The course is aimed for the Lecturers, students, analysts and researchers who have already undergone a first course in Functional Analysis, Complex Analysis and Analysis of single and Multivariable functions. We propose the following topics which are supposed to be covered in this workshop.
Ordered vector spaces and vector lattices, Various norms related to these spaces, Extension and decomposition of linear maps, Proof of Kakutani theorem for M-spaces.(Will be covered by Dr. Anil Karn)
Weak, weak*-topologies on Banach spaces, Banach Alaoglu theorem and applications to reflexivity/separability of Banach spaces. Applications of Hahn Banach theorem: Markov Kakutani theorem, Banach limits. Lipschitz functions on Banach spaces and their extensions, Lipschitz retracts and nonlinear geometry of Banach spaces.(Will be covered by Dr. Amin Sofi)
Two lectures on basic notions of Convexity and Smoothness in Normed linear spaces will be given. Various generalizations of Convexity and Smoothness.(Will be covered by Dr.Tanmoy Paul)
Three lectures will be given on the development of basic theory of vector measures leading to the celebrated Bartle, Dunford and Schwartz representation theorem of weakly compact operators whose domain is the space of continuous functions on a compact space. This classical result is an extension of the well known Riesz representation theorem. The aim is also to discuss the recent result of Rao and Roy that extends Bartle-Dunford and Schwartz's theorem to domains whose duals are isometric to L1(μ) space.(Will be covered by Dr. TSSRK Rao)
It is quite well known that in his classic work P. Korovkin very skillfully used positivity of linear maps to unify many classical approximation processes of scalar functions. Later on W. Arveson extended a part of it to operator algebras, where complete positivity has been used to deal with the non commutativity. During the last two decades, probabilistic methods in approximation theory have been developed and Feller's limit theorem is a clear indication of that. However it is known that the classical Korovkin's theorem(first theorem)and Feller's theorem are equivalent for compact intervals. Three lectures aiming at introducing the above classical results and applications in approximation theory.(Will be covered by Dr. Namboodiri)
A set of three lectures will be given on distinguished varieties of the bidisc in ℂn . These are algebraic objects that play a very important role in Hilbert space operator theory. A complete characterization and relations to Ando’s classical inequality will be explained.(Will be covered by Dr. Tirthankar Bhattacharyya).
Let G be a graph with n vertices denoted by {1,...,n} and be the set of its edges. Following Noga Alon, Assaf Naor and many others, define the Grothendieck constant of the graph G, denoted by K(G), to be the smallest constant K such that sup{|∑{i,j∈E}aij⟨xi,yj⟩|:‖xi‖=1=‖yj‖}⩽Ksup{∑{i,j∈E}aijsitj:|si|=1=|tj|}
holds true for any real matrix A=((aij)).
The original Grothendieck inequality is the particular case that corresponds to the bipartite graphs (i.e. of chromatic number 2) and, as a consequence,
KG=Supn∈ℕ{K(G):Gis a biparticular graph on n vertices}
We will discuss several connection of the Grothendieck inequality with several problems including the MAX CUT problem:A cut in a undirected graph G = (V, E) is defined as partition of the vertices of G into two sets; and the weight of a cut is the number of edges that has an end point in each set, that is, the edges that connect vertices of one set to the vertices of the other. The max-cut is the problem of finding a cut in G with maximum weight.(Will be covered by Dr. Gadadhar Misra
Click here for schedule of the event.