Mathematics

Geometry and analysis of low dimensional manifolds

Speaker: 
Gang Tian
Date: 
Fri, Aug 7, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 
In this talk, I will start with a brief tour on geometrization of 3-manifolds. Then I will discuss recent progresses on geometry and analysis of 4-manifolds.

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.

Raising the Floor and Lifting The Ceiling: Math For All (Slides)

Author: 
Sharon Friesen
Date: 
Fri, Apr 29, 2011
Location: 
SFU, Vancouver, Canada
Conference: 
Changing the Culture
Abstract: 
Slides to accompany lecture notes.
Notes: 
application/pdf iconpims_math_friesen.pdf

Raising the Floor and Lifting The Ceiling: Math For All

Author: 
Sharon Friesen
Date: 
Fri, Apr 29, 2011
Location: 
SFU-Vancouver
Conference: 
Changing the Culture
Abstract: 
Perhaps more than any other discipline, the teaching of mathematics lends itself to procedural recipes where students memorize and duplicate procedures by rote: if it looks like this, do that to it. “If one believes that mathematics is mostly a set of procedures—rules and truths—and the goal is to help students become proficient executors of the procedures, then it is understandable that mathematics would be learned best by mastering the material incrementally, piece by piece” (Stigler and Hiebert, 1999, p.90). Teaching practices that commonly flow from this view are demonstration, repetition and individual practice. In addition to being a misunderstanding of the discipline of mathematics itself, this belief also colors people’s views about who can learn mathematics. Curricula and teaching practices are often based on what Mighton calls a destructive ignorance “that leads us, even in this affluent age, to neglect the majority of children by educating them in schools in which only a small minority are expected to naturally love or excel at learning” (2007, p.2) particularly mathematics. He insists that too many students lose faith in their own intelligence, and too much effort is directed at creating artificial differences between fast and slow, gifted and “special”, advanced and delayed. And worse yet, procedural approaches to the teaching of mathematics that create problems of understanding and engagement are applied with even more vigor in remedial programs designed to help those very students for whom such practices did not work in the first place. A growing number of researchers argue that other approaches are needed to help students learn mathematics. “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” (Research Points, 2006, p.1). But are they? Is it possible to design learning that engages the vast majority of students in higher mathematics learning? To answer these questions, I designed a research study to determine whether the principles of Universal Design for Learning (UDL) resulted in increased student mathematical proficiency and achievement for all students in a typical Grade 7 classroom. Was it possible, in a regular classroom to lift the ceiling and raise the floor?
Notes: 
application/pdf iconMath For All.pdf
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