Apart from the other aspects of the theory, we will try to touch on Fourier theory over finite fields and some applications.

In the first half of the seventeenth century, the French mathematician Fermat gave a simple criterion to know when a number can be written as a sum of two squares, thus answering a question raised by the Greek mathematician Diophantus. Since then, it has occupied a central place in mathematics, involving almost all disciplines within it and with contributions from Euler, Legendre, Gauss, Dirichlet, Kummer, Kronecker, Galois, Hilbert, Artin, Langlands, Ramanujan, Hecke, Harish-Chandra, Serre, Tate, Drinfeld amongst many others. I will try to touch upon some of the central themes giving shape to our modern understanding of the reciprocity laws in a language accessible also to non-mathematicians.

Why it is important to point out the most obvious things in the most systematic manner to check for truth. Here is a very attractive probability problem that questions simple, Yet for the answer a saga of tales , paradoxes and confusion ensues .

In this talk, we discuss some well-known examples of fractals and introduce the notion of fractional dimension. We first look at the definition of the Cantor set and prove some of its properties. We then take a brief look at Koch curve and Sierpi ́nski triangle.

We begin by presenting a spectral characterization theorem that settles Chevreau's problem of characterizing the class of absolutely norming operators --- operators that attain their norm on every closed subspace. We next extend the concept of absolutely norming operators to several particular (symmetric) norms and characterize these sets. In particular, we single out three (families of) norms on $\mathcal B(\mathcal H, \mathcal K)$: the ``Ky Fan $k$-norm(s)", ``the weighted Ky Fan $\pi, k$-norm(s)", and the ``$(p,k)$-singular norm(s)", and thereafter define and characterize the set of absolutely norming operators with respect to each of these three norms.

We propose and analyse finite volume Godunov type methods based on discontinuous flux for a 2 ×2 system of non-linear partial differential equations proposed by Hadeler and Kuttler to model the dynamics of growing sand piles generated by a vertical source on a flat bounded rectangular table. The scheme is made well-balanced by modifying the flux function locally by including source term as a part of the convection term. Its extension to multi-dimensions is not straightforward for which an approach has been introduced here based on Transport Rays. Numerical experiments are presented to illustrate the efficiency of the proposed scheme for steady state calculations.

Once we have a new mathematical structure (Lie algebra) in hand, it is natural to ask, can we classify them upto isomorphism? To be more specific, can we classify all finite dimensional complex Lie algebras g upto isomorphism? But then the next question is how to approach the problem. As Lie algebras are vector spaces, we may look at the dimension for such a classification. For each non-negative integer n, there is a unique abelian Lie algebra upto isomorphism (abelian Lie algebras are well understood). For non-abelian Lie algebras, we will further look at their derived subalgebras and centre as they are preserved under Lie algebra isomorphism. Using these ideas, we will classify Lie algebras of dimension upto 3.

We begin by a rough motivation on why to study Lie algebras. Then we mainly focus on concrete examples of Lie algebras, namely the general linear Lie algebra gl(n). We discuss how to construct new ones with a given Lie algebra g, i.e. subalgebras, ideals, and quotients of Lie algebras. The general linear Lie algebra serves as the building block example of Lie algebras as any abstract Lie algebra g is sitting inside gl(n) as subalgebra. Finally, we introduce the structure preserving maps (homomorphisms) between Lie algebras.

Sarnak's conjecture links dynamical systems with arithmetic’s: it asserts that every sequence produced by a zero-entropy topological dynamical system is (asymptotically) orthogonal to the Mobius function; After presenting the different objects involved, and discussing on the validity of the conjecture, I shall give a survey on recent results, including personal contributions, and focusing on two famous families of dynamical systems, interval exchanges and substitutions.

1. Background information related to Minkowski's bound. 2. Statement of Minkowski's bound and its application (e.g-to show class number is finite, evaluate class group of some quadratic fields). 3. Proof of theorem related to Minkowski's bound I shall give a survey on recent results, including personal contributions, and focusing on two famous families of dynamical systems, interval exchanges and substitutions.

