Department of Mathematics and Statistics

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

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.

Department Colloquium - Natalia Stepanova (Carleton University)

Natalia Stepanova (Carleton University)

Friday, February 7th, 2020

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

Speaker: Natalia Stepanova (Carleton University)

Title: Goodness-of-fit tests based on sup-functionals of weighted empirical processes.

Abstract: A large class of goodness-of-fit test statistics based on sup-functionals of weighted empirical processes is proposed and studied. The weight functions employed are Erdos-Feller-Kolmogorov-Petrovski upper-class functions of a Brownian bridge. Based on the result of M. Csorgo, S. Csorgo, Horvath, and Mason obtained for this type of test statistics, we provide the asymptotic null distribution theory for the class of tests in hand, and present an algorithm for tabulating the limit distribution functions under the null hypothesis. A new family of nonparametric confidence bands is constructed for the true distribution function and it is found to perform very well. The results obtained, together with a new result on the convergence in distribution of the higher criticism statistic, introduced by Donoho and Jin, demonstrate the advantage of our approach over a common approach that utilizes a family of regularly varying weight functions. Furthermore, we show that, in various subtle problems of detecting sparse heterogeneous mixtures, the proposed test statistics achieve the detection boundary found by Ingster and, when distinguishing between the null and alternative hypotheses, perform optimally adaptively to unknown sparsity and size of the non-null effects. This is joint work with Tatjana Pavlenko (Sweden).

Natalia Stepanova is a Professor of Statistics in the School of Mathematics and Statistics at Carleton University. She has a Ph.D. degree in Statistics from St. Petersburg State University. From 2001--2003 Natalia Stepanova was a Postdoctoral Fellow at Queen's University, supervised by Prof. Boris Levit. Her research interests lie mainly in the area of nonparametric statistics, including high-dimensional statistical inference.

Department Colloquium - Dengdeng Yu (University of Toronto)

Dengdeng Yu (University of Toronto)

Thursday, February 6th, 2020

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

Speaker: Dengdeng Yu (University of Toronto)

Title: Causal Inference with 2D Treatment: An Application to Alzheimer’s Studies

Abstract: Alzheimer's disease is a progressive form of dementia that causes problems with memory, thinking and behavior. It is important to identify the changes of certain brain regions that lead to behavioral deficits. In this paper, we study how hippocampal atrophy affects behavioral deficits using data from the Alzheimer's Disease Neuroimaging Initiative. The special features of the data include a 2D matrix-valued imaging treatment and more than $6$ million of potential genetic confounders, which bring significant challenges to causal inference. To address these challenges, we propose a novel two-step causal inference approach, which can naturally account for the 2D treatment structure while only adjusting for the necessary variables among the millions of covariates. Based on the analysis of the Alzheimer's Disease Neuroimaging Initiative dataset, we are able to identify important biomarkers that need to be accounted for in making causal inference and located the subregions of the hippocampus that may affect the behavioral deficits. We further evaluate our method using simulations and provide theoretical guarantees.

Dengdeng Yu is a postdoctoral fellow at the Department of Statistical Sciences at the University of Toronto and the Canadian Statistical Sciences Institute (CANSSI). He obtained his Ph.D. degree in Statistics from the University of Alberta in 2017. His research interests include high-dimensional and functional data analysis, neuroimaging and imaging genetics data analysis, and quantile regression.

Department Colloquium - Gregory G. Smith (Queen's University)

Gregory G. Smith (Queen's University)

Friday, January 31st, 2020

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

Speaker: Gregory G. Smith (Queen's University)

Title: The geometry of closed subsets.

Abstract: How can we understand the spaces embedded in a fixed projective space? There are different ways to answer this question. After examining a few, we will focus on a geometric approach. Ultimately, we aim to determine when the natural parameter space (the so called Hilbert scheme) is smooth.

Gregory G. Smith is a Professor of Mathematics at Queen's University. His research interests include algebraic geometry, commutative algebra, computer algebra and combinatorics. He was elected a fellow of the Canadian Mathematical Society in 2018. He also received the 2012 Coxeter-James Prize and 2007 André Aisenstadt Prize.

