# Welcome to Krishan's website!

I'm a PhD student in the Mathematics department at the University of Toronto working with Professor Israel Michael Sigal.

### Mini-Bio

Aside from my mathematical interests, I'm also an amateur web developer. I've used Joomla and Bootstrap to develop this website.

### Research Interests

My research interests are in the broad area of applied mathematics and mathematical physics. More specifically, I'm interested in partial differential equations and many body quantum mechanics.

I'm currently studying partial differential equations related to anyons and the fractional quantum hall effect. One method for studying these equations is to find bifurcations. Some nice pictures of symmetry-breaking bifurcations in the Rayleigh-Benard system from fluid dynamics can be found here.

See below for a description of the area I worked in during my Master's at Waterloo.

Besides the analytic approach to partial differential equations, I'm also interested in the differential geometric approach, where ideas such as symmetries and conserved quantities are used to obtain exact solutions to PDEs.

The particular method I'm most familiar with is that of obtaining exact solutions of certain scalar valued PDEs of mathematical physics using separation of variables. This method can be given a nice theory using tools from (psuedo-)Riemannian geometry, which I'll briefly describe.

First, this theory covers the following PDEs defined on an open subset of $\mathbb{R}^n$

#### Helmholtz Equation

$$\Delta u + V(x) u = 0$$

#### Klein-Gordon Equation

$$\Box u + V(x) u = 0$$

The geometric approach to the separation of variables of the above partial differential equations reframes the problem of finding separable coordinates for a given PDE in to a problem in Riemannian geometry. Namely it says that any one of the above PDEs admit a separable solution if and only if the the Riemannian manifold on which it's defined admits a special type of Killing tensor, $K$, which satisfies $$\operatorname{d}(K \operatorname{d} V) = 0$$ Thus in order to separate the above PDEs, one needs to find a special type of Killing tensor satisfying the above equation. It's a remarkable fact, which even Jacobi (one of the invetors of the method) himself didn't think was possible, that one can use this reframing of the problem to develop a tractable algorithm to separate these PDEs.