Jason Bramburger, University of Ottawa
Area Of Research: Dynamical Systems
Title: Spiral wave solutions to lambda-omega lattice dynamical systems
Abstract: In this talk we inspect an infinite system of coupled lambda-omega differential equations indexed by the two-dimensional integer lattice and show there exists a spiral wave solution. It is demonstrated that the infinite-dimensionality of the system poses unique problems in that tradi- tional techniques from systems of finitely many differential equations cannot be directly applied. In particular, the existence proof requires extensive results from Banach space theory and a ”Hard” Implicit Function Theorem. Some brief remarks on the history and relevance of lambda-omega systems are also provided to properly frame the results.
Xin Yang Lu, McGill University
Area Of Research: Geometric optimization, PDE
Title: Centroidal Voronoi tessellations and Gersho’s conjecture in 3D
Abstract: Centroidal Voronoi Tessellations (CVT) are tessellations using Voronoi regions of their centroids. CVTs are useful in data compression, optimal quadrature, optimal quantization, clustering, and optimal mesh generation. Many patterns seen in nature are closely approximated by a CVT. Examples of this include the Giant’s Causeway, the cells of the cornea, and the breeding pits of the male tilapia.
This is closely related to Gersho’s conjecture, which states that there exists an asymptotically optimal CVT whose Voronoi regions are all rescaled copies of the same polytope. Straightforward in 1D, and proven in 2D, Gersho’s conjecture is still open for higher dimensions. One of the main difficulties is that Gersho’s conjecture is a strongly nonlocal, infinite dimensional minimization problem (even in 3D). In this talk we will present some recent results which reduce Gersho’s conjecture to a local, finite dimensional problem in 3D. Joint work with Rustum Choksi.
Melkior Ornik, University of Toronto
Area Of Research: Control Theory
Title: An Overview of Reach Control Theory
Abstract: The behaviour of controlled mechanical systems is commonly interpreted in terms of ordinary differential equations, where the equation guiding the system trajectories in some way depends on the control input applied on the system. A classical problem in the area is that of controllability, i.e., of determining whether there exists a control input which drives the system trajectory from one point in the state space to another. Standard methods in control apply only to Euclidean state spaces, and do not consider the existence of physical obstacles or other constraints affecting the system behaviour. Reach control theory is an approach to solving the problem of controllability on a constrained state space. The method it uses is to triangulate the state space into simplices, and then design a simple controller on each of the simplices to drive the system trajectory from the starting point to a desired ending point. Reach control admits a substantial mathematical structure, with a top layer based on graph theory, computational geometry, and formal logic, and a lower layer driven by dynamical equations. In this talk, I will give an overview of the current state of the art in reach control theory, and outline the theoretical challenges it is still facing. Finally, I will illustrate the method and use of reach control by presenting an application to the adaptive cruise control feature in vehicles. Along with an appealing real-world use, this application motivates a number of new theoretical problems in reach control.
Kaveh Mousavand, UQAM
Area of Interest: Representation Theory of Associative Algebras
Title: Application of -tilting theory in preprojectivization
Abstract: Preprojective algebras were firstly introduced by Gelfand and Ponomarev, in 1979 via a combinatorial description, and a few years later by Baer, Geigle and Lenzing (1987) through a homological approach. In 1998, M. Ringlel showed that these seemingly different constructions give rise to the same algebra, which led into applications in various areas, including representation theory of associative algebras, algebraic geometry, combinatorics, lattice theory, Coxeter groups and cluster theory, with growing linkage between other subjects.
On the other hand, in 2013, Adachi, Iyama and Reiten introduced the notion of -tilting theory, as a natural generalization of the well-known tilting theory, in which the Auslander-Reiten translation plays a crucial role. This has provided an efficient tool in the study of module category of associative algebras and related areas.
In this talk, we look at the connections between -tilting theory and preprojective algebras and will show how a good knowledge of Auslander-Reiten translation in the module category of a given algebra paves the way for preprojectivization of non-hereditary algebras. We use gentle algebras to depict our direction, as this class of algebras embodies the right framework for this approach.
