Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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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.

Probability Seminar - Jamie Mingo (Queen's University)

Monday, January 21st, 2019

Time: 10:00-11:30 a.m.  Place: Jeffery Hall 319

Speaker: Jamie Mingo (Queen's University)

Title: Tensor Products of Modules and Vector Spaces.

Abstract:  This talk will review the construction of the tensor product of modules and its universal property. This property can be used to establish associativity, commutativity over direct sums, and a check for linear independence when our modules are vector spaces. I also hope to give a brief introduction to tensor norms.

Free Probability and Random Matrices Seminar Webpage:

Special Colloquium - Michelle Miranda (University of Texas)

Michelle Miranda (University of Texas)

Monday, January 21st, 2019

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

Speaker: Michelle Miranda (University of Texas)

Title: Modeling Modern Data Objects: Statistical Methods for Ultra-high Dimensionality and Intricate Correlation Structures.

Abstract: Advances in technology have been generating data with increased complexity. Modern data objects are often high-dimensional and can also lay in 2D, 3D and even 4D Euclidean and sometimes non-Euclidean planes. One example are data generated from large-scale multi-site studies such as the Human Connectome Project and the Alzheimer's Disease Neuroimaging Initiative. In these scenarios, complexity comes in the form of brain images such as functional magnetic resonance imaging (fMRI), a 4D object that is high-dimensional (number of voxels is around 100K for each time point and each subject), with measurements that are correlated both in time and in space. Another example are data coming from an instrument specifically designed to measure scleral strain in donor eyes. The device generates functional data with functions defined on a non-Euclidean 2D partial spherical domain. Associating these complex data with clinical, environmental, and genetic variables can be challenging and classical statistical tools need to evolve to keep up with the complexity of these new data types. In this talk I address some of the challenges brought by modern data objects and show a few solutions to some important statistical questions in this context.

Michelle Miranda obtained her Ph.D. degree at the University of North Carolina at Chapel Hill in 2014. She is currently a Postdoctoral Fellow in Biostatistics at The University of Texas MD Anderson Cancer Center. Her research interests include Bayesian analysis with focus on high-dimensional settings, functional data analysis, and methods for correlated data objects.

Department Colloquium - Lee Mosher (Rutgers University)

Lee Mosher (Rutgers University)

Friday, January 18th, 2019

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

Speaker: Lee Mosher (Rutgers University)

Title: The automorphism group of a rank 2 free group, and other matters.

Abstract: As a lead up to discussing current research into the geometry of the automorphism and outer automorphisms groups of finite rank free groups, I’ll examine in some detail the geometry of the automorphism group of a rank 2 free group.

Professor Lee Mosher completed his PhD in Princeton in 1983 under the direction of Bill Thurston. He then went on to Harvard, and was a member of the IAS, before accepting a position at Rutgers University, where he is now a Distinguished Professor. Although trained as a topologist, Prof.~Mosher's broad research interests also cover geometry, geometric group theory and dynamical systems. His work has been published in the top mathematical journals, including Annals of Mathematics, Acta Mathematica and Inventiones, and has been continuously supported by the NSF since 1990.

Special Colloquium - Long Feng (Yale)

Long Feng (Yale)

Wednesday, January 16th, 2019

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Long Feng (Yale)

Title: Sorted Concave Penalized Regression.

Abstract: The Lasso is biased. Concave penalized lease squares estimation (PLSE) takes advantage of signal strength to reduce this bias, leading to sharper error bounds in prediction, coefficient estimation and variable selection. For prediction and estimation, the bias of the Lasso can be also reduced by taking a smaller penalty level than what selection consistency requires, but such smaller penalty level depends on the sparsity of the true coefficient vector. The sorted L1 penalized estimation (Slope) was proposed for adaptation to such smaller penalty levels. However, the advantages of concave PLSE and Slope do not subsume each other. We propose sorted concave penalized estimation to combine the advantages of concave and sorted penalizations. We prove that sorted concave penalties adaptively choose the smaller penalty level and at the same time benefits from signal strength, especially when a significant proportion of signals are stronger than the corresponding adaptively selected penalty levels. A local convex approximation, which extends the local linear and quadratic approximations to sorted concave penalties, is developed to facilitate the computation of sorted concave PLSE and proven to possess desired prediction and estimation error bounds. We carry out a unified treatment of penalty functions in a general optimization setting, including the penalty levels and concavity of the above mentioned sorted penalties and mixed penalties motivated by Bayesian considerations. Our analysis of prediction and estimation errors requires the restricted eigenvalue condition on the design, not beyond, and provides selection consistency under a required minimum signal strength condition in addition. Thus, our results also sharpens existing results on concave PLSE by removing the upper sparse eigenvalue component of the sparse Riesz condition.

Long Feng obtained his Ph.D. degree at the Department of Statistics and Biostatistics, Rutgers University, in 2017. He is currently a Postdoctoral Associate at Yale University. His research interests include high-dimensional statistics, variable selection, empirical Bayes methods, tensor regression, imaging data analysis, and convex/non-convex optimizations.

Number Theory - Siddhi Pathak (Queen's University)

Tuesday, January 15th, 2019

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

Speaker: Siddhi Pathak (Queen's University)

Title: Dedekind zeta function at odd positive integers.

Abstract: Let $\zeta(s)$ denote the Riemann zeta-function. Thanks to Euler's evaluation and Lindemann's theorem on transcendence of $\pi$, we understand that $\zeta(2k)$ is transcendental for any positive integer $k$. However, the arithmetic nature of the values of $\zeta(s)$ at odd positive integers remains a mystery. Recently, significant progress was made concerning the irrationality of these values, with perhaps the most remarkable theorem being that infinitely many of $\zeta(2k+1)$ are irrational, which was shown by T. Rivoal in 2000.

Similarly, one can inquire regarding the arithmetic nature of values of the Dedekind zeta-function $\zeta_K(s)$ attached to a number field $K$. When $K$ is totally real, the values $\zeta_K(2k)$ were proven to be algebraic multiples of powers of $\pi$ by Siegel and Klingen, independently. But this question remains unsolved in all other cases. In this talk, we discuss how our current knowledge allows us to deduce certain irrationality results for $\zeta_K(2k+1)$, where $K$ is an imaginary quadratic field. This is joint work with Prof. M. Ram Murty.