# Mathematics

## Changing the Culture of Homework

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.

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

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.

## Sparse Optimization Algorithms and Applications

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.

## Virtual Lung Project at UNC: What's Math Got To Do With It?

A group of scientists at the University of North Carolina, from theorists to clinicians, have coalesced over the past decade on an effort called the Virtual Lung Project. There is a parallel VLP at the Pacific Northwest Laboratory, focused on environmental health, but I will focus on our effort. We come from mathematics, chemistry, computer science, physics, lung biology, biophysics and medicine. The goal is to engineer lung health through combined experimental-theoretical-computational tools to measure, assess, and predict lung function and dysfunction. Now one might ask, with all due respect to Tina Turner: what's math got to do with it? My lecture is devoted to many responses, including some progress yet more open problems.

## Approximating Functions in High Dimensions

This talk will discuss mathematical problems which are challenged by the fact they involve functions of a very large number of variables. Such problems arise naturally in learning theory, partial differential equations or numerical models depending on parametric or stochastic variables. They typically result in numerical difficulties due to the so-called ''curse of dimensionality''. We shall explain how these difficulties may be handled in various contexts, based on two important concepts: (i) variable reduction and (ii) sparse approximation.

## Quantum Magic in Secret Communication

In this talk, we shall tell the tale of the origin of Quantum Cryptography from the birth of the first idea by Wiesner in 1970 to the invention of Quantum Key Distribution in 1983, to the first prototypes and ensuing commercial ventures, to exciting prospects for the future. No prior knowledge in quantum mechanics or cryptography will be expected.

## Expanders, Group Theory, Arithmetic Geometry, Cryptography and Much More

This is a lecture given on the occasion of the launch of the PIMS CRG in "L-functions and Number Theory".

The theory of expander graphs is undergoing intensive development. It finds more and more applications to diverse areas of mathematics. In this talk, aimed at a general audience, I will introduce the concept of expander graphs and discuss some interesting connections to arithmetic geometry, group theory and cryptography, including some very recent breakthroughs.

## Emerging Aboriginal Scholars Summer Camp

From July 4 to August 5, 2011, the UBC First Nations House of Learning and PIMS ran a summer camp for grade 10 and 11 students with First Nations backgrounds. The camp combined academics and cultural components. In this video we meet some of the camp organizers and participants. Videography by Elle-Maija Tailfeathers.

## Perfect Crystals for Quantum Affine Algebras and Combinatorics of Young Walls

In this talk, we will give a detailed exposition of theory of perfect crystals, which has brought us a lot of significant applications. On the other hand, we will also discuss the strong connection between the theory of perfect crystals and combinatorics of Young walls. We will be able to derive LLT algorithm of computing global bases using affine paths. The interesting problem is how to construct affine Hecke algebras out of affine paths.

## Regular Permutation Groups and Cayley Graphs

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.