The protagonist of this course is a mathematical object called the Metric Tensor ($g$).
In a standard coordinate system, distance is simple: $ds^2 = dx^2 + dy^2$. But on a curved surface (like the surface of a sphere or a crumpled piece of paper), this formula fails. The metric tensor is a machine that allows you to calculate distances, angles, and areas on any surface, no matter how bizarrely curved.
Math 6644 teaches you to wield this tool. You learn that a Riemannian manifold is essentially a topological space equipped with this metric "ruler" everywhere you go. math 6644
Not "I don't understand Girsanov," but rather "In the Cameron-Martin theorem, why can't we shift Brownian motion by a non-square-integrable drift?"
We all love the simplicity of the Forward Euler method for time integration. It’s explicit, it’s easy, and it looks beautiful in code. But as we saw when solving the heat equation ( u_t = \alpha u_xx ), setting your time step ( \Delta t ) even 1% too large doesn’t just give you a slightly inaccurate answer—it gives you an apocalypse. The protagonist of this course is a mathematical
Within 20 time steps, your temperature profile looks like the seismograph of an earthquake. The solution isn't wrong; it’s infinite. This isn't a bug; it's a feature of the mathematics. Von Neumann taught us that the amplification factor ( G(\theta) ) must satisfy ( |G| \le 1 ). For Forward Euler on the diffusion equation, that gives us the infamous constraint:
[ \Delta t \le \frac\Delta x^22\alpha ]
Notice that ( \Delta t ) scales with ( \Delta x^\mathbf2 ). Want double the resolution? You must take four times the time steps. This is the brutality of explicit methods.
Your professor will assign one or two of these classics: We all love the simplicity of the Forward