions of $ X_i $ are within an abstract subset of distributions $ \{F_q(x):q \in Q\} $, called the distribution uncertainty of $ X_i $, with $ ['(m)]=[\^(\mathbf{E})][X_i]=\sup_q\int_{-\infty}^\infty xF_q(dx) $ and $ m=-[\^\,(\mathbf{E})][-X_i]=\inf_q \int_{-\infty}^\infty x F_q(dx) $.
  2. Any realization of $ X_1, \ldots, X_n $ does not change the distributional uncertainty of $ X_{n+1} $ (a new type of `independence' ).

Our new LLN is: for each linear growth continuous function $ j $ we have

$$\lim_{n\to\infty} \^{\mathbf{E}}[j(S_n/n)] = \sup_{m\leq v\leq ['(m)]} j(v)$$

Namely, the distribution uncertainty of $ S_n/n $ is, approximately, $ \{ d_v:m \leq v \leq ['(m)]\} $.

In particular, if $ m=['(m)]=0 $, then $ S_n/n $ converges strongly to 0. In this case, if we assume furthermore that $ ['(s)]2=[\^\,(\mathbf{E})][X_i^2] $ and $ s_2=-[\^\,(\mathbf{E})][-X_i^2] $, $ i=1, 2, \ldots $. Then we have the following generalization of the CLT:

$$\lim_{n\to\infty} [j(Sn/\sqrt{n})]= \^{\mathbf{E}}[j(X)], L(X)\in N(0,[s^2,\overline{s}^2]).$$

Here $ N(0, [s^2, ['(s)]^2]) $ stands for a distribution uncertainty subset and $ [\^(E)][j(X)] $ its the corresponding upper expectation. The number $ [\^(E)][j(X)] $ can be calculated by defining $ u(t, x):=[^(\mathbf{E})][j(x+\sqrt{tX})] $ which solves the following PDE $ \partial_t u= G(u_{xx}) $, with $ G(a):=[1/2](['(s)]^2a^+-s^2a^-). $

An interesting situation is when $ j $ is a convex function, $ [\^\,(\mathbf{E})][j(X)]=E[j(X_0)] $ with $ X_0 \sim N(0, ['(s)]^2) $. But if $ j $ is a concave function, then the above $ ['(s)]^2 $ has to be replaced by $ s^2 $. This coincidence can be used to explain a well-known puzzle: many practitioners, particularly in finance, use normal distributions with `dirty' data, and often with successes. In fact, this is also a high risky operation if the reasoning is not fully understood. If $ s=['(s)]=s $, then $ N(0, [s^2, ['(s)]^2])=N(0, s^2) $ which is a classical normal distribution. The method of the proof is very different from the classical one and a very deep regularity estimate of fully nonlinear PDE plays a crucial role.

A type of combination of LLN and CLT which converges in law to a more general $ N([m, ['(m)]], [s^2, ['(s)]^2]) $-distributions have been obtained. We also present our systematical research on the continuous-time counterpart of the above `G-normal distribution', called G-Brownian motion and the corresponding stochastic calculus of Itô's type as well as its applications.

Regular Permutation Groups and Cayley Graphs

Speaker: 
Cheryl E. Praeger
Date: 
Fri, Jul 10, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 
Regular permutation groups are the `smallest' transitive groups of permutations, and have been studied for more than a century. They occur, in particular, as subgroups of automorphisms of Cayley graphs, and their applications range from obvious graph theoretic ones through to studying word growth in groups and modeling random selection for group computation. Recent work, using the finite simple group classification, has focused on the problem of classifying the finite primitive permutation groups that contain regular permutation groups as subgroups, and classifying various classes of vertex-primitive Cayley graphs. Both old and very recent work on regular permutation groups will be discussed.
Syndicate content