Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Department Colloquium

Department Colloquium - Emine Yildirim (Queen’s University)

Emine Yildirim (Queen’s University)

Friday, October 23rd, 2020

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Emine Yildirim (Queen’s University)

Title: Graphs and Combinatorics in Representation Theory of Algebras.

Abstract: Representation theorists of finite dimensional algebras often use quivers, also known as directed graphs, and many other combinatorial tools associated with these quivers. This is because we understand the module category of algebras via representation of quivers. On the other hand, we also capture the beautiful combinatorics of cluster algebras via the same representation theory. In this talk, I will outline how this machinery works along with some recent results on Cluster Categories we obtained joint with Charles Paquette using the combinatorics and representations theory of quivers.

Emine Yildirim is a Coleman Research Fellow within the Department of Mathematics and Statistics at Queen’s University. She obtained her Ph.D. in Mathematics from the Universite du Quebec a Montreal in 2018. She is mainly interested in representation theory of algebras, specifically path algebras, incidence algebras, and their representations. She also works on cluster algebras, their categorification and related combinatorics.

Department Colloquium - Rafael Potrie (U de la Republica-Uruguay)

Rafael Potrie (Universidad de la Republica-Uruguay)

Friday, October 16th, 2020

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Rafael Potrie (Universidad de la Republica-Uruguay)

Title: Anosov flows and the fundamental group.

Abstract: The hairy ball theorem says that a vector field in the sphere must have some singularity. How does the dynamics interact with the topology of the underlying manifold in higher dimensions? We will discuss some instances of this question for dynamics in 3-manifolds, featuring a beautiful result due to Margulis and Plante-Thurston. Time permitting, we will touch upon more recent developments on the interactions between the topology and dynamics in 3 dimensions.

Rafael Potrie is an Associate Professor at the Universidad de la Republica-Uruguay. He obtained his Ph.D. in Mathematics from Universite Paris 13/Universidad de la Republica-Uruguay in 2012. He was an invited speaker of the Dynamical Systems session at the International Congress of Mathematicians (ICM) in 2018. His research mainly concerns the topological classification of partially hyperbolic systems in three-dimensional manifolds and its dynamical consequences. Other interests include smooth dynamics, ergodic theory, discrete subgroups of Lie groups, and the geometry of foliations and laminations.

Department Colloquium - Farouk Nathoo (University of Victoria)

Farouk Nathoo  (University of Victoria)

Friday, October 9th, 2020

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Farouk Nathoo (University of Victoria)

Title: Spatial Statistical Modeling for Neuroimaging Data.

Abstract: I will describe three projects involving the analysis of neuroimaging data and the development of hierarchical spatial Bayesian models for each. In the first, we develop an approach for determining the location and dynamics of brain activity from combined magnetoencephalography and electroencephalography data. The resulting inverse problem is ill-posed and we propose a distributed solution based on a Bayesian spatial finite mixture model that incorporates the Potts model to represent the spatial dependence in an allocation process that partitions the cortical surface into a small number of latent states. In the second project, we consider statistical modelling of functional magnetic resonance imaging (fMRI) data which is challenging in part as the data are both spatially and temporally correlated. Motivated by an event‐related fMRI experiment, we propose a novel hierarchical Bayesian model with automatic selection of the auto‐regressive orders of the noise process that vary spatially over the brain. In the third project, we develop a Bayesian bivariate spatial model for multivariate regression analysis applicable to studies examining the influence of genetic variation on brain structure. Our model is motivated by an imaging genetics study of the Alzheimer's Disease Neuroimaging Initiative, where the objective is to examine the association between images of volumetric and cortical thickness values summarizing the structure of the brain as measured by magnetic resonance imaging (MRI) and a set of 486 SNPs from 33 Alzheimer's Disease (AD) candidate genes obtained from 632 subjects. A bivariate spatial process model is developed to accommodate the correlation structures typically seen in structural brain imaging data and we develop a mean-field variational Bayes algorithm and a Gibbs sampling algorithm to fit the model. We compare the new spatial model to an existing non-spatial model in our motivating application.

Department Colloquium - Elliot Paquette (McGill University)

Elliot Paquette (McGill University)

Friday, October 2nd, 2020

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Elliot Paquette (McGill University)

Title: Random perturbations of non-normal matrices.

Abstract: Suppose one wants to calculate the eigenvalues of a large, non-normal matrix. For example, consider the matrix which is 0 in most places except above the diagonal, where it is 1. The eigenvalues of this matrix are all 0. Similarly, if one conjugates this matrix, in exact arithmetic one would get all eigenvalues equal to 0. However, when one makes floating point errors, the eigenvalues of this matrix are dramatically different. One can model these errors as performing a small, random perturbation to the matrix. And, far from being random, the eigenvalues of this perturbed matrix nearly exactly equidistribute on the unit circle of the complex plane. This talk will give a probabilistic explanation of why this happens and discuss the general question: how does one predict the eigenvalues of a large, non-normal, randomly perturbed matrix?

