# Keynote Speakers

**Kelly Burkett**, University of Ottawa**Title:**Markov chain Monte Carlo sampling of gene genealogies, with application to genetic association mapping**Abstract:**The gene genealogy is a tree structure describing the ancestral relationships among chromosomal sequences sampled from individuals not known to be related. Knowledge of the tree is useful for inference of population-genetic parameters such as the mutation or recombination rate. It also has potential application in gene mapping, as individuals with similar trait values will tend to be more closely related genetically at the location of a trait-influencing mutation. However, since the time scale of the genealogy is on the order of thousands of years, the true tree cannot be known.One way to incorporate genealogical trees in genetic applications is to sample them conditional on observed genetic data. In this presentation, I will describe a Markov chain Monte Carlo based genealogy sampler that we have implemented. I will also describe convergence performance and methods to improve convergence. Finally, I will discuss the application of the sampler to genetic association mapping.

**Alistair Savage**, University of Ottawa**Title:**A gentle introduction to categorification**Abstract:**This will be an expository talk concerning the idea of categorification and its role in representation theory. We will begin with some very simple yet beautiful observations about how various ideas from basic algebra (monoids, groups, rings, representations etc.) can be reformulated in the language of category theory. We will then explain how this viewpoint leads to new ideas such as the ``categorification'' of the above-mentioned algebraic objects. We will conclude with a brief synopsis of some current active areas of research involving the categorification of quantum groups. One of the goals of this idea is to produce four-dimensional topological quantum field theories. Very little background knowledge will be assumed.

# Other Speakers

**Alena Antipova**, University of Western Ontario**Title:**Colloidal discs and their pairs in a nematic liquid crystal**Abstract:**In the present work the motion of discs in a nematic liquid crystal was simulated. Under the action of a rotating magnetic field the colloidal disc with perpendicular surface anchoring immersed in a nematic liquid crystal experiences a torque and continues turning following the field. However, when the disc reaches some critical position when the director field around it is highly distorted, the disc suddenly flips to minimize the free energy. We analyze this motion and the behaviour of two discs placed close together and examine the possible uses of this flip behaviour.

**Mohammad Bardestani**, University of Ottawa**Title:**Solving polynomials over finite fields.**Abstract:**We describe Chebotarev density theorem and apply it to some interesting questions concerning solving polynomials over finite fields.

**Cyril Joël Batkam**, Université de Sherbrooke**Title:**Multiplicity result for a class of indefinite elliptic systems**Abstract:**We consider a class of symmetric elliptic systems which can be formulated variationally, but the associated Euler-Lagrange functionals are strongly indefinite in the sense that they are neither bounded from above nor from below, even constrained on subspaces of finite co-dimension. By applying a recent generalization of the symmetric mountain pass theorem, we will show that theses problems have infinitely many solutions.

**Gabriel Bergeron-Legros**, University of Ottawa**Title:**Weil Representation and Loop Symplectic Groups**Abstract:**The Weil Representation is of significant importance in the study of theta functions and modular forms. In this talk, we will introduce the standard Weil representation as a natural consequence of the Stone von Neumann theorem. Then, we will adapt it to infinite dimensional symplectic groups, which will require some symplectic geometry and finish the talk by discussing its continuity.

**Jason Bramburger**, University of Ottawa**Title:**Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback**Abstract:**In this talk we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback.

**Yue Dong**, University of Ottawa**Title:**On the Problem of Universal Consistency of the kNN Classifier in Banach Spaces**Abstract:**The kNN classifier is one of the oldest natural classification for labeling data. The classical Stone’s theorem says that this algorithm is universal consistent in finite Euclidean space. Since then, the result has been generalized in many directions. In particular, it will show that the algorithm is no longer universal consistent in infinite dimensional space. Of certain interest is the problem of universal consistency in more general infinite-dimensional Banach space(the so called functional learning). In this talk, we will survey and discuss what is known in this problem.

**Pierre Yves Gaudreau Lamarre**, University of Ottawa**Title:**Applications of Free Probability in Random Matrix Theory**Abstract:**This talk will be about the applications of tools and techniques developed in Free Probability to solve problems in Random Matrix Theory. Firstly, we will briefly introduce the field of random matrix theory by going over basic definitions as well as classical results. Secondly, we will introduce the theory of Free Probability and explain how it can be viewed as a generalization of the classical measure theoretic formulation of probability in which one can naturally study noncommutative random variables. Lastly, we will see how methods in Free Probability can be used to solve problems in random matrix theory, and time permitting, we will discuss some related ongoing research projects.

**Gaël Giordano**, University of Ottawa**Title:**A new feature selection technique: Mass Transportation Score**Abstract:**The presentation introduces and defines the notion of learning algorithms and feature selection. Then we focuses on a new feature selection technique called the Mass Transportation Score (MTS), developed by the research team of Dr. Pestov at the University of Ottawa. The MTS is based on the notion of distance between two finitely supported measures. We first rigorously construct the MTS, then investigate the performance of the MTS for an individual feature, using a genetic dataset from the UOHI.

**Adina Goldberg**, University of Toronto**Title:**The privacy/security tradeoff across jointly designed secure sketch biometric systems**Abstract:**In the area of secure biometrics, work has been done to build an information theoretic framework characterizing privacy and security of single biometric systems. People have worked extensively on designing such systems, some more cryptographic in nature, and some more tied to error correcting codes. However, there is still little known about security and privacy across multiple jointly designed systems.This work will focus on the privacy/security tradeoff across multiple secure sketch biometric systems. Secure sketch is a type of biometric system architecture related to error-correcting codes where a system is characterized by a parity-check matrix over a finite field, or equivalently by a subspace of a vector space over that same field.

