Department of Mathematics and Statistics

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

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.

Department Colloquium - Yi Xiong (Simon Fraser University)

Yi Xiong (Simon Fraser University)

Thursday, January 16th, 2020

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

Speaker: Yi Xiong (Simon Fraser University)

Title: Statistical Issues in Forest Fire Control.

Abstract: This talk presents statistical issues arising from forest fire (wildfire) control with a particular focus on studying the duration times in the presence of missing origin. A new methodology is proposed to tackle the issue of missing time origin with the aid of available longitudinal measures.  I present an intuitive and easy-to-implement estimator for the distribution together with a method to conduct semi-parametric regression analysis. The estimation procedure is also extended to accommodate the spatial correlation in the data. A collection of wildfire records from Alberta, Canada is used for illustration and motivation. The finite-sample performances of proposed approaches are examined via simulation. On-going work and future directions to overcome other challenges of making inference on the underlying wildfire process will be discussed.

Yi Xiong is a Ph.D. student in the Department of Statistics at Simon Fraser University, under the supervision of Dr. Joan Hu and Dr. John Braun. She is interested in developing statistical methods to analyze complex data including missing data, censored lifetime data and spatio-temporal data.

Department Colloquium - Michael Gallaugher (McMaster University)

Michael Gallaugher (McMaster University)

Monday, January 13th, 2020

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

Speaker: Michael Gallaugher (McMaster University)

Title: Clustering and Classification of Three-Way Data.

Abstract: Clustering and classification is the process of finding and analyzing underlying group structure in heterogenous data and is fundamental to computational statistics and machine learning. In the past, relatively simple techniques could be used for clustering; however, with data becoming increasingly complex, these methods are oftentimes not advisable, and in some cases not possible. One such such example is the analysis of three-way data where each data point is represented as a matrix instead of a traditional vector. Examples of three-way include greyscale images and multivariate longitudinal data. In this talk, recent methods for clustering three-way data will be presented including high-dimensional and skewed three-way data. Both simulated and real data will be used for illustration and future directions and extensions will be discussed.

Michael Gallaugher is a Ph.D. candidate in the Department of Mathematics and Statistics at McMaster University, working under the supervision of Dr. Paul D. McNicholas. His research interests lie in the area of clustering and classification which aims to find underlying group structure in heterogenous data.

Department Colloquium - Dimitris Koukoulopoulos (U Montreal)

Dimitris Koukoulopoulos (Universite de Montreal)

Friday, January 10th, 2020

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

Speaker: Dimitris Koukoulopoulos (Université de Montréal)

Title: On the Duffin-Schaeffer conjecture.

Abstract: Given any real number $\alpha$, Dirichlet proved that there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q|\le 1/q^2$. Can we get closer to $\alpha$ than that? For certain "quadratic irrationals" such as $\alpha=\sqrt{2}$ the answer is no. However, Khinchin proved that if we exclude such thin sets of numbers, then we can do much better. More precisely, let $(\Delta_q)_{q=1}^\infty$ be a sequence of error terms such that $q^2\Delta_q$ decreases. Khinchin showed that if the series $\sum_{q=1}^\infty q\Delta_q$ diverges, then almost all $\alpha$ (in the Lebesgue sense) admit infinitely many reduced rational approximations $a/q$ such that $|\alpha-a/q|\le \Delta_q$. Conversely, if the series $\sum_{q=1}^\infty q\Delta_q$ converges, then almost no real number is well-approximable with the above constraints. In 1941, Duffin and Schaeffer set out to understand what is the most general Khinchin-type theorem that is true, i.e., what happens if we remove the assumption that $q^2\Delta_q$ decreases. In particular, they were interested in choosing sequences $(\Delta_q)_{q=1}^\infty$ supported on sparse sets of integers. They came up with a general and simple criterion for the solubility of the inequality $|\alpha-a/q|\le\Delta_q$. In this talk, I will explain the conjecture of Duffin-Schaeffer as well as the key ideas in recent joint work with James Maynard that settles it.

Dimitris Koukoulopoulos is an Associate Professor of Mathematics at the Université de Montréal. He received his PhD from the University of Illinois in 2010. He works in analytic number theory, especially probabilistic and multiplicative aspects of the subject. Among his accolades, he was the cowinner of the 2013 Paul R. Halmos - Lester R. Ford Award. He is the author of the recent book The Distribution of Prime Numbers, published by the AMS.

Department Colloquium - Matthew Pratola (OSU)

Matthew Pratola (OSU)

Friday, November 29th, 2019

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

Speaker: Matthew Pratola (OSU)

Title: Bayesian Additive Regression Trees for Statistical Learning.

