# Advice for picking a supervisor

• Have they published with their students?
• Where have their students ended up after the PhD?
• Get feedback from their previous students
• Work on a project they are interested in

Also,

[knot theory] [invariants]

The writhe of an oriented knot is invariant under change of orientation. i.e. the writhe is an invariant of unoriented knots. (NOTE: But it’s not invariant under change of orientation for links!)

Hence the normalized Kauffman bracket polynomial of a knot is invariant under change of orientation.

[lin.alg] [tricks]

$A = \begin{bmatrix}0 & 1 & 0 & \cdots & 0 \\ \vdots & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 1 \\ -a_{n-1} & -a_{n-2} & \cdots & \cdots & -a_{0} \end{bmatrix}$

has characