# Research Impacts

#### Image Denoising Using Block Matching Tensor Approximation (BMTA)

*Elena Braverman (University of Calgary)*

*Bin Han (University of Alberta)*

*Yi Shen (University of Calgary and University of Alberta)*

...

Figure 1: (a) Original color image of kodim03, (b) Noisy image with σ = 40, (c) Denoised image by CBM3D, (d) Denoised image by our proposed BMTA.

#### Dynamic Graph Algorithms

*Bruce Kapron (University of Victoria)*

*Valerie King (University of Victoria)*

*Ben Mountjoy (University of Victoria)*

...

#### Calculus for Functions That Don't Have Derivatives

*Heinz H. Bauschke (University of British Columba Okanagan)*

*Warren L. Hare (University of British Columbia Okanagan)*

*Yves Lucet (University of British Columbia Okanagan)*

One of the most successful mathematical discoveries of all time is the differential calculus by Newton and Leibniz. It is an indispensable tool in applied mathematics. For example, in optimization - which aims to solve mathematically optimal allocation of resources - one of the key techniques is finding critical points via derivatives.

#### Estimating Immanants of Special Unitary Matrices by Interferometry and Photon Counting

*Barry Sanders (Institute for Quantum Information Science - University of Calgary)*

*Si-Hui Tan (Data Storage Institute - Singapore)*

*Hubert de Guise (Lakehead University - Thunder Bay)*

...

#### The Byzantine Generals' Problem (or Byzantine agreement)

*Valerie King (University of Victoria)*

There is an army of generals who want to plan a common attack. Some traitors among them may lie to the others about what they know. Exchanging only messages, what decision-making protocol can the loyal generals use to arrive at a consensus on which plan to adopt? This problem was first posed over thirty years ago, to enable computation in a network where some of the nodes are faulty. Shortly after the problem was formulated, it was shown to be impossible to solve with a deterministic protocol when messages could be arbitrarily delayed.

#### The Non-Commutative Geometry of Algebraic Integers

*Marcelo Laca (University of Victoria)*

In one of the main outcomes of the PIMS CRG20 Operator Algebras and Non-Commutative Geometry, Joachim Cuntz, Christopher Deninger and Marcelo Laca associated Toeplitz-type C*-algebras to the rings of integers of algebraic number fields. In earlier work Cuntz and Li had produced purely infinite simple C*-algebras from rings of algebraic integers, and Laca and Raeburn had introduced a Toeplitz type C*-algebra for the natural numbers.

#### Analysis and geometry of optimal transportation

*Young-Heon Kim (University of British Columbia)*

In optimal transport theory, one wants to understand the phenomena arising when a mass distribution is transported to another in a most efficient way, where efficiency is measured by a given transportation cost function. For example, consider the problem of how to match water resources and towns that are distributed over a region.

#### Numerical Solution of the Monge-Ampere Equation

*Adam Oberman (Simon Fraser University)*

*Brittany Froese (Simon Fraser University)*

The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail.

#### Multidimensional Symbolic Systems: Growth, Entropy and Computability

*Tom Meyerovitch (University of British Columbia)*

Suppose you have are to tile the plane by placing colored square tiles on a grid. Each of the tile has a color, which we number from 1 to n. You are given some local adjacency constraints on placing colored tiles: There are pairs of colors i and j so that a tile of color i can not be placed directly to the left of a tile of color j, and other pairs l and m so l can not be placed directly below a tile with color m. A tiling is called admissible if it doesn’t violate the constraints.

#### Harry Potter's Cloak

*Gunther Uhlman (University of Washington)*

Invisibility has been a subject of human fascination for millenia, from the Greek legend of Perseus versus Medusa to the more recent The Invisible Man, The Invisible Woman, Star Trek and Harry Potter, among many others.