Abstract: Regression trees are flexible non-parametric models that are well suited to many modern statistical learning problems. Many such tree models have been proposed, from the simple single-tree model (e.g.~Classification and Regression Trees -- CART) to more complex tree ensembles (e.g.~Random Forests). Their nonparametric formulation allows one to model datasets exhibiting complex non-linear relationships between predictors and the response. A recent innovation in the statistical literature is the development of a Bayesian analogue to these classical regression tree models. The benefit of the Bayesian approach is the ability to quantify uncertainties within a holistic Bayesian framework. We introduce the most popular variant, the Bayesian Additive Regression Trees (BART) model, and describe recent innovations to this framework such as improved Markov Chain Monte Carlo sampling and a heteroscedastic variant (HBART). We conclude with some of the exciting research directions currently being explored.

Dr. Matthew Pratola is an associate professor of statistics at the Ohio State University. His research program is focused on two areas of statistical methodology: (1) statistical models and methodology for calibrating complex simulation models to real-world observations for parameter estimation, prediction and uncertainty quantification; and (2) statistical models and methodology for computationally scalable and flexible Bayesian non-parametric regression models for high-dimensional "big data" and parallel computation. His work is motivated by applied collaborations and has worked with researchers at the National Center for Atmospheric Research, Los Alamos National Laboratories, the Biocomplexity Institute of Virginia Tech, King Abdullah University of Science and Technology and the JADS Institute.

Lorne Campbell Lectureship - Olgica Milenkovic (UIUC)

Olgica Milenkovic (UIUC)

This is the second in a lecture series named in honour of Lorne Campbell, emeritus professor in the department, made possible by a generous donation from alumnus Vijay K. Bhargava, Professor of Electrical and Computer Engineering at the University of British Columbia.

Wednesday, November 27th, 2019

Time: 3:30 p.m.  Place: Jeffery Hall 127

Speaker: Olgica Milenkovic (UIUC)

Title: String reconstruction problems in molecular storage.

Abstract: String reconstruction problems frequently arise in many areas of genomic data processing, molecular storage, and synthetic biology. In the most general setting, they may be described as follows: one is given a single or multiple copies of a coded or uncodedstring, and the string copies are subsequently subjected to some form of (random) processing such as fragmentation or repeated transmission through a noise-inducing channel. The goal of the reconstruction method is to obtain an exact or approximate version of the string based on the processed outputs. Examples of string reconstruction questions include reconstruction from noisy traces, reconstruction from substrings and k-decks and reconstruction from compositional substring information. We review the above and some related problems and then proceed to describe coding methods that lead to strings that can be accurately reconstructed from their noisy traces, substrings and compositions. (This is a joint work with Ryan Gabrys, Han Mao Kiah, Srilakshmi Pattabiraman and Gregory Puleo.

Olgica Milenkovic is a professor of Electrical and Computer Engineering at the University of Illinois, Urbana-Champaign (UIUC), and Research Professor at the Coordinated Science Laboratory. She obtained her PhD from the University of Michigan, Ann Arbor. Her research interests include coding theory, bioinformatics, machine learning and signal processing.

Among her accolades, she received an NSF CAREER grant, the DARPA Young Faculty Award, the Dean’s Excellence in Research Award, and several best paper awards. She was elected a UIUC Center for Advanced Study Associate and Willett Scholar (2013) and became a Distinguished Lecturer of the Information Theory Society (2015). She is an IEEE Fellow and has served as Associate Editor and Guest-Editor-in-Chief of several leading IEEE journals.

Images from Olgica Milenkovic's Lecture - Nov. 28th, 2019

Lorne Campbell Lectureship - Olgica Milenkovic (UIUC)
Lorne Campbell Lectureship - Olgica Milenkovic (UIUC)
Lorne Campbell Lectureship - Olgica Milenkovic (UIUC)
Lorne Campbell Lectureship - Olgica Milenkovic (UIUC)

Department Colloquium - Undergraduate Summer Projects

Undergraduate Summer Projects

Friday, November 22nd, 2019

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

Speaker: Multiple Speakers

Title: Undergraduate Summer Projects.

This week colloquium will consists of ten 9 minutes presentations, starting at 2:30pm and
in the following order, by:

  • Rebecca Carter - RabbitMath animations: bringing dynamical systems to high school mathematics.
  • Daniel Cloutier - Computation of Beta Invariants on Toric Varieties I.
  • Keenan McPhail - Computation of Beta Invariants on Toric Varieties II.
  • Luca Sardellitti - Modelling and analyzing antibiotic resistance using artificial life
  • Adam Gronowski - Deep Variational Information Bottleneck.
  • Paul Wilson - Error control codes for two-way multiplying channels.
  • Ian Hogeboom-Burr - Comparison of information structures for zero-sum games and Blackwell ordering in standard Borel spaces.
  • Matt Spragge - Differential Equations Driven by Rough Paths.
  • Shikai Liu - Extensions of the Kalman lter I.
  • Linke Li - Extensions of the Kalman lter II.

