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.

Perhaps surprisingly, the study of degenerate curves plays a crucial role in our understanding of a general smooth curve. One of the first successes of this idea was the theory of limit linear series developed by Griffiths and Harris which they used to prove the Brill-Noether theorem. The analogous theory for degenerate curves of non-compact type falls in the realm of tropical geometry where it takes the shape of metric graphs (or tropical curves) and divisors on them. This leads to a rich interplay between graph theory and algebraic geometry of curves. After explaining the central ideas we will discuss some applications to Brill-Noether theory and curves of large theta Characteristic.

We continue the study of resolutions of monomial ideals. We start with a short proof of the exactness of the Koszul complex. We then generalize this to free resolutions of any monomial ideal. We'll conclude with the proof of the Hilbert syzygy theorem and some more examples of monomial ideals.

A Higgs bundle on a compact Kahler manifold M consists of a holomorphic vector bundle E together with a holomorphic 1-form with values in End(E), say \phi, such that \phi^\phi = 0 as a 2-form with values in End(E). It turns out that there is a one to one correspondence between irreducible representations of fundamental group of M and stable Higgs bundles on M with vanishing Chern classes. This can be seen as the analogue of the Narasimhan-Seshadri theorem connecting irreducible unitary representations of the fundamental group with stable, flat vector bundles.

Huneke and Itoh independently proved a celebrated result on integral closure of powers of an ideal generated by a regular sequence. As a consequence of this theorem, one can find the Hilbert-Samuel polynomial of the integral closure filtration of I if the normal reduction number is at most 2. We prove Hong and Ulrich's version of the intersection theorem.

Consider a graded ideal in the polynomial ring in several variables. We shall discuss criterion for the graded ideal and its power to have linear resolution. Then we focus our attention to study linear resolution of monomial ideals. Monomial ideals are the bridge between commutative algebra and the combinatorics. Monomial ideals are also significant because they appear as initial ideals of arbitrary ideals. Since many properties of an initial ideal are inherited by its original ideal, one often adopt this strategy to decipher properties of general ideals. The first talk is meant for covering the preliminary results on resolution and regularity of monomial ideal. The aim of this series of talk is to present the result in ArXiv:1709.05055 .

We will study free resolutions of monomial ideals via the concept of labelled simplicial complexes due to Bayer, Peeva and Sturmfels. We will derive a criterion for a labelled complex to define a free resolution. As applications, we will obtain the exactness of the Koszul complex and the Hilbert syzygy theorem. If time permits, we will obtain a formula for Betti numbers of a monomial ideal in terms of a corresponding labelled simplicial complex.

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.

There are many situations where one expects an ordering among K>2 experimental groups or treatments. Although there is a large body of literature dealing with the analysis under order restrictions, surprisingly very little work has been done in the context of the design of experiments. Here, we provide some key observations and fundamental ideas which can be used as a guide for designing experiments when an ordering among the groups is known in advance. Designs maximizing power as well as designs based on single and multiple contrasts are discussed. The theoretical findings are supplemented by numerical illustrations.

Recurrence relations of moments which are useful to reduce the amount of direct computations quite considerably and usefully express the higher order moments of order statistics in terms of the lower order moments and hence make the evaluation of higher order moments easy. We have derived recurrence relations for single, double (product) and higher moments of various ordered random variables, like ordinary order statistics, progressively censored order statistics, generalized order statistics and dual generalized order statistics from some specific continuous distributions. It also deals with L-moments and TL-moments which are analogous of the ordinary moments. We have derived L-moments and TL-moments for some continuous distributions. These results have been applied to find the L-moment estimators and TL-moment estimators of the unknown parameters for some specific continuous distributions.