http://www.math.iitb.ac.in/~seminar/colloquium/colloq-02-nov-16.pdf

http://www.math.iitb.ac.in/~seminar/algebra/watanabe-02-nov-16.pdf

http://www.math.iitb.ac.in/~seminar/algebra/watanabe-02-nov-16.pdf

Consider the following extremal (geometric) problem: Suppose we have a set of lines in R^d such that the angle between any two of these is the same. How many lines can there be? As it turns out, one can an upper bound that is quadratic in the dimension d without too much difficulty. There are also (several) examples of sets of lines of size $\Omega(d^2)$ in $R^d$ that form an equi-angular set. However, in all these examples, the angle between a pair of lines goes to 0 as d goes to infinity. If we insist that the angle is fixed then the bound becomes linear. We shall see a proof of this due to B. Bukh.

We discuss second and fourth order energy space based Dirichlet control problems. We propose an alternative formulation which enables us to obtain a smoother optimal control, which in turn provides a better convergence rate for the discrete optimal control than earlier approaches. In the first part we discuss conforming finite element analysis for the second order problem, where optimal order a priori error estimates are derived for the optimal control, optimal state and adjoint states in energy and L2 norms. Subsequently a residual based reliable and efficient (locally) error estimators are derived for a posteriori error control. In the second part we extend this alternative formulation for the fourth order problem. In this case the resulting optimality system is discretized using C0 interior penalty method. We obtain an optimal order error estimate for the optimal control, optimal state and adjoint state in Energy and L2 norms. At the end we present numerical experiments which illustrate our theoretical findings.

In this talk we present the main problems and some recent results on Teichmüller spaces of surfaces with boundary.

Let c(n,k,q) be the number of simultaneous similarity classes of k-tuples of commuting n x n matrices over a finite field of order q. Wridaye determine the asymptotic behaviour of c(n,k,q) as a function of k, for a fixed n and q.

In this presentation we review some methods for evolution of curves and surfaces \Omega_t. The methods are 1. The level set method by Sethian, 2. The fast marching method of Osher and Sethian, 3. Method based on introduction of ray coordinates associated with \Omega_t and 4. Kinematical Conservation Laws. We present some results and suggest comparison of these results by intensive numerical computation. Introduction of ray coordinates simplifies tracking of the successive positions of the surface. Formulation in terms of KCL has further advantages. The first two methods have been discussed intensively both theoretically and with numerical results on many practical problems. However, these methods can not reproduce appearance of a special singularity on \Omega_t, namely kinks across which the normal to \Omega_t and the velocity of \Omega_t have finite jumps. Further the computational efficiency of last two methods are equally good. We also highlight a case when further theoretical development is Required in order to make the last two methods applicable to curvature driven ? t .