# One week in

So, I’ve officially survived the first week of my (second) second year as a Ph.D. student. It’s hard to imagine sometimes that I’ve been a grad student for over three years.

I’m such an old man.

It’s been even less easy than I’d anticipated, too, which is nicht spaß; sadly, too, this is the first week where teaching responsibilities are actually a thing (we get week 1 off from such things), so now it’s difficult and time-consuming. Jajajajaja.

Rather than making this entry all about Yours Truly, I figured I’d pit stop in and use some of my (*gasp*) unoccupied time write up a little spiel I saw for the first time in my Riemannian geometry class last week.

In topology, there’s an idea called invariance of dimension which can be stated in many different contexts, situations, etc. This can be modified in the case of manifolds with differential structures, and because the idea of the proof seemed a bit cool, I decided to throw it up here for you guys. Throughout, the notation $M^m$ denotes a manifold $M$ of dimension $m$ with an associated differential structure.

# Movin’ on up (and down) (and up) (and down)….

I decided to spend as much time as possible today studying after a few days of being nonchalant with it. I went to bed early-ish last night, woke up early-ish this morning, and hit the books with very few breaks in between.

As it turns out, this recipe gave me ample opportunity to learn new things. Who woulda thunk?

I started with my professor’s paper on $M$-conformal Cliffordian mappings. I made it through a couple more pages of that guy, verifying theorems and assertions as I went along. Then, right as I was on the precipice of real math, I realized how mentally taxing my morning had been and shifted direction a bit.

My new direction: Dummit and Foote. I started section 15.2 on Radicals and Affine Varieties. About 2/3 of the way through that section, I realized I really really need to learn some stuff about Gröbner Bases, so I decided to forego that and keep the ball rolling. I spent a few minutes flipping through Osborne’s book on Homological Algebra and upon realizing I’m far too underwhelming to tackle that guy, I shifted focus again to Kobayashi and Nomizu.

Of course, K&N has kind of worn out its welcome around here, and upon reading a page or two, I decided to break out a different Differential Stuff book instead. My target? Warner’s book Foundations of Differentiable Manifolds and Lie Groups. This book is a nice amalgam of Geometry and Topology, as evidenced by its somewhat nonstandard definition of tangent vectors. Maybe I’ll share some of that later.

Finally, I decided to shift my focus back towards Algebraic Geometry, whereby I broke out Eisenbud and Harris’s book The Geometry of Schemes and tried to stay afloat. Much to my own surprise, I was able to make it through fifteen-or-so pages without floundering completely and/or ripping all my hair out, so I’m hoping that maybe the information I’ve picked up in other places has done me some good. We’ll see for sure moving on.

Overall, I think I cranked out about 45-50 pages of reading today – and all (well, most) on material that’s completely new. It ain’t a Fields Medal, but it ain’t a flop either.

Until next time….

# Hodge Dual part deux

You may recall that I deemed yesterday Differential Geometry Sunday and posted a small expository thing on the Hodge Star/Dual operator. Apparently in my cloudy haze of mathematical mediocrity, I concluded my post without having touched on the derivations I actually intended to touch on.

Sometimes I feel like I need a vacation from my vacation.

In any case, I’m going to take a stab at saying some of the things I’d meant to say yesterday, but in order to ensure we’re all on the same page, I’m going to recall what exactly the Hodge Star/Dual operator is. Then, after the break, I’m going to show some of the cool derivations that come about because of it.

Let $M$ be a manifold of dimension $\dim(M)=n$ and let $\omega=\sum_I f_I dx_I$ be a $k$-form on $M$ for $k\leq n$. Here, $I$ denotes a multi-index which – without loss of generality – can be assumed to be increasing. Then the hodge star operator is the operator which takes $\omega$ to the $(n-k)$-form $*\omega =\sum_I f_I(*dx_I)$ where $*dx_I=\varepsilon_I dx_{I^C}$, where $I^C$ is the multi-index consisting of all numbers $1,\ldots,n$ not in $I$, and where $\varepsilon_I=\pm 1$ denotes the sign of $dx_I dx_{I^C}$.

So what we said is that for the most commonly-recognized example $\mathbb{R}^3$ with orthogonal 1-forms $dx,dy,dz$ and the usual metric $ds^2=dx^2+dy^2+dz^2$, the Hodge Star operator sends $1$ to $dx\wedge dy\wedge dz$ and vice versa, it sends $dx,dy,dz$ to $dy\wedge dz, dz\wedge dx,dx\wedge dy$ respectively, and it sends $(dx\wedge dy)\mapsto dz$, $(dy\wedge dz)\mapsto dx$, and $(dx\wedge dz)\mapsto -dy$. But the question then remains: Why does anybody care?