Elliot Paquette is an Assistant Professor within the Department of Mathematics and Statistics at McGill University. He obtained his Ph.D.~in Mathematics from the University of Washington in 2013. He was an NSF Postdoctoral Fellow at the Weizmann Institute of Science from 2013-2016, and an Assistant Professor at the Ohio State University from 2016-2020. His research is in probability theory, with a focus on random matrix theory and on problems with geometric and topological inspirations.

Department Colloquium - Abdalrazzaq Zalloum (Queen's)

Abdalrazzaq Zalloum  (Queen's University)

Friday, September 25th, 2020

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Abdalrazzaq Zalloum (Queen's University)

Title: Negative curvature in geometric group theory.

Abstract: Geometric group theory studies the interplay between the algebraic/combinatorial properties of infinite groups and the geometries of the spaces on which they act. A naive example of this phenomena is the following theorem: "An infinite group is free if and only if it admits a free action on some tree". In the previous theorem, the geometric property of the tree containing no loops informed the algebraic/combinatorial property of the group being free and vice versa; this is a theme in geometric group theory. A metric space $X$ is said to be hyperbolic if there exists a number $\delta$ such that for any geodesic triangle in $X$, the union of the {$\delta$-nbhd} of any two of its three sides contains the third, see the attached image. Group actions on hyperbolic spaces tend to be particularly informative. A fundamental (and almost defining) property of hyperbolic spaces is that infinite geodesics satisfy a \textit{local to global} property: to check whether an infinite path is a geodesic in $X$, you need only to check that in uniformly small windows. Given a nice action of an infinite group $G$ on a hyperbolic space $X$, Cannon showed that the local-to-global property of geodesics in $X$ is reflected in the combinatorial structure of $G$. In particular, he observed that the local-to-global property of $X$ is inherited by $G$ in the sense that all the combinatorial and growth information of the \textbf{infinite} group $G$ can be encoded using only a $\textbf{finite}$ amount of data: a finite graph. I will discuss recent work where we study groups acting on spaces satisfying a similar local-to-global property, and we will see the interplay between the geometric local-to-global properties of the space, and the combinatorial structure of the acting group. Some of the results I will discuss are joint with Cordes, Russell and Spriano.

Abdalrazzaq Zalloum is a Coleman post-doctoral fellow within the Department of Mathematics and Statistics at Queen's University, working with Thomas Barthelmé and Francesco Cellarosi. He obtained his Ph.D. in Mathematics from the SUNY Buffalo in 2019. He is mainly interested in geometric group theory, which studies the interplay between the algebraic structures of groups and the geometries of the spaces on which they act.

Department Colloquium - Serdar Yuksel (Queen's University)

Serdar Yuksel (Queen's University)

Friday, September 11th, 2020

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Serdar Yuksel (Queen's University)

Title: Geometry of Information Structures, Strategic Measures and Associated Control Topologies.

Abstract: In many areas of applied mathematics (including control theory, information theory, game theory) decentralization of information among several decision makers is unavoidable. Information and correlation structures determine who knows what information and how the decisions may be dependent leading to various problems on the geometry of correlation structures among decisions/controls. We define information structures, place various topologies on them, and study closedness and compactness properties on the (strategic) measures induced by decentralized control/decision policies under varying degrees of relaxations with regard to access to private or common randomness. Ultimately, we present existence and approximation results for optimal decision/control policies. We then discuss various upper and lower bounding techniques, through realizable and classically non-realizable (such as quantum correlations and non-signaling) convex relaxations and quantization. For each of these, we review or establish closedness and convexity properties and present a hierarchy of correlation structures. As a second main theme, we review or introduce various topologies on decision/control strategies defined independently from information structures, but for which information structures determine whether the topologies entail utility in arriving at existence, compactness, convexification or approximation results. These approaches, which we will term as the strategic measures approach (where the induced joint measure is considered) and the control topology approach (where a product space of individual control policy spaces is considered), lead to complementary results and solution methods in optimal stochastic control. (Joint work primarily with Prof. Naci Saldi, other collaborators will also be acknowledged).

Department Colloquium - Tai-Peng Tsai (University of British Columbia)

Tai-Peng Tsai (University of British Columbia)

Friday, March 13th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Tai-Peng Tsai (University of British Columbia)

Title: Discretely self-similar solutions of incompressible Navier-Stokes equations and the local energy class.