Department Colloquium - Hok Kan Ling (Columbia University)

Hok Kan Ling (Columbia University)

Monday, January 27th, 2020

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

Speaker: Hok Kan Ling (Columbia University)

Title: Shape-constrained Estimation and Testing.

Abstract: Shape-constrained inference has been gaining more attention recently. Such constraints are sometimes the direct consequence of the problem under investigation. In other times, they are used to replace parametric models while retaining qualitative shape properties that exist in problems from diverse disciplines. In this talk, I will first discuss the estimation of a monotone density in s-sample biased sampling models, which has been long missing in the literature due to certain non-standard nature of the problem. We established the asymptotic distribution of the maximum likelihood estimator (MLE) and the connection between this MLE and a Grenander-type estimator. In the second part of the talk, a nonparametric likelihood ratio test for the hypothesis testing problem on whether a random sample follows a distribution with a decreasing, k-monotone or log-concave density is proposed. The obtained test statistic has a surprisingly simple and universal asymptotic null distribution, which is Gaussian, instead of the well-known chi-square for generic likelihood ratio tests. We also established rates of convergence of the maximum likelihood estimator under weaker conditions than the existing literature that are of independent interest.

Hok Kan (Brian) Ling is a Ph.D. candidate in the Department of Statistics at Columbia University, working under the supervision of Dr. Zhiliang Ying. His research interests primarily lie in the areas of multivariate statistics, latent variable models, event history analysis, nonparametric estimation, semiparametric models and shape-restricted statistical inference.

Department Colloquium - Siliang Gong (University of Pennsylvania)

Siliang Gong (University of Pennsylvania)

Thursday, January 23rd, 2020

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

Speaker: Siliang Gong (University of Pennsylvania)

Title: Per-family Error Rate Control for Gaussian Graphical Models via Knockoffs.

Abstract: Driven by many real applications, the estimation of and inference for Gaussian Graphical Models (GGM) are fundamentally important and have attracted much research interest in the literature. However, it is still challenging to achieve overall error rate control when recovering the graph structures of GGM. In this paper, we propose a new multiple testing method for GGM using the knockoffs framework. Our method can control the overall finite-sample Per-Family Error Rate up to a probability error bound induced by the estimation errors of knockoff features. Numerical studies demonstrate that our method has competitive performance compared with existing methods. This is joint work with Qi Long and Weijie Su.

Siliang Gong is a postdoctoral fellow in the Department of Biostatistics at the University of Pennsylvania. She completed her Ph.D. in statistics at the University of North Carolina at Chapel Hill in 2018. She works on high-dimensional data analysis and statistical machine learning.

Department Colloquium - Kasun Fernando (University of Toronto)

Kasun Fernando (University of Toronto)

Friday, January 17th, 2020

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

Speaker: Kasun Fernando (University of Toronto)

Title: Error terms in the Central Limit Theorem.

Abstract: Expressing the error terms in the Central Limit Theorem as an asymptotic expansion (commonly referred to as the Edgeworth expansion) goes back to Chebyshev. In the setting of sums of independent identically distributed (iid) random variables, sufficient conditions for the existence of such expansions have been extensively studied. However, there is almost no literature that describe this error when the expansions fail to exist. In this talk, I will discuss the case of sums of iid non-lattice random variables with $d+1$ atoms. It can shown that they never admit the Edgeworth expansion of order d. However, using tools from Homogeneous Dynamics, it can shown that for almost all parameters the Edgeworth expansion of order $d-1$ holds and the error of the order $d-1$ Edgeworth expansion is typically of order $n^{-d/2}$ but the order $n^{-d/2}$ terms have wild oscillations (to be made precise during the talk). This is a joint work with Dmitry Dolgopyat.

Kasun Fernando is a postdoctoral fellow in the Department of Mathematics at the University of Toronto. He completed his Ph.D. in 2018 at the University of Maryland, College Park. His research is primarily focused on possible extensions of this theory of asymptotic expansions to more general settings that are not included in the classical theory, including the case of random variables arising as observations of chaotic dynamical systems.

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