Speaker: Bikram Bir, IIT Bombay

Date and Time (of the first lecture): Thursday 16 March 2023 at 02.30PM

Venue: Ramanujan Hall, Department of Mathematics.

Title: Oldroyd Model of Order One: Theory and Numerics

Abstract:  In these lectures, we discuss about a few finite element methods
for the equations of motion arising in the two-dimensional Oldroyd model
of order one; a model that represents linear viscoelastic fluid flows and
that can be viewed as an integral perturbation of the Navier-Stokes
equations.
First, we study existence and uniqueness of the continuous and weak
solution. Then, we discretize the space variable based on standard
Galerkin finite element method and the temporal variable based on
different time-discrete schemes. We next employ different finite element
methods like the two-grid method, the penalty method, the grad-div
stabilization method, the projection method and the nonconforming finite
element method. In all these cases, our main aim will be to obtain an
optimal error estimate for the velocity and the pressure. Finally, we
present some
numerical experiments to validate our theoretical findings.
 

Statistics and Probability  seminar

Thursday, 16 March 2023, 3 pm

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Venue: Online, link will be sent later

Host: Debraj das

Speaker: Sandipan Roy

Affiliation: University of Bath

Title: Statistical Inference in Complex Data with Network Structure

Abstract: New technological advancements have allowed the collection of datasets of large volume and different levels of complexity. Many of these datasets have an underlying network structure. Networks are capable of capturing dependence relationships among a group of entities and hence analyzing these datasets unearth the underlying structural dependence among the individuals. Examples include gene regulatory networks, understanding stock markets, protein-protein interaction within the cell, online social networks etc. We present two important aspects of large high-dimensional data with network structure. The first one focuses on a data with a network structure that evolves over time. Examples of such data sets include time course gene expression data, voting records of legislative bodies etc. Traditionally, the estimation of Gaussian graphical models (GGM) is performed in an i.i.d setting. More recently, such models have been extended to allow for changes in the distribution, but primarily where changepoints are known a priori. In this work, we study the Group-Fused Graphical Lasso (GFGL) which penalizes partial correlations with an L1 penalty while simultaneously inducing block-wise smoothness over time to detect multiple changepoints. The other aspect that we examine is heterogeneity in a network structure and how we can use such heterogenous features in a predictive model. We use a linear latent variable model viz. PCA and its extensions to learn an underlying network structure from data varying over time. We then employ the learned network as a feature in a predictive model to perform the downstream task in the test data. Neuroimaging-driven prediction of brain age, defined as the predicted biological age of a subject using only brain imaging data, is an exciting avenue of research. In this work, we seek to build models of brain age based on functional connectivity while prioritizing model interpretability and understanding. This way, the models serve to both provide accurate estimates of brain age as well as allow us to investigate changes in functional connectivity that occur during the aging process.

Commutative algebra seminar

Thursday, 16 March 2023, 4 pm

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Venue: Ramanujan Hall

Host: Tony Puthenpurakal

Speaker: Prof. R. V. Gurjar

Affiliation: Former Professor, IIT Bombay

Title: Positively Graded domains

Abstract:  I will continue my lectures on this topic. Following results will be discussed.  1. Demazure's construction of normal affine positively graded domains. Some applications of this will be discussed.  2. Flenner and Keiichi Watanabe's rationality of singularities criterion for positively graded affine domains.  3. A very general result I conjectured around 1990 and proved by O.Mathieu In 2002 will be discussed. It has some new consequences for rings of invariants of reductive algebraic group action on an affine space.  4. Divisor Class Groups of positively graded domains. Works of Brieskon Flenner, Samuel, Scheja-Storch, Anurag Singh, etc, will be mentioned. The connection with the Topology of these results will be discussed.

Geometry seminar

Thursday, 16th March 2023, 5 pm

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Venue: Ramanujan Hall

Host: Mayukh Mukherjee

Speaker: Mitul Islam

Affiliation: Heidelberg University

Title: Relatively hyperbolic groups and convex projective structures

Abstract: Studying discrete subgroups of linear groups using a preserved geometric structure has a long tradition. For instance, using real hyperbolic geometry to study discrete subgroups of SO(n,1). Convex projective structures, a generalization of real hyperbolic structures, has recently received much attention in the context of studying discrete subgroups of PGL(n). In this talk, I will discuss convex projective structures and discuss results (joint with A. Zimmer) on relatively hyperbolic groups that preserve convex projective structures. In particular, I will discuss a complete characterization of relative hyperbolicity in terms of the geometry of the projective structure.