#### Evolution and Symmetry of Multipartite Entanglement

*Gilad Gour (University of Calgary)*

University of Calgary mathematician Gilad Gour discovered a simple factorization law for multipartite entanglement evolution of a composite quantum system with one subsystem undergoing an arbitrary physical process. Gour introduced the "entanglement resilience factor", which uniquely determines the multipartite entanglement decay rate. Well known bipartite entanglement evolution emerges as a special case, and this theory also readily reveals whether a permuted version of a given entangled state can be obtained by local operations.

#### An Efficient Test for Product States with Applications to Quantum Merlin-Arthur Games

*Aram Harrow (University of Bristol)*

*Ashley Montanaro (University of Washington)*

At the University of Washington, Aram Harrow and collaborator Ashley Montanaro proved the validity of an important, simple, efficient test of whether or not a quantum state is entangled. Entanglement is a key resource for quantum communication and quantum computation so this test is quite valuable to ascertain the value of a quantum state. One important consequence of this result is that the tensor optimization problem is not efficiently solvable, even approximately.

#### Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource

*Robert Raussendorf (University of British Columbia)*

*Tzu-Chieh Wei (University of British Columbia)*

*Ian Affleck (University of British Columbia)*

...

Robert Raussendorf and his colleagues Tzu-Chieh Wei and Ian Affleck at the University of British Columbia showed that Affleck-Kennedy-Lieb-Tasaki states, which are ground states of a simple, highly symmetric Hamiltonian, can serve as a universal resource for quantum computation by local measurement. Their result, which makes use of percolation theory, opens the possibility that universal computational resources could be obtained simply by cooling.

#### Analyzing the Motion of Immune Cell Surface Molecules

*Daniel Coombs (University of British Columbia)*

*Raibatak Das (University of British Columbia)*

*Christopher W. Cairo (Univesity of Alberta)*

Many important biological processes begin when a target molecule binds to a cell surface receptor protein. This event leads to a series of biochemical reactions involving the receptor and signalling molecules, and ultimately a cellular response. Surface receptors are mobile on the cell surface and their mobility is influenced by their interaction with intracellular proteins. We wanted to understand the details of these interactions and how they are affected by cellular activation.

#### The Kinetics of Immune Cell Receptor Binding Influence Immune Cell Signaling

*Daniel Coombs (University of British Columbia)*

*Omer Dushek (Oxford University)*

*Milos Aleksic (Oxford University)*

...

T cells are essential players in the immune response to pathogens such as viruses and bacteria. They can be activated to respond when they recognize molecular signatures of infection (antigens) on the surface of antigen-presenting-cells of the immune system. The T cell response is highly specific (a particular T cell responds to only the right antigen), sensitive (a T cell will respond to as few as 1–10 antigens on a single cell) and speedy (antigen binding may induce signalling within just a few seconds).

#### Mathematical Modelling of the Immune System and Disease

*Daniel Coombs (University of British Columbia)*

*Jessica M. Conway (University of British Columbia)*

While on successful drug treatment, routine testing does not usually detect virus in the blood of an HIV patient. However, more sensitive techniques can detect extremely low levels of virus. Occasionally, routine blood tests show “viral blips”: short periods of elevated, detectable viral load. In work with postdoctoral fellow Jessica Conway, we explored the hypothesis that residual low-level viral load can be largely explained by re-activation of cells that were infected before the initiation of treatment, and that viral blips can be viewed as occasional statistical events.

#### Inferring individual rules from collective behavior

*Leah Keshet (University of British Columbia)*

*Ryan Lukeman (Saint Francis Xavier University)*

Research can be challenging and arduous, but sometimes a discovery arises by serendipity. Such was the case with a project carried out by former IGTC PhD student Ryan Lukeman and his co-supervisors (LEK and Yue Xian Li). Ryan had been writing a PhD thesis on mathematical models for schools and flocks. He derived conditions under which such flocks form perfect structures, with equally-spaced individuals, all moving coherently. He had already found interesting results and had material ready for a thesis.