• Vishwambhar Pati

Articles written in Proceedings – Mathematical Sciences

• The Laplacian on algebraic threefolds with isolated singularities

We give a complete description of the induced Fubini-Study metric (up to quasi-isometry) in a neighbourhood of an isolated complex projective threefold singularity, by using a sufficiently high resolution of singularities. This is then used to prove the self-adjointness of the corresponding Laplacian acting on square integrable functions, on the non-compact smooth locus of complex projective threefolds with isolated singularities.

• Seiberg-Witten invariants — An expository account

The Seiberg-Witten monopole equations, and a new invariant for 4-manifolds which results from these equations, are introduced in this paper.

• The theorem of Kronheimer-Mrowka

A proof of the conjecture of Thom (algebraic curves in the complex projective plane minimize genus within their homology class) due to Kronheimer-Mrowka is presented. The proof uses the mod-2 Seiberg-Witten invariants.

• Unitary tridiagonalization inM(4, ⫳)

A question of interest in linear algebra is whether alln xn complex matrices can be unitarily tridiagonalized. The answer for alln ≠ 4 (affirmative or negative) has been known for a while, whereas the casen = 4 seems to have remained open. In this paper we settle then = 4 case in the affirmative. Some machinery from complex algebraic geometry needs to be used.

• Subanalytic bundles and tubular neighbourhoods of zero-loci

We introduce the natural and fairly general notion of a subanalytic bundle (with a finite dimensional vector spaceP of sections) on a subanalytic subsetX of a real analytic manifoldM, and prove that whenM is compact, there is a Baire subsetU of sections inP whose zero-loci inX have tubular neighbourhoods, homeomorphic to the restriction of the given bundle to these zero-loci.

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019