Department Colloquium - Alexei Novikov (Penn State)

Alexei Novikov (Penn State)

Friday, November 15th, 2019

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

Speaker: Alexei Novikov (Penn State)

Title: The Noise Collector for sparse recovery in high dimensions.

Abstract: The ability to detect sparse signals from noisy high-dimensional data is a top priority in modern science and engineering. A sparse solution of the linear system $Ax=b$ can be found efficiently with an $l_1$-norm minimization approach if the data is noiseless. Detection of the signal's support from data corrupted by noise is still a challenging problem, especially if the level of noise must be estimated. We propose a new efficient approach that does not require any parameter estimation. We introduce the Noise Collector (NC) matrix $C$ and solve an augmented system $Ax+Cy=b+e$, where $e$ is the noise. We show that the $l_1$-norm minimal solution of the augmented system has zero false discovery rate for any level of noise and with probability that tends to one as the dimension of $b$ increases to infinity. We also obtain exact support recovery if the noise is not too large, and develop a Fast Noise Collector Algorithm which makes the computational cost of solving the augmented system comparable to that of the original one. I'll introduce this new method and give its geometric interpretation.

Prof. Alexei Novikov obtained his Ph.D.~from Stanford in 1999 and then held postdoctoral positions at the IMA and at CalTech before joining the Pennsylvania State University where he is now a Professor in the Department of Mathematics. Prof. Novikov specializes in applied analysis and probability. His research has been supported by the NSF since 2006, as well as by the US--Israel Binational Science Foundation from 2005-2009.

Department Colloquium - Ari Arapostathis (UT Austin)

Ari Arapostathis (UT Austin)

Friday, November 8th, 2019

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

Speaker: Ari Arapostathis (UT Austin)

Title: Lower bounds on the rate of convergence for heavy-tailed driven SDEs motivated by large scale stochastic networks.

Abstract: We show that heavy-tailed Levy noise can have a dramatic effect on the rate of convergence to the invariant distribution in total variation. This rate deteriorates from the usual exponential to strictly polynomial under the presence of heavy-tailed noise. To establish this, we present a method to compute a lower bound on the rate of convergence. We should keep in mind that standard Foster-Lyapunov theory furnishes only an upper bound on this rate. To motivate the study of such systems, we describe how L\'evy driven stochastic differential equations arise in the study of stochastic queueing networks. This happens when the arrival process is heavy-tailed, or the system suffers asymptotically negligible service interruptions. We identify conditions on the parameters in the drift, the Levy measure and/or covariance function which result in subexponential and/or exponential ergodicity, and we show that these conditions are sharp. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence to the stationary distribution in total variation is polynomial, and we provide a sharp quantitative characterization of this rate via matching upper and lower bounds. We conclude by presenting analogous results on convergence in the Wasserstein distance.

This talk is based on joint work with Hassan Hmedi, Guodong Pang and Nikola Sandric.

Ari Arapostathis is a professor in the Department of Electrical and Computer Engineering at The University of Texas at Austin, and holds the Texas Atomic Energy Research Foundation Centennial Fellowship in Electrical Engineering. He received his BS from MIT and his PhD from U.C. Berkeley, in 1982. He is a Fellow of the IEEE, and was a past Associate Editor of the IEEE Transactions on Automatic Control and the Journal of Mathematical Systems and Control. His research has been supported by several grants from the National Science Foundation, the Air-Force Office of Scientific Research, the Army Research Office, the Office of Naval Research, DARPA, the Texas Advanced Research/Technology Program, Samsung, and the Lockheed-Martin Corporation.

Department Colloquium - Atabey Kaygun (Istanbul Tech University)

Atabey Kaygun (Istanbul Technical University)

Friday, November 1st, 2019

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

Speaker: Atabey Kaygun (Istanbul Technical University)

Title: Noncommutative Geometry for Fun and Profit.

Abstract: There are whole fields of mathematics devoted to transferring problems of geometry and topology to commutative algebra (and vice versa) and solving them. This practice produced different "dictionaries'" of terms that tell us which type of objects in the realm of geometry or topology correspond to which other types of objects in the realm of algebra. In this talk, I am going to describe such a dictionary from the perspective of a homological algebraist who forgoes commutativity "for fun and profit" going through K-theory, cyclic and Hochschild homology, Hopf algebras, and quantum groups.

Prof. Atabey Kaygun works on homological and homotopical algebra in the context of noncommutative geometry. He obtained his Ph.D. from The Ohio State University in 2005. He was a postdoctoral fellow at the University of Western Ontario, KMMF-Warsaw University, Max-Plank-Institut fur Mathematik and University of Buenos Aires before joining the faculty of Bahcesehir University in 2009. He has been an associate professor at the Istanbul Technical University since 2016. Prof. Kaygun is currently on sabbatical and visiting Queen's University.