Department Seminars & Colloquia
When you're logged in, you can subscribe seminars via e-mail
A network represents a way of interconnecting any pair of users or nodes by means of some meaningful links. Thus, it is quite natural that its structure can be represented, at least in a simplified form, by a connected graph whose vertices represent nodes and whose edges represent their links.
As an efficient method to investigate dynamical phenomena on networks such as electrical flow on a circuits, chemical reaction between molecules, behavior of biological individuals in their societies and so on, in a systematic way, we introduce the theory of discrete partial differential equations on networks. In order to do this, the calculus on networks is introduced, at first, after defining the partial derivatives at each nodes. Being based on this calculus, we discuss the various types of partial differential equations on networks. In particular, the solvabilities of (nonlinear) elliptic PDE and parabolic PDE on networks will be discussed.
Understanding the parameter spaces of dynamical systems has long been the dream of the greatest mathematicians. Even Newton asked: what initial conditions(positions, velocities masses) lead to a stable solar system?
There are exceedingly few cases where we can answer such questions: no one knows anything about the parameter space for the 3-body problem. But for the simplest nonlinear dynamical system, z 7! z2 + c with parameter c, we do understand the parameter space.
The crucial object in parameter space is the Mandelbrot set: it features some very delicate combinatorics, which can be written exactly.
In my lecture I will attempt to describe these combinatorial laws, and sketch where they come from.
Calculus of variations is a critical point theory for functionals. First, I would like to introduce some classical problems in view of the calculus of variations and its relation to differential equations. Last one third of my talk will be devoted to introducing my recent development of some variational methods for some singularly perturbed elliptic partial differential equations.
The derived category of bounded complexes of coherent sheaves on an algebraic variety is an interesting invariant of the algebraic variety. There is more symmetry than the varieties themselves in the sense that there are different varieties with equivalent derived categories. There is a surprising parallelism between the minimal model program and the semi-orthogonal decompositions of derived categories. I will review some old and new results in this direction.
We discuss about a result of Littlewood on the horizontal distribution of the zeros of the Riemann zeta-function ζ(s) in the critical strip and further we discuss about the progress made on the zeros of ζ(s) locally in the neighbourhood of the critical line. (An old work of mine jointly done with Professor K. Ramachandra).