Omid Makhmali, McGill University
Area Of Research: Differential Geometry
Title: Differential Geometric Aspects of Causal Structures
Abstract: In this talk I will define causal structures as a field of tangentially non-degenerate projective hypersurfaces in the projectivized tangent bundle of a manifold. As a result, causal structures are a generalization of conformal pseudo-Riemannian structures. The local equivalence problem of causal structures on manifolds of dimension at least four is solved using Cartan?s method of equivalence, which allows one to geometrically interpret the essential local invariants of such geometries as a natural generalization of the sectional Weyl curvature and the Fubini cubic forms of the fibers. After giving examples, in dimension four, a generalization of the notion of self-duality for indefinite conformal structures to causal structures with ruled null cones will be given. This is carried out by extending Penrose’s original realization about self-duality to causal structures, i.e., the existence of a 3-parameter family of surfaces whose tangent planes at each point rule the null cone.
Amir El-Aooiti, Ryerson University
Area of Interest: Computational complexity, universal algebra, and theoretical computer science
Title: Circuit Complexity of Solving Constraint Satisfaction Problems with Few Subpowers
Abstract: Although combinatorial problems represented in the form of Constraint Satisfaction Problems (CSPs) are generally known to be NP-complete, placing restrictions on the constraint template can yield tractable subclasses. By studying the operations in the polymorphism of algebras of the constraint languages, we can construct algorithms which solve our CSP in polynomial time. It was shown by (Bulatov and Dalmau, 2006) that constraint languages with polymorphisms containing Mal’tsev operations are tractable, and it was shown later by (Idziak et al, 2007) that constraint languages with polymorphisms containing -edge operations (equivalent to having Few subpowers) for integers (a general form of the Mal’tsev operation for arities greater than 3) are also tractable. We seek to improve on this result by presenting an algorithm which can solve ‘k-edge’ constraints more efficiently with circuit complexity in NC. By doing this, we can potentially expand on the Dichotomy Conjecture presented by (Feder and Vardi, 1999) by hypothesizing that tractable CSP subclasses are either P-complete or in NC.
First, we perform a logspace reduction of our CSP to an equivalent CSP for directed graphs with binary constraints as done by (Bulin et al, 2014). The algebra formed by taking the polymorphism of the edge relations of the digraph will only have strictly simple subalgebras of type 2, 3, and 4 in the sense of the Tame Congruence Theory. Having set the stage for our main result, we prove that the solution set of the digraph CSP contains a subalgebra which itself is a subdirect product of a set of algebras which are subalgebras of the algebra formed by taking the polymorphism of the digraph edge relations. With this property, we can then arrange the logical formulas describing the CSP into a binary tree where each leaf represents a constraint in the CSP. As we travel up to the root of the binary tree, we take the conjunction of the constraint formulas yielding partial solutions at every step until we are left with a set of solutions at the root of the tree which satisfy all the constraints.
Clemonell Lord Baronat Bilayi-Biakana, University of Ottawa
Area Of Research: Probability and Statistics
Title: Tail empirical processes for stochastic volatility models
Abstract: We consider the tail empirical process (TEP) related to a distribution with a regularly varying tail. This is an important tool used in nonparametric estimation of extremal quantities, like the Hill estimator of the index of regular variation, or various risk measures. In this talk, we consider a long memory stochastic volatility model of interest in finance. We first start by investigating some probabilistic properties of this model. We establish central and non-central limit theorems for the TEP and apply these results to investigate the asymptotic behaviour of the aforementioned extremal quantities. Our theoretical results are illustrated by simulation studies. This talk is based on joint work with Rafal Kulik and Gail Ivanoff.
Noé Aubin-Cadot, Université de Montréal
Area Of Research: Gauge theory / symplectic topology
Title: Introduction to Atiyah-Floer conjecture
Abstract: In 1987, M. F. Atiyah conjectured an eventual isomorphism between instanton Floer homology and Lagrangian Floer homology. After a short/superficial reminder about gauge theory and symplectic topology, I’ll state the conjecture and summarize various attempts that happened since.