Abstract: In this talk, we first review several concepts of solutions of the incompressible Navier-Stokes equations and the questions of regularity and uniqueness. We then introduce forward and backward self-similar solutions and their variants, and the similarity transform. We next sketch a few constructions of forward discretely self-similar (DSS) solutions for arbitrarily large initial data in weak $L^3$ and $L^2$ local. We finally explain their connection to the theory of local energy solutions.

Tai-Peng Tsai graduated from the University of Minnesota under the supervision of Vladimir Sverak. He was a Courant Instructor at the New York University and a Member of the Institute for Advanced Study before he joined the University of British Columbia. He works on the analysis of fluid and dispersive PDEs, including the regularity problem and self-similar solutions of Navier-Stokes equations, the asymptotic behavior of multi-solitons of Schrödinger and gKdV equations, and the regularity of energy critical Schrödinger maps.

Department Colloquium - Tim Hoheisel (McGill University)

Tim Hoheisel (McGill University)

Friday, March 6th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Tim Hoheisel (McGill University)

Title: Cone-Convexity and Composite Functions.

Abstract: In this talk we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone-convexity, is straightforward. The results are established under a verifiable Slater-type condition, with relaxed monotonicity and without lower semicontinuity assumptions on the functions in play. The versatility of our findings is illustrated by a series of applications in optimization and matrix analysis, including conic programming, matrix-fractional, variational Gram, and spectral functions.

Tim Hoheisel is an Assistant Professor within the Department of Mathematics and Statistics at McGill University. He obtained his Ph.D. in Mathematics from the University of Wuerzburg in 2009. His research lies at the intersection of continuous optimization and nonsmooth analysis and therefore between applied and pure mathematics. The problems on which he works on can be motivated by concrete applications as well as purely conceptual interest.

Department Colloquium - Anup Dixit (Queen's University)

Anup Dixit (Queen's University)

Friday, February 28th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Anup Dixit (Queen's University)

Title: The generalized Brauer-Siegel conjecture.

Abstract: One of the principal objects of study in number theory are number fields $K$, which are finite field extensions of $\mathbb{Q}$. The ring of integers of $K$ is analogous to integers $\mathbb{Z}$ in $\mathbb{Q}$. A natural question to investigate is if the ring of integers of $K$ is a unique factorization domain. The answer lies in the study of the invariant class number of $K$, which captures how far the ring of integers of $K$ is from having unique factorization. The origins of this problem can be traced back to Gauss, who conjectured that there are finitely many imaginary quadratic fields with this property. This was proved in the mid-twentieth century, independently by Baker, Heegner and Stark. A more intricate question is to understand how class number varies on varying number fields. In this context, the generalized Brauer-Siegel conjecture, formulated by M. Tsfasman and S. Vl\u{a}du\c{t} in 2002, predicts the behavior of class number times the regulator over certain families of number fields. In this talk, we will discuss recent progress towards this conjecture, in particular, establishing it in special cases.

Anup Dixit is a Coleman Postdoctoral Fellow at Queen's University under the supervision of M. Ram Murty. He obtained his Ph.D. in Mathematics from the University of Toronto in 2018. He is interested in analytic as well as algebraic number theory. He has worked on families of L-functions, behaviour of the class number on varying number fields, infinite extensions of number fields, universality of functions and Euler-Kronecker constants.

Department Colloquium - Bill Ralph (Brock University)

Bill Ralph (Brock University)

Friday, February 14th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 128

Speaker: Bill Ralph (Brock University)

Title: Can Mathematics Recognize Great Art?

Abstract: Is there an objective truth hiding within great works of art that only mathematics can detect? In this talk, I'll present evidence for a mathematical aesthetic shared by many great artists across the centuries. We'll look at several striking works of art by artists ranging from Tintoretto to Picasso and use a new statistic to see that they are creating bell curves with their brushes. I’ll also show some of my own attempts to create visual art based on a variety of mathematically oriented techniques. Examples of my work can be viewed at www.billralph.com.

Bill Ralph is in the Faculty of Mathematics and Statistics at Brock University. His mathematical research began in algebraic topology with the study of exotic homology and cohomology theories and their connections with Banach Algebras. After that, he developed a transfer for finite group actions and studied a number that appears in the transfer that he calls the "coherence number" of the group. Lately, he has also been using the Hausdorff dimension of the orbits of dynamical systems to generate mathematical art. The following is an excerpt from the curator's notes from Prof. Ralph's Rodman Hall Museum show:
It is perhaps not surprising that some of the images have a painterly feel to them since the mixing of paint on the palette and the action of the brush on a surface are both processes that can be modeled as chaotic dynamical systems. In a sense, each image is a window into the intersection of the two great universes of mathematics and fine art.

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