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Exotic Manifolds and Some Applications in Physics and Neuroscience

Indranil Mitra

New Alipore College, Kolkatta


Venue: Bhaktivedanta Institute, Juhu Road, Juhu, Mumbai
November 6, 2004

Abstract

Given two topological spaces, how do we know whether the spaces are equivalent or not? Alternatively, given two manifolds, how can we ascertain whether they are diffeomorphic or not? In general, the questions are highly nontrivial and can only be understood by a construction of a suitable map. Amidst all these it will be seen that D = 4 is a highly exceptional case in these regards.

The work of Donaldson et al has shown that in four dimensions there are uncountably many non-equivalent manifolds. This is the case of the cherished exotic manifolds. The existence of such exotic structures is evidently counterintuitive. It means that although each of these manifolds is topologically equivalent to R4 there is no local coordinate patch structure in which the global topological coordinates, ordered sets of four numbers are everywhere smooth.

In this talk we will be mainly concerned with discussing these novel properties of exotic manifolds, but as it can be understood the journey will obviously include developing the heavy mathematical machinery involving tools from algebra, algebraic topology, differential geometry, operator algebras, index theorems, K-theory and surgery theory.

As far as applications of these ideas are concerned, the original motivation of Donaldson from Nonabelian Gauge theory of SU(2) and moduli space of Yang mills Instantons studied in the context of Seiberg-Witten Theory has got good realization in this context.

So these issues will be touched upon depicting the importance of physics in the classification of manifolds and vice-versa. It is quite premature to try to develop these ideas in the areas of neuroscience but recent works of Amari have given rise to information geometry and open the flood gates of a geometrical exploration and thereby a clue to develop a cortical geometry and neuro-manifold structure inside the brain.

If the sensory experiences are mapped inside the brain to a surface with defined geometrical properties, then a well posed question to ask is whether there are any exotic properties associated with it. We investigate these questions and discuss some issues as to how geometry can improve our understanding of brain modeling.

About the Speaker

Indranil Mitra is a lecturer in physics at New Alipore College, Kolkatta. He is about to complete his Ph.D. in theoretical physics (string theory) from the Saha Institute of Nuclear Physics, Kolkatta.

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