Abstracts

Abstract: Self-adjoint unitaries on Hilbert spaces are known as symmetries. These symmetries have a very simple structure as they have their spectrum contained in {-1, 1}. In 1958 Halmos and Kakutani showed that every unitary on an infinite dimensional complex Hilbert space is a product of four symmetries. We show that in type II von Neumann algebras every unitary is a product of six symmetries. In this setting at least four symmetries are needed but we don’t know whether four are enough. This talk is based on a joint work with Soumyashant Nayak and P. Shankar.

Abstract: In this work numerical schemes are proposed for approximating the solutions, possibly measure-valued with concentration (delta shocks), for a class of non-strictly hyper-bolic systems. These systems are known to model physical phenomena such as the collision of clouds and dynamics of sticky particles, for example. The scheme is constructed by extending the theory of discontinuous flux for scalar conservation laws, to capture measure-valued solutions with concentration. The numerical approximations are shown to be entropy stable in the framework of Bouchut. Numerical performance of these schemes are presented to show their performance in one and two dimensions. This is a joint work with Aekta Aggarwal and Ganesh Vaidya.

Abstract: In this talk, “How do Scientists develop models?” and “How do Scientists obtain Mathematical models?” will be discussed. Using these basic concepts, the Bioheat equation, basically partial differential equations, will be modelled to solve a particular cancer treatment, namely minimally invasive cancer treatment. The aim of this bioheat model is to help interventional radiologists (IRs) to monitor the treatment online and decide the correct treatment beforehand. In the clinical environment, the total treatment of this minimal invasive cancer treatment takes between 20 minutes and 1 hour. Obtaining the numerical solution of these PDEs using a minimum of 1 million finite element mesh delays the decision making by IRs. Therefore, we developed a fast finite element bioheat solver to predict the ablation with the help of the GPU. Our solver produces the simulated lesion within 3 to 5 minutes for treatment duration of 26 minutes to 1 hour and predicted more accurate ablations for more than 90% of the cases.

Abstract: In PDEs, quite often it is become necessary to approximate the PDEs by a family of PDEs involving a parameter which may go to 0 or ∞. It can also happen the other way that the physical systems produces a family of PDEs and we may need to find the limiting PDE. It can be due to several reasons, for example due to the presence of multi-scales. Multi scales arise in many physical and industrial problems, and industrial constructions include very complicated structures. Homogenization is a branch of science where one try to understand the microscopic structures via a macroscopic medium by taking care of the various scales involved in the problem which appears through the solutions.

But mathematically these solutions lie in an infinite dimensional space like Sobolev space. The standard procedure is to obtain a-priori estimates which only produces weak convergence which in turn averages out in the limit (hence disappears) the interesting and relevant physical phenomena, namely the rapid oscillations and concentrations. So one need to understand the hidden, but lost oscillations (similarly concentrations) due to weak convergence to pass to the limit. There are several methods and a couple of recent, but powerful methods are two-scale method and method of unfolding operators. The main aim of this talk is to introduce the unfolding operators, its development by our group and its applications to homogenization problems. We do not present a particular problem and the details, rather we wish to go through some of the important developments in the last 10 years.

Abstract: Genuinely ramified maps are those dominant maps with surjective etale fundamental groups. We will discuss many bundle theoretic consequences of such maps.

Abstract: I will explain a result proved jointly with N. V. Trung in 2007 that esablishes a connection between mixed volumes of a lattice polytopes and Hilbert functions of multigraded algebras. This approach enables us to find an algorithm for computation of mixed volumes and mixed multiplicities of ideals. This algorithm has recently been implemented in Macaulay2. This is joint work with Kriti Goel, Vivek Mukundan and Sudeshna Roy.

Abstract: In this talk, we introduce a notion of a principal 2-bundle over a Lie groupoid. For such principal 2-bundles, we produce a short exact sequence of VB-groupoids, namely, the Atiyah sequence. Two notions of connection structures viz. strict connections and semi-strict connections on a principal 2-bundle arising respectively, from a retraction of the Atiyah sequence and a retraction up to a natural isomorphism will be introduced. An existence criterion for the connections on a principal 2-bundle over a proper, étale Lie groupoid will be discussed. Further we will study the action of the 2-group of gauge transformations on the category of strict and semi-strict connections.

This is a joint work with Adittya Chowdhury and P Koushik.