he main result of this colloquium is the equality of the number of K-rational points with the signature of the trace form of a finite K-algebra over a real closed field K. The main tools are symmetric bilinear forms, Hermitian forms, trace forms, generalized trace forms and their types and signatures. Further, we prove a criterion for the existence of K-rational points by using generalized trace forms. As an application we prove the Pederson-Roy-Szpirglas theorem about counting common real zeros of real polynomial equations.

In this seminar, I will talk about the classification of tuples of commuting matrices over a finite field, upto simultaneous conjugation.

In a few lectures, I will introduce some of the main problems in Algebraic Complexity theory and some of the techniques that have been used to make progress on them. The techniques are closely related to Hilbert functions and Young flattenings.

The task of manipulating randomness has been a subject of intense investigation in computational complexity with dispersers, extractors, pseudorandom generators, condensers, mergers being just a few of the objects of interest. All these tasks consider a single processor massaging random samples from an unknown source. In this talk I will talk about a less studied setting where randomness is distributed among different players who would like to convert this randomness to others forms with relatively little communication. For instance players may be given access to a source of biased correlated bits, and their goal may be to get a common random bit out of this source. Even in the setting where the source is known this can lead to some interesting questions that have been explored since the 70s with striking constructions and some suprisingly hard questions. After giving some background, I will describe a recent work which explores the task of extracting common randomness from correlated sources with bounds on the number of rounds of interaction. Based on joint work with Mitali Bafna (Harvard), Badih Ghazi (Google) and Noah Golowich (Harvard).

In a few lectures, I will introduce some of the main problems in Algebraic Complexity theory and some of the techniques that have been used to make progress on them. The techniques are closely related to Hilbert functions and Young flattenings.

In a few lectures, I will introduce some of the main problems in Algebraic Complexity theory and some of the techniques that have been used to make progress on them. The techniques are closely related to Hilbert functions and Young flattenings.

: We have considered the problem of making statistical inferences for different lifetime models on the basis of progressive type-II censored samples. In particular, we have derived various estimates of parameters using both classical and Bayes methods. The associated MLEs are computed using the EM algorithm. We also compute the ob- served Fisher information matrices and based on these computations, the asymptotic confidence intervals of parameters are constructed. Bootstrap intervals are also dis- cussed. We also derive Bayesian estimates of parameters against different loss func- tions. Most of these estimates appear in analytically intractable forms and so we have used different approximation methods like importance sampling, Lindley, Tier- ney and Kadane procedures to compute the Bayes estimates. In sequel, we have also constructed highest posterior density intervals of parameters. We have also derived predictive inference for censored observations under frequentist and Bayesian frame- works. In particular, we obtain best unbiased predictor, conditional median predictor from frequentist perspective. Among prediction intervals, we construct pivotal in- terval, highest conditional density interval, equal tail interval and HPD interval for future observations. Determination of optimal plans is one of the primary objective in many life test studies. We have obtained such plans again using both frequentist and Bayesian approaches under progressive censoring. We also consider estimation of multicomponent stress-strength reliability under progressive censoring. We have numerically compared the proposed methods using simulations for each problem. We have also discussed real life examples in support of studied methods. We have provided relevant information in each chapter of the thesis.

Let L be a cyclic extension of a number field K. Hasse’s theorem says that an element of K is a norm from L if it is a norm locally at all completions of K. Examples of failure of similar local global principle if L is not cyclic were also known. We survey recent results on obstructions to local global principle for norm equations over number fields.

The Theory for vector bundles in topology shaped the research in projective modules in algebra, consistently. This includes Obstruction Theory. The algebra has always been trying to catch up. To an extent, this fact remained under appreciated. For an affine scheme $X=\spec{A}$, and a projective $A$-module $P$, our objective would be to define an obstruction class $\varepsilon(P)$ in a suitable obstruction house (preferably a group), so the triviality of $\varepsilon(P)$ would imply $P \equiv Q \oplus A$. One would further hope the obstruction house is an invariant of $X$; not of $P$. We would report on what is doable. We detect splitting $P \equiv Q \oplus A$ by homotopy.

Let K be a field of characteristic zero and A_n(K) be the nth-Weyl algebra over K. In this talk, we discuss strongly generalised Eulerian $A_n(K)$-modules and their properties. We prove that if M is a strongly generalized Eulerian $A_n(K)$-module, then so is the graded Matlis dual of M. We also prove that Ext functor of strongly generalized Eulerian modules is strongly generalized Eulerian $A_n(K)$-module. As a consequence, we prove the following conjecture: Let M and N be non-zero, left, holonomic, graded generalized Eulerian $A_n(K)$-modules. Then the graded K-vector space $Ext^i_{A_n(K)}(M, N)$ is concentrated in degree zero for any i >=0.