IITG Mathematics Seminar Series

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Lecture Number:292
Title:Absence of the Riemann mapping theorem in higher dimensions
Speaker:Dr. Diganta Borah
Affiliation:IISER Pune
Date: 14th February, 2017 (Tuesday)
Time: 4:00 PM

Abstract: The Riemann mapping theorem states that any proper simply connected domain in the complex plane is holomorphically equivalent to the unit disc. H. Poincare discovered that this theorem fails spectacularly in higher dimensions. We will discuss a proof of this surprising phenomenon. This will be an elementary talk and should be accessible to anyone with some familiarity with basic complex analysis in one variable.

Lecture Number:291
Title:Studies of human microbiome and the role of statistics
Speaker:Dr. Siddhartha Mandal
Affiliation:Research Scientist, Public Health Foundation of India
Date: 25th January, 2017 (Wednesday)
Time: 4:00 PM

Abstract: Human microbiome plays a crucial role in health and diseases with recent research unravelling the myriad ways in which these associations are manifested. While traditional microbiology focused on microbial pathogens and few beneficial bacteria, advances in high-throughput technologies have presented evidence of a broader spectrum of functions in relation to chronic diseases, brain function and neurodevelopmental outcomes. Understanding factors regulating our microbiota and the impact of microbiota on health requires appropriate statistical methodology. Microbial communities associated with the human body sites are complex communities, with unknown interactions and functions. In this talk, we shall discuss some of the statistical designs and methodologies that are useful to study questions related to microbiome. Few major questions of interest in studies of microbiome may be "How do microbial communities cluster between different groups ?" or "Which microbial taxa are differentially abundant between these groups?". Data generated from microbial surveys are relative abundances of microbial taxa (which are themselves members of a phylogenetic tree), while the actual abundances are unobservable quantities. This results in compositional data, where observations on each subject are multivariate vectors belonging to a simplex.

To answer the first question, we shall explore the Unifrac distance (Lozupone et al. 2005), a phylogenetic distance based metric that is used to separate microbial communities according to groups. Existing approaches to detect differentially abundant microbes either discount the underlying compositional structure in the microbiome data or use inappropriate probability distributions including the multinomial and Dirichlet-Multinomial that may potentially increase false discovery rate. For the second question, we introduced a novel statistical framework called Analysis of Composition of Microbiomes (ANCOM, Mandal et al. 2015) that accounts for the underlying compositional structure in the data, and, unlike existing approaches, can compare the composition of microbiomes across populations. ANCOM makes no distributional assumptions, and is sufficiently general to enable easy adjustment for covariates. ANCOM also scales well to compare samples involving thousands of taxa. We shall illustrate these methodologies using publicly available microbial datasets in the human gut. In addition, we shall explore some of the recent developments in microbiome research and how these may be relevant in the case of Indian populations and related health problems.

Lecture Number:290
Title:Control of compressible Navier-Stokes system
Speaker:Dr. Debanjana Mitra
Affiliation:Post Doctoral Fellow, Department of Mathematics, Virginia Tech, Blacksburg, USA
Date: 23rd January, 2017 (Monday)
Time: 4:00 PM

Abstract: We consider the one dimensional compressible Navier-Stokes system near a constant steady state with the periodic boundary conditions. The linearized system around the constant steady state is a hyperbolic-parabolic coupled system. We discuss some of the properties of the linearized system and its spectrum. Next we study some controllability results of the system.

Lecture Number:289
Title:Numerical Solution for a Transient Incompressible Viscous Fluid Flow in a Cavity
Speaker:Prof. Dambaru Bhatta
Affiliation:Professor of Mathematics, School of Mathematical & Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, Texas
Date: 4th Jauary, 2017 (Wednesday)
Time: 11:30 AM

Abstract: We consider time dependent flow for a Newtonian, viscous and incompressible fluid in a rectangular cavity. The governing system consists of the conservation of mass and momentum equations. The momentum equations considered here are linearization of the Navier-Stokes Equations. We derive the weak formulation for the governing system. Using the Galerkin method, we obtain matrix form at element level. The elements considered here are of Taylor-Hood type elements. Time dependency part is solved by using the Crank-Nicolson method. Numerical results for the velocity for a square cavity are presented.

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