Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Number Theory - Ahmet Muhtar Guloglu (Bilkent University, Turkey)

Tuesday, February 5th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Ahmet Muhtar Güloğlu (Bilkent University, Turkey)

Title: Cyclicity of elliptic curves modulo primes in arithmetic progressions (joint work with Yildirim Akbal).

Abstract: Let E be an elliptic curve defined over the rational numbers. We investigate the cyclicity of the group of F_p-rational points of the reduction of E modulo primes in a given arithmetic progression as a special case of Serre's Cyclicity Conjecture.

Special Colloquium - Haoran Li (UC Davis)

Haoran Li (UC Davis)

Monday, February 4th, 2019

Time: 4:30 p.m.  Place: Jeffery Hall 234

Speaker: Haoran Li (UC Davis)

Title: High-Dimensional General Linear Hypothesis Tests via Spectral Shrinkage.

Abstract: In statistics, one of the fundamental inferential problems is to test a general linear hypothesis of regression coefficients under a linear model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA and others as special cases. The testing problem is well-studied when the sample size is much larger than the dimension but remains underexplored under high dimensional settings. Various classical invariant tests, despite their popularity in multivariate analysis, involve the inverse of the residual covariance matrix, which is inconsistent or even singular when the dimension is at least comparable to the degree of freedom. Consequently, classical tests perform poorly.

In this talk, I seek to regularize the spectrum of the residual covariance matrix by flexible shrinkage functions. A family of rotation-invariant tests is proposed. The asymptotic normality of the test statistics under the null hypothesis is derived in the setting where dimensionality is comparable to the sample size. The asymptotic power of the proposed test is studied under a class of local alternatives. The power characteristics are then utilized to propose a data-driven selection of the spectral shrinkage function. As an illustration of the general theory, a family of tests involving ridge-type regularization is constructed.

Haoran Li is a Ph.D. candidate in Statistics at the University of California, Davis.
His research interests include high dimensional statistics, random matrix theory, and high dimensional time series.

Special Colloquium - Yanglei Song (UIUC)

Yanglei Song (UIUC)

Friday, February 1st, 2019

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

Speaker: Yanglei Song (UIUC)

Title: Asymptotically optimal multiple testing with streaming data.

Abstract: The problem of testing multiple hypotheses with streaming (sequential) data arises in diverse applications such as multi-channel signal processing, surveillance systems, multi-endpoint clinical trials, and online surveys. In this talk, we investigate the problem under two generalized error metrics. Under the first one, the probability of at least $k$ mistakes, of any kind, is controlled. Under the second, the probabilities of at least $k_1$ false positives and at least $k_2$ false negatives are simultaneously controlled. For each formulation, we characterize the optimal expected sample size to a first-order asymptotic approximation as the error probabilities vanish, and propose a novel procedure that is asymptotically efficient under every signal configuration. These results are established when the data streams for the various hypotheses are independent and each local log-likelihood ratio statistic satisfies a certain law of large numbers. Further, in the special case of iid observations, we quantify the asymptotic gains of sequential sampling over fixed-sample size schemes.

Yanglei Song is a Ph.D. Candidate in Statistics at the University of Illinois, Urbana-Champaign. His current research interests include multiple testing with streaming data, sequential change-point detection and high dimensional U-statistics.

Math Club - Peter Taylor (Queen's University)

Thursday, January 31st, 2019

Time: 5:30 - 6:30 p.m Place: Jeffery Hall 118

Speaker: Peter Taylor (Queen's University)

Title: Is Bacon Shakespeare?

Abstract: We will look at a sampling of the cryptographic "proofs" that have been put forward for Francis Bacon's authorship of Shakespeare's plays, and then analyze these using Claude Shannon's fundamental ideas based on the information content of the English language.

Number Theory - Neha Prabhu (Queen's University)

Tuesday, January 29th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Neha Prabhu (Queen's University)

Title: A probabilistic approach to analytic number theory.

Abstract: In 2005, Allan Gut showed that the distribution of a certain random variable, constructed using a zeta-distributed random variable, is compound Poisson. Exploiting this property, he reproved some well-known facts about the Riemann zeta function, and Selberg’s identity, using probability theory. In this talk, I present these results.

Special Colloquium - Chenlu Shi (SFU)

Chenlu Shi (SFU)

Monday, January 28th, 2019

Time: 1:30 p.m.  Place: Jeffery Hall 234

Speaker: Chenlu Shi (SFU)

Title: Space-filling Designs for Computer Experiments and Their Application to Big Data Research.

Abstract: Computer experiments provide useful tools for investigating complex systems, and they call for space-filling designs, which are a class of designs that allow the use of various modeling methods. He and Tang (2013) introduced and studied a class of space-filling designs, strong orthogonal arrays. To date, an important problem that has not been addressed in the literature is that of design selection for such arrays. In this talk, I will first give a broad introduction to space-filling designs, and then present some results on the selection of strong orthogonal arrays. The second part of my talk will present some preliminary work on the application of space-filling designs to big data research. Nowadays, it is challenging to use current computing resources to analyze super-large datasets. Subsampling-based methods are the common approaches to reducing data sizes, with the leveraging method (Ma and Sun, 2014) being the most popular. Recently, a new approach, information-based optimal subdata selection (IBOSS) method was proposed (Wang, Yang and Stufken, 2018), which applies the design methodology to the big data problem. However, both the leveraging method and the IBOSS method are model-dependent. Space-filling designs do not suffer this drawback, as shown in our simulation studies.

Chenlu Shi is a Ph.D. candidate in Statistics at Simon Fraser University. Her research interests include experimental design and analysis with applications to big data.

Special Colloquium - Qian Qin (University of Florida)

Qian Qin (University of Florida)

Friday, January 25th, 2019

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

Speaker: Qian Qin (University of Florida)

Title: Convergence complexity analysis of MCMC.

Abstract: Convergence complexity analysis is the study of how Markov chain Monte Carlo (MCMC) algorithms used in Bayesian statistics scale with the size of the underlying data set. To conduct this type of analysis, one needs tools to construct convergence bounds for high-dimensional Markov chains. I will review a few classical techniques of Markov chain convergence analysis (in particular, drift and minorization), and discuss their applicability and limitations in high-dimensional settings. I will then present a result concerning the convergence complexity of Albert and Chib's algorithm for Bayesian probit regression.

Qian Qin is a Ph.D. Candidate in Statistics at the University of Florida. His research interests include Markov chain Monte Carlo, Bayesian statistics, and high-dimensional statistics.

Math Club - Mike Roth (Queen's University)

Thursday, January 24th, 2019

Time: 5:30 - 6:30 p.m Place: Jeffery Hall 118

Speaker: Mike Roth (Queen's University)

Title: The primes from analysis.

Abstract: The prime numbers are the multiplicative building blocks of the integers, and as such appear to be creatures of algebra. This talk will explain a way in which the prime numbers arise naturally out of a question in analysis.

Number Theory - Steven Spallone (Indian Institute)

Tuesday, January 22nd, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Steven Spallone (Indian Institute of Science Education and Research, Pune)

Title: A Chinese Remainder Theorem for Young Diagrams.

Abstract: Given a natural number 't' and a Young Diagram 'Y', there is a notion of a "remainder of Y upon division by t", called the t-core of Y. Let s,t be relatively prime, and consider the map taking a given st-core Y to the pair consisting of its s-core and t-core. The fibres of this map are infinite. More precisely, we have proven that the cardinality of the set of length k members of a given fibre is a quasipolynomial in k, of degree st-s-t. This is joint work with K. Seethalakshmi.

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