In this talk, we will present the existence of nice patterns in some subsets of the natural numbers N. In this context, we shall disuss additive and multiplicative patterns in syndetic sets and the set of sparsely totient numbers. Details of these subsets are the following: (i) Syndetic set: A subset S of the natural numbers is called syndetic set if there exists l ∈ N such that S intersects every set of l consecutive natural numbers. In 2006, Beiglb¨ock, Bergelson, Hindman and Strauss posed an open question in relation to syndetic sets as to whether or not every syndetic set contains arbitrarily long geometric progressions. Even two term geometric progression with integer common ratio in a syndetic set is not entirely understood. We will discuss the presence of two term geometric progression in syndetic set where common ratio of progression is an integer. (ii) Sparsely totient number: A positive integer N is called sparsely totient number if φ(y) > φ(N) for every y ∈ N with y > N where φ is Euler’s totient function. In this talk, we will discuss some infinite families of sparsely totient number and the presence of additive and multiplicative patterns in the set of sparsely totient numbers.

In this talk, I shall talk about some symplectic and holomorphic aspects of moduli spaces of finite dimensional semistable representations of finite quivers. Namely, I shall describe the construction of a natural Hermitian holomrphic line bundle on the stratified moduli space of semistable representations, which has the property that the Chern form of this line bundle on each stratum of the moduli space is a scalar multiple of the Kahler form of the stratum. I shall then define the tensor products of two quivers and of their representations, and discuss their semistability. Moreover, I shall also give a relation among the natural line bundles on the moduli spaces of these quivers.

Let L be a line bundle of degree d on an elliptic curve C and ϕ : C → P n is a morphism given by a sub-linear system of the complete linear system |L| of dimension n + 1. When d = 4, n = 2, we prove that ϕ ∗TPn is semi-stable if deg(ϕ(C)) > 1. Moreover, we prove that ϕ ∗TPn is isomorphic to direct sum of two isomorphic line bundles if and only if deg(ϕ(C)) = 2. Conversely, for any rank two semi-stable vector bundle E on an elliptic curve C of degree 4, there is a non-degenerate morphism ϕ :C → P n such that ϕ ∗TPn (−1) = E. More precisely, E is isomorphic to direct sum of two isomorphic line bundles if and only if deg(ϕ(C)) = 2. Further E is either indecomposable or direct sum of non-isomorphic line bundles if and only if deg(ϕ(C)) = 4. When d = 5, n = 3, we compute the Harder-Narasimhan filtration of ϕ ∗TP.

In this talk, I am going to discuss about the ever-evolving advances in thin-film materials and devices, which have fueled many of the developments in the field of flexible and stretchable electronics. After the discovery of conducting polymers, the question arose as to whether organic materials would also find applications as organic semiconductors. Today, owing to the constant improvements of the particular properties of molecular materials – including the synthesis of new compounds – organic semiconductors have made their way for the fabrication of devices, which are flexible in nature. On the process, there have been substantial development and optimization over many decades on the growth of these materials on hard and flexible substrates as thin-films and fabrication of various devices with different functionalities by exploiting their easily tunable electronic properties (e.g. organic field-effect transistors, OFETs) and optical properties (e.g. organic light emitting diodes, OLEDs). I will discuss about the issues related to the growth of organic thin films and underlying growth mechanisms in comparison with the growth of inorganic semiconducting materials. Different strategies will be discussed as to how one can make these films mechanically flexible and integrate it into electronic devices. Various materials will be introduced as candidate of flexible electronics suitable for various applications. Effort will be made to understand the fundamental properties of their flexibility at the molecular level. At the end, I am going to introduce the latest developments on flexible and stretchable electronics and their potential in the field in wearable healthcare applications including some of the latest results came out from our laboratory at IIT Kharagpur.

This talk will develop three axes: first, we will show that prime number.

The FTIC is the most basic and useful theorem in Riemann Integration in one dimension. Its reformulation in two or more dimensions leads to many interesting consequences . A further reformulation allows many far-reaching applications .