Given a set of systems (a design), we introduce worst-case measures of privacy leakage and security in the case that a subset of them become compromised. It turns out that more secure designs are necessarily less private and vice versa. We study the tradeoff between privacy and security in the following ways.

[1] We consider a restricted design space, and then relax the problem to packing subintervals of a continuous interval. As the system sizes (i.e. the dimensions of the subspaces) grow, solutions to the relaxed problem become feasible as designs. We can currently find optimal designs for small examples with linear programs.

[2] We examine the algebraic structure of the problem, using the Grassmannian graph and properties of projective space.

Both of these approaches generate bounds on achievable privacy/security pairs.

**Kevin Green**, University of Ontario Institute of Technology**Title:**Dynamic tiling patterns in a 2D neural field**Abstract:**Neural field models aim to describe the bulk properties of synaptically coupled neuronal networks through spatial and/or temporal averaging. This averaging shifts the view of the system, from that of discrete neurons with discrete connections to viewing it in terms of the densities of neurons and their connections. The present work focuses on how to do nonlinear analysis of the resulting integro-differential equation that incorporates spatio-temporal delays in two dimensions. In particular, it is shown how to reduce the model to its normal form for a Hopf bifurcation with the square lattice tiling pattern ($D_4\ltimes T^2$ symmetry). Solutions branching from this Hopf bifurcation are demonstrated by numerical simulation.

**Stan Hatko**, University of Ottawa**Title:**Borel Dimensionality Reduction of Data and Supervised Learning**Abstract:**In this talk we discuss Borel dimensionality reduction of datasets with the purpose of subsequently applying supervised learning algorithms. We will start by introducing the notions of a classifier, learning algorithm (in particular the k-NN learning algorithm), and universal consistency of a learning algorithm. Any universally consistent learning algorithm, for instance k-NN, remains so after an injective Borel map is applied. This means we can reduce the dimensionality of a high dimensional dataset by applying an injective Borel map to a lower dimensional space and subsequently apply a supervised learning algorithm. We will give some concrete examples of applying Borel dimensionality reduction to actual datasets. We will see how selecting a different Borel map at each step, depending on the sample, is equivalent to choosing from a family of metrics on the domain at each step for the k-NN learning algorithm. We would like to determine under what conditions this will produce a universally consistent classifier and avoid the problem of overfitting. We will show that k-NN with regards to norms changed depending on a sample is universally consistent provided the family of norms satisfies certain conditions.

**Eric Jalbert**, University of Guelph**Title:**Numerical computation of sharp travelling waves of a degenerate diffusion-reaction equation arising in biofilm modelling**Abstract:**Many degenerate diﬀusion-reaction equations permit sharp travelling wave solutions that describe the propagation of an interface with ﬁnite speed. If the equation is at least double degenerate, the derivative of the travelling wave solution can blow up at the interface, which poses considerable challenges for the computation of the travelling wave speed. We propose a numerical method for this problem that is based on the idea to approximate the multiple degenerate problem by a family of simple degenerate problems. For the latter we propose an interval-bracketing algorithm based on the theory of Sanchez-Garduno and Maini. The travelling wave speed of the original problem is obtained as the limit of the travelling wave speeds of the auxiliary problems. The performance of the method is investigated in a numerical simulation experiment for a problem that arises in the mathematical modelling of bioﬁlm processes.

**Melkior Ornik**, University of Toronto**Title:**Orbits and Controllability of Coupled Driftless Bilinear Systems**Abstract:**Coupled bilinear control systems is a class of driftless systems defined by having a one-to-one correspondence, in some way, between the state variables and the control inputs. This talk will examine the questions of controllability and orbits of such systems. In particular, after a short introduction to control systems, we will give a simple graph-theoretical criterion for determining whether a given coupled driftless system is controllable. Furthermore, we will show that, under a certain genericity condition, orbits of coupled systems can be fully determined using a somewhat more difficult graph-theoretical structure. An algorithm for determining the orbit of a given state will be given, and its complexity shown to be polynomial.

**Jeffrey Pike**, University of Ottawa**Title:**A Category-Theoretic Approach to Quivers**Abstract:**The theory of quivers and their representations is a topic that has received a lot of attention since the middle of the twentieth century due to its relationship with many other areas of mathematics including Lie theory, invariant theory, and algebraic geometry. In this talk we will give an introduction to this subject with a focus on how the language of category theory can simplify many of the combinatorially heavy classical definitions. We will develop the well-known relationship between representations of quivers and representations of associative algebras, and as an application we will show how this relationship can be used to study a family of three dimensional Lie algebras that depend upon a continuous parameter.

**Marianne Wilcox**, University of Guelph**Title:**Regulation and Interactions of Cytokines During a Cytokine Storm**Abstract:**Cytokine storms are a potentially fatal exaggerated immune response consisting of an uncontrolled positive feedback loop between immune cells and cytokines, leading to an elevated level of cytokines in the body. Symptoms include pyrexia, fatigue, swelling and nausea, which are evident in many species, including humans. The problem is that researchers are encountering cytokine storms during oncolytic viral therapy. That is, instead of the virus having positive effects (helping abolish the tumour), the body becomes overloaded and severe negative effects occur. Researchers at the Ontario Veterinary College have been doing work with mice. In one trial consisting of eight mice, after a virus was injected all eight mice experienced a cytokine storm and were dead within the first 30 hours. A linear ordinary differential equation model of the 23 recorded cytokines was produced in order to determine how cytokines interact. Using optimization techniques, cytokine interaction parameters were determined and cytokines with the largest regulatory effects were further studied. Results provide insight into the complex mechanism that drives the storm, and this will hopefully lead to understanding how to prevent such immune responses from happening.

**Jun Yang**, Queen's University**Title:**Lower bounds on the probability of a finite union of events**Abstract:**here.