Priyabrata Senapati, Ryerson University
Area Of Research: Computational Biology
Title: Effective Multilevel Monte Carlo Methods for Stochastic Biochemical Kinetics
Abstract: Stochastic mathematical models are critical for an accurate representation of biochemical processes in a single cell. The effect of random fluctuations at the molecular level may be significant when some species have low population numbers. While exact stochastic simulation methods exist, they are prohibitively expensive on most systems of interest. The high computational cost of such numerical simulations for complex stochastic models motivated the search for more effective strategies. Often, the expected value of some function of the final time solution of the stochastic model is of interest. Then, the approach employing multi-level Monte Carlo methods is more efficient than traditional techniques. In this talk, we present multi-level Monte Carlo (MLMC) schemes for a reliable and effective simulation of stochastic models of biochemical kinetics. The advantages of these MLMC strategies are illustrated on several biochemical models arising in applications.
Aram Dermenjian, Université du Québec a Montréal
Area Of Research: Combinatorics
Title: The facial weak order and its lattice quotients
Abstract: We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W. We call order the facial weak order. We first provide two alternative characterizations of this poset : a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bjrner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of their classes. As application, we describe the facial boolean lattice on the faces of the cube and the facial Cambrian lattice on the faces of the corresponding generalized associahedron.
Gustavo Felisberto Valente, University of Ottawa
Area Of Research: Dynamical Systems
Title: Bratteli diagrams and Dynamical Systems
Abstract: Bratteli diagrams are graphs with special properties that can be translated into Operator Algebras and Dynamical Systems settings. We use these diagrams to visualize a direct sequence of finite-dimensional algebras and help describe properties of direct limits of such algebras. In this talk I’ll give an overview of these concepts making an introductory approach to the topics.
Damien Rioux Lavoie, Université de Montréal
Area Of Research: Numerical analysis
Title: A penalty method applied to the fractional heat equation
Abstract: Since Abel’s success in solving the tautochrone curve problem two centuries ago, fractional differential equations (FDE) have found a great deal of applications in the domain of physics and engineering.
Sadly, due to the fact that the fractional order differential operators are non-local, it is not easy to find analytical solutions to most of the important FDE. Therefore, it is necessary to develop efficient numerical tools to solve these equations.
In this talk, we will present an active penalty method coupled with a high-order Fourier spectral method to solve the fractional heat equation with non-trivial boundaries and obstacles.
Becem Saidani, University of Ottawa
Area Of Research: Probability and Statistics
Title: Multivariate stable distributions and stable CLT
Abstract: The stable distribution is a natural generalization of the normal distribution with which it shares the property of being stable under addition, a property that is needed in actuarial sciences for continuously compounded returns. Furthermore, stable distributions are capable of capturing excess kurtosis shown by historical stock returns. The goal of this report is to give a characterization of stable distributions using Levy-Khinchine and spectral representations and to prove the Stable Central Limit Theorem.
Keith O’Neill, University of Ottawa
Area Of Research: Category Theory
Title: Smoothness in Codifferential Categories
Abstract: In noncommutative geometry, it is useful to invoke notions of smoothness which are compatible with ostensibly pathological spaces; in this way one may apply the tools of differential geometry to spaces for which even topological structure is elusive. In this talk we explore a class of conjectures around which we may orient our varied conceptions of smoothness. Two crucial examples are highlighted.
Evans Boadi, Brock University
Area Of Research: Mathematical Physics and General Relativity
Title: Symmetric Polynomials, Combinatorics and Mathematical Physics
Abstract: Abstract Symmetric polynomials have interesting connection with Mathematical Physics. The Jack polynomials are connected to the Hamiltonian of the quantum n body systems. This work present the link between Jack polynomials and the CMS operator. The Kerov’s map translates the Jack functions into super Jack polynomials. The super symmetric version of the Jack polynomials turn out to the eigenfunctions of the deformed CMS operators.