The compact torus S^1�S^1 has a structure of Riemann surface and therefore is a complex projective manifold. On product of odd dimensional spheres S^{2p+1}�S^{2q+1 with p > 0 or q > 0, complex structures were obtained by H. Hopf (1948) and Calabi-Eckmann (1953). These complex manifolds are one of the first examples of non-K ?ahler, and hence non-projective, compact complex manifolds. The aim of this talk is to describe construction of a new class of non-Kahler compact complex manifolds. Let K be an even dimensional compact Lie group and G be its universal complexification. We will show that if a smooth K -principal bundle EK---->M over complex manifold M, is obtained after reduction of the structure group of a holomorphic G-principal bundle EG---->M, then the total space EK admits a complex structure. If K is non-abelian then the complex manifold EK will be non-Kahler. In certain special cases we will discuss Picard group, deformation and algebraic dimension of EK.This talk is based on an ongoing joint work with Mainak Poddar.

We will define principal bundles, connections and curvature. With the basics defined, we will construct the Chern-Weil homomorphism. Let E be a principal G-bundle on M with a connection D. Let F be the curvature of D, and g=Lie(G). The Chern-Weil homomorphism associates to each Ad-invariant polynomial on g, a well defined cohomology class in the de Rham cohomology H_{dR}^*(M). Let P be an Ad-invariant homogeneous polynomial of degree k on g. The Chern-Weil image of P is given by the closed 2k-form P(F^{2k}). Its class in H^{2k}_{dR}(M) does not depend on the choice of the connection. This class is functorial. We will conclude with a few examples.

We will define principal bundles, connections and curvature. With the basics defined, we will construct the Chern-Weil homomorphism. Let E be a principal G-bundle on M with a connection D. Let F be the curvature of D, and g=Lie(G). The Chern-Weil homomorphism associates to each Ad-invariant polynomial on g, a well defined cohomology class in the de Rham cohomology H_{dR}^*(M). Let P be an Ad-invariant homogeneous polynomial of degree k on g. The Chern-Weil image of P is given by the closed 2k-form P(F^{2k}). Its class in H^{2k}_{dR}(M) does not depend on the choice of the connection. This class is functorial. We will conclude with a few examples.

We shall indicate a solution to a problem posed by Euclid: Prove or disprove the parallel postulate assuming only the first 4 axioms of Euclidean geometry. The solution leads to a new geometry called hyperbolic geometry. We shall only assume some familiarity with calculus.

Within crystallization theory, two interesting PL invariants for d manifolds have been introduced and studied, namely gem-complexity and regular genus. In this talk, we prove that, for any closed connected PL 4-manifold M, its gem-complexity k(M) and its regular genus G(M) satisfy: k(M) >= 3\Chi(M)+10m-6 and G(M)>=2\Chi(M)+5m-4, where rk(Pi_1(M))=m. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of PL 4-manifolds and a large class of PL 4-manifolds satisfy these bounds. We also provide upper bounds for regular genus and gem-complexity of manifold bundles over circle.

The goal of this talk will be to prove Thom�s theorem that the oriented cobordism ring when tensored with rational numbers is a polynomial algebra over Q generated by the cobordism classes of complex projective spaces.

Time-like metric spaces are topological spaces equipped with an order relation < and a distance function, where the distance from x and y is defined only when x

A question of Stillman asks: Is there a bound for the projective dimension of every homogeneous ideal generated in a polynomial ring, which depends solely on the knowledge of the degrees of the generators, and not on the number of variables? In joint work with A. Banerjee, we show that for every fixed degree sequence, the correctness of a conjectural bound can be proved by an implementable algorithm.

The goal of this talk will be to prove Thom�s theorem that the oriented cobordism ring when tensored with rational numbers is a polynomial algebra over Q generated by the cobordism classes of complex projective spaces.