Our project, dealing with the study of singular MASAs in von Neumann algebras,addresses two main objectives- 1) the global study of MASAs and their generic properties in a given von Neumann algebra, and 2) the investigation of explicit MASAs in quantum group von Neumann algebras, possibly including type III examples. As usual, in Pure Mathematics, we will attack both objectives simultaneously. We expect indeed cross-fertilization between the research to be conducted towards each of these objectives. For each objective, we have identified different tasks/activities, of which some can be attacked immediately.

We investigate certain analytic and geometric aspects of quantization. This deals with establishing a mathematically precise correspondence between quantum, classical and scattering resonances in uniform ways on spaces admitting large (non-compact Lie) groups of symmetries.

We study of completely positive maps and completely bounded maps at different contexts. It includes the investigation of some of the mathematical aspect of Quantum Channels, Gaussian Channels, Semigroup theory of completely bounded maps etc.

• Research article in international quality journals.
• Presentations and seminars.
• Highly trained graduate students and postdocs.

Research Articles:

• Mapping cones of k-entanglement breaking maps, R. Devendra, Nirupama Mallick,K. Sumesh
• On bounded co-ordinates in GNS spaces, K. Mukherjee
• Further investigation is on.

International Collaboratiors

• Jan Cameron, Vassar College, New York.
• Jon Bannon, Siena College, New York.
• Ken Dykema, Texas A&M.
• Roger Smith, Texas A&M.
• Pierre Fima, University of Paris 7.
• Francois Le Maitre, University of Paris 7.

As pure mathematicians, the only societal impact locally with respect to time the CoE can make is to generate and share knowledge and create would-be flag bearers in the subject for India to carry on. Without a strong school in the subjects depicted above, a nuclear-armed nation is unheard of and inconsistent. The audacity of our hope attempts to sow the seeds.

Sustenance statement

We propose a centre of learning and research continuing continuously. Promising research in Mathematics never happens without researchers always training and updating themselves. The research will continue with whatever funds we can manage from NBHM, DST, under various schemes. We will also try to obtain CEFIPRA, Fullbright, Indo-US, DFGn and other external funds by ourselves and our collaborators in India rotating as PIs. We do believe that the support we require beyond the stipulated times depends on two components:

1. our hard work and
2. the general attitude of the public sector. The former is in our hands, but for the later, we would make a point.

New work done in the project

The class of k-entanglement breaking maps was recently introduced in the context of quantum information theory. We studied the mapping cone structure of k-entanglement breaking maps. We established various characterization of k-entanglement breaking maps. We also characterized completely positive which strictly reduce the Schmidt-number of non-separable quantum states. We studied many concrete examples, determine parameter regions where these maps are k-entanglement breaking, and discussed some of their significance in separability of quantum states. Further extended a spectral majorization result for separable states.

On a current study we studied the C*-convex set of entanglement breaking maps and give a complete description of C*-extreme points.

Infrastructure developments

Output

1. Mapping cone of k-entanglement breaking maps, Preprint 2021, (Jointly with R. Devendra and N. Mallick).

2. C^*-extreme points of entanglement breaking maps, Preperint 2022, (Jointly with BVR Bhat, R. Devendra and N. Mallick)

Visits planned for PI, co-PIs, international collaborators and students (both inbound and outbound)

Industrial Engagement

University Engagement

Relevant Updates

Jointly with T.C John and R. Sengupta started a new project on Gaussian channels, and is under progress.