Excursions of random fields is an increasingly important topic within data analysis in medicine, cosmology, materials science, etc. In this talk, we will discuss some detailed results concerning their Betti numbers. Specifically, we shall consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local, boundedness, and decay rate conditions. We will discuss a way to model its excursion set using a Cech complex. For each Betti number of this complex, we shall then prove various limit theorems in different regimes based on how fast the window size and excursion level grow to infinity. These include asymptotic mean estimates, a vanishing to non-vanishing phase transition with a precise estimate of the transition threshold, and a weak law in the non-vanishing regime. We shall further see a Poisson and a central limit theorem close to the transition threshold. The expected vertex degree asymptotically vanishes in the regimes we shall deal with. This places all our above results in the so-called `sparse' regime. Our proofs combine tools from both extreme value theory and combinatorial topology. This is joint work with Sunder Ram Krishnan.

We will give proofs of Cellular approximation and then discuss fibrations and Blaker-Massey Homotopy Excision thorem.

Topological K-theory was one of the first instances of a generalized cohomology theory being used to successfully resolve classical problems involving very concrete objects like vector fields and division algebras. In this talk, we will briefly review some properties of vector bundles, introduce the complex K groups, and discuss some of their properties - including the all-important Bott periodicity theorem.

We will talk about the classifying theorem of principal G-bundles for a topological group G. For every group G, there is a classifying space BG so that the homotopy classes of maps from a space X to BG are in bijective correspondence with the set of isomorphism classes of principal G-bundles over X. We will be outlining the construction, due to Milnor, of a classifying space for any group G.

We will talk about relative homotopy groups, long exact sequence in homotopy and cellular approximation theorem.

Fractional calculus (FC) is witnessing rapid development in recent past. Due to its interdisciplinary nature, and applicability it has become an active area of research in Science and Engineering. Present talk deals with our work on fractional order dynamical systems (FODS), in particular on local stable manifold theorem for FODS. Further we talk on bifurcation analysis and chaos in the context of FODS. Finally we conjecture a generalization of Poincare-Bendixon for fractional systems.

Let $M$ be a compact Kahler manifold and $\Omega (M)$ the canonical $2$-form on $M$. When $M$ is projective $n$-space $\P^n(\C)$, $H^2(M,\C)$ is of dimension 1. It follows that for any Kahler metric on the projective space, the cohomology class $[\Omega (M)$ of the canonical $2$-form is a multiple of the (unique up to sign) of a generator of $H^2(M,\Z)$. It is immediate from this that if $M$ is a complex sub-manifold of $\P^n(\C)$ for some $n$, then for the Kahler metric on $M$ induced from one on $\P^n(\C)$, it is clear that $\Omega(M)] \in $\C \cdot H^2(M, Z)$. Kodaira's theorem is a converse to this fact: If a complex manifold $M$ admits a Kahler metric such that the class of $\Omega(M)$ is a multiple of an integral class, then $M$ can be embedded in some projective space. This result was conjectured by W V D Hodge.

Let $M$ be a compact Kahler manifold and $\Omega (M)$ the canonical $2$-form on $M$. When $M$ is projective $n$-space $\P^n(\C)$ , $H^2(M,\C)$ is of dimension 1. It follows that for any Kahler metric on the projective space, the cohomology class $[\Omega (M)$ of the canonical $2$-form is a multiple of the (unique up to sign) of a generator of $H^2(M,\Z)$. It is immediate from this that if $M$ is a complex sub-manifold of $\P^n(\C)$ for some $n$, then for the Kahler metric on $M$ induced from one on $\P^n(\C)$, it is clear that $[\Omega(M)] \in $\C \cdot H^2(M, Z)$. Kodaira's theorem is a converse to this fact: If a complex manifold $M$ admits a Kahler metric such that the class of $\Omega(M)$ is a multiple of an integral class, then $M$ can be embedded in some projective space. This result was conjectured by W V D Hodge.

Singularities occur naturally everywhere around us, may it be an eye of a cyclone where there is no wind at all, or the north pole where the different time zones converge. The purpose of this talk is to study these patterns mathematically. Hairy ball theorem precisely states that: An even dimensional sphere does not possess any continuous nowhere vanishing tangent vector field". The basic notions of tangent vector field, fundamental groups, some concepts of point set topology will be discussed (at least intuitively) and then a geometric proof of the theorem will be studied. It will be followed by a few applications in the end.

I will give introduce the basic ideas in homotopy theory, along the way state some classical theorems and if time permits some recent results. The talk will be expository and will have few or no proofs possibly.