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 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,wherer k(\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 PL4-manifolds satisfy these bounds. We also provide upper bounds for regular genus and gem-complexity of manifold bundles over circle. The bound for regular genus is attained by an orientable (resp., non-orientable) manifold bundle over circle when the manifold is S^n, RP^3, S^2×S^1 or g(S1×S1)×S1 for n,g \geq 1. In particular, we prove that there exists an orientable (S2×S1)-bundle over S^1 with regular genus 6. This disproves a conjecture of Spaggiari which states that regular genus six characterizes the topological product RP3×S1 among closed connected prime orientable 4-manifolds.

Among the mathematical development inspired by Arnold's conjecture, about the lower bound of fixed points of a symplectomorphism, Conley-Zehnder's work of 1983 is notable. Their idea gave rise to many further developments like the Floer homology. The proposed commentary will lead to the idea of the proof by Conley-Zehnder.

In this talk, we will try to understand Topology of a quotient of the complex Stiefel manifold, which we call $m$-projective Stiefel manifold. More precisely, we consider quotients of complex Stiefel manifolds by finite cyclic groups whose action is induced by the scalar multiplication on the corresponding complex vector space. We will obtain a description of their tangent bundles, compute their mod $p$ cohomology and obtain estimates for their span (with respect to their standard differentiable structure). We will discuss the Pontrjagin and Stiefel-Whitney classes of these manifolds and give applications to their stable parallelizability. This is part of joint work with P. Sankaran.

In this talk I will construct manifolds and orbifolds with quasitoric boundary. I'll show that these manifolds and orbifolds with boundary has a stable complex structure. These induce explicit (orbifold) complex cobordism relations among quasitoric manifolds and orbifolds. In particular, we show that a quasitoric orbifold is complex cobordant to some copies of fake weighted projective spaces. The famous problem of Hirzebruch is that which complex cobordism classes contain connected nonsingular algebraic varieties? I'll give some sufficient conditions to show when a complex cobordism class may contain an almost complex quasitoric manifold. Andrew Wilfong give some necessary condition of this problem up to dimension 8.

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 < y. The notion was introduced by H. Busemann, and the motivation comes from the theory of relativity in physics. In this talk, I will describe the basics of this theory and give some examples.

In this seminar, we will study the structure of linear algebraic groups over complex numbers and over real numbers. The first lecture will be elementary.

One of the most useful analogies in mathematics is the fundamental group functor (also known as the Galois Correspondence) which sends a topological space to its fundamental group while at the same time sending continuous maps between spaces to corresponding homomorphisms of groups in such a way that compositions of maps are preserved. A an obvious question one might ask is whether the fundamental group functor is "onto", that is: (1) is every group the fundamental group of a space and (2) every homomorphism the image of a continuous map between corresponding spaces? The easy answer to (1) is "yes" and the nonobvious answer to (2) is "it depends on the spaces". We'll introduce the harmonic archipelago as the shining example of a space with a strange fundamental group, define an archipelago of groups as a group theoretic product operation and finally describe how such products are (almost) all isomorphic to the fundamental group of the harmonic archipelago. We will study examples showing that there are group homomorphisms that cannot be induced by continuous maps on certain spaces and how the fundamental group of the harmonic archipelago factors through all such "discontinuous homomorphisms", how none of the examples is constructible (or even understandable in any reasonable way) and how one might detect spaces whose fundamental group allows them to be the codomain of such weird homomorphisms (the conjecture is that they contain the rational numbers or torsion). We'll talk a bit about the notion of cotorsion groups from classical Abelian group theory and how that notion can be generalized to non-Abelian groups by requiring certain types of systems of equations have solutions and then mention how countable groups which have solutions to such systems are always images of the fundamental group of an archipelago. In the end we're lead to an example of a countable group which we can prove is either the rational numbers or gives a counterexample to a nearly 50 year old conjecture in number theory: the Kurepa conjecture. So there is a little topology, a little homotopy theory, some group theory, a pinch of logic and a wisp of number theory in the talk. This is a distillation of work I've published recently with Hojka and Meilstrup (Proc AMS) and work that is still being written up with Hojka and Herfort.

In 1986, Margulis, using methods from dynamics, proved A. Oppenheim's 1929 conjecture that for every indefinite irrational quadratic form in at least three variables, the values it takes at integer lattice points form a dense subset of the real line. Subsequently, Eskin-Margulis-Mozes proved an associated counting result, giving polynomial asymptotics for the number of lattice points of norm at most $T$ which get mapped to a fixed interval. In joint work with Margulis, we give an effective version if this result for almost every quadratic form.