• Volume 104, Issue 3

August 1994,   pages  435-595

• 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.

• The Hoffman-Wielandt inequality in infinite dimensions

The Hoffman-Wielandt inequality for the distance between the eigenvalues of two normal matrices is extended to Hilbert-Schmidt operators. Analogues for other norms are obtained in a special case.

• Rigidity problem for lattices in solvable Lie groups

The paper concerns rigidity problem for lattices in simply connected solvable Lie groups. A lattice Γ⊂G is said to be rigid if for any isomorphism ϕ:Γ→Γ′ with another lattice Γ′⊂G there exists an automorphism$$\hat \phi :G \to G$$ which extends ϕ. An effective rigidity criterion is proved which generalizes well-known rigidity theorems due to Malcev and Saito. New examples of rigid and nonrigid lattices are constructed. In particular, we construct: a) rigid lattice Γ⊂G which is not Zariski dense in the adjoint representation ofG, b) Zariski dense lattice Γ⊂G which is not rigid, c) rigid virtually nilpotent lattice Γ in a solvable nonnilpotent Lie groupG.

• Some remarks on the Jacobian question

This revised version of Abhyankar's old lecture notes contains the original proof of the Galois case of then-variable Jacobian problem. They also contain proofs for some cases of the 2-variable Jacobian, including the two characteristic pairs case. In addition, proofs of some of the well-known formulas enunciated by Abhyankar are actually written down. These include the Taylor Resultant Formula and the Semigroup Conductor formula for plane curves. The notes are also meant to provide inspiration for applying the expansion theoretic techniques to the Jacobian problem.

• On polynomial isotopy of knot-types

We have proved that every knot-type ℝ↪ℝ3 can be uniquely represented by polynomials up to polynomial isotopy i.e. if two polynomial embeddings of ℝ in ℝ3 represent the same knot-type, then we can join them by polynomial embeddings.

• Row-reduction and invariants of Diophantine equations

To any Diophantine equation with integral coefficients we associate a finitely generated abelian group. The analysis of this group by row-reduction generally leads to simpler equations which are equivalent to the original but often dramatically easier to solve. This method of studying equations is useful over finite fields as well as over Q. Some applications and an example are discussed.

• Positive values of non-homogeneous indefinite quadratic forms of type (1,4)

Let Γr,n-r denote the infimum of all numbers Γ&gt;0 such that for any real indefinite quadraticQ inn variables of type (r, n−r), determinantD≠0 and real numbersc1,…,cn there exist (x1,…,xn)≡(c1,…,cn) (mod 1) satisfying$$0&lt; Q(x_1 ,...,x_n ) \leqslant (\Gamma \left| D \right|)^{1/n} .$$. All the values of Γr,3 are known except Γ1,4. It is shown that$$8 \leqslant \Gamma _{1,4} \leqslant 16.$$.

• Extended Kac-Akhiezer formulae and the Fredholm determinant of finite section Hilbert-Schmidt kernels

This paper deals with some results (known as Kac-Akhiezer formulae) on generalized Fredholm determinants for Hilbert-Schmidt operators onL2-spaces, available in the literature for convolution kernels on intervals. The Kac-Akhiezer formulae have been obtained for kernels, which are not necessarily of convolution nature and for domains in ℝn.

• A proof of Howard’s conjecture in homogeneous parallel shear flows

A rigorous mathematical proof of Howard's conjecture which states that the growth rate of an arbitrary unstable wave must approach zero, as the wave length decreases to zero, in the linear instability of nonviscous homogeneous parallel shear flows, is presented here for the first time under the restriction of the boundedness of the second derivative of the basic velocity field with respect to the vertical coordinate in the concerned flow domain.

• # Proceedings – Mathematical Sciences

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