Research Highlights of Basudeb Datta

2017: We show that every semi-equivelar map on the torus is a quotient of an Archimedean tiling on the plane. We also show that the number of Aut(X)-orbits of vertices for any semi-equivelar map X on the torus is at most six.

2017: We have proved that the Pachner graph of n-vertex flag 2-spheres distinct from the double cone is connected. In contrast, we proved that the Pachner graph of n-vertex stacked 2-spheres has at least as many connected components as there are trees of maximum degree at most 4 on [(n-5)/3] vertices.

2017: We have shown that a connected, closed, homology manifold of dimension d > 1 is stacked if and only if it is in the Walkup's class H ^d+1.

2017: We have proved that, for any field F, a triangulation of a closed 3-manifold is F-tight if and only if it is F-orientable, neighbourly and stacked.

2015: We have proved that the separation index of a triangulated 2-sphere is maximal if and only if it is stacked. We also proved that, amongst all n-vertex triangulated 2-spheres, the separation index is minimised by some flag sphere for n > 5.

2014: We have introduced the k-stellated spheres and studied the class W_k(d) of triangulated d-manifolds all whose vertex links are k-stellated. We proved that, when d is not 2k+1, all the (k+1)-neighbourly members of W_k(d) are tight.

2013: We have constructed an infinite series of tight triangulated manifolds. This is the first infinite series of examples of tight triangulated manifolds after 26 years.

2008: We have presented a self-contained combinatorial proof of the lower bound theorem for normal pseudomanifolds.

2008: We have proved that any triangulation of a non-simply connected closed manifold of dimension d > 2 requires at least 2d+3 vertices, and there is a unique such triangulation on 2d+3 vertices, triangulating a sphere-bundle over the circle.

2005: We have proved that any degree-regular triangulation of the torus is vertex-transitive. We have shown that there exists an n-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number > 8.

1998: We have introduced the notion of primitive pseudomanifolds and prove that all pseudomanifolds (without boundary) are built out of the primitive ones by a canonical procedure. This theory is used to explicitly determine and count all the pseudomanifolds of dimension d > 0 on at most d + 4 vertices.

1997: We have proved that the perimeter of any convex n-gons of diameter 1 is at most 2n×sin(p/2n). Equality is attained here if and only if n has an odd factor.

1994: We have shown that if an n-vertex combinatorial d-manifold M satisfies complementarity then d = 0, 2, 4, 8 or 16 with n = 3d/2 + 3 and |M| is a `manifold like a projective plane'.

1994: We have presented an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of CP².

1990: We have proved that S ^2p × S ^2q allows an almost complex structure if and only if (p, q) = (1, 1), (1, 2), (2, 1),(1,3), (3, 1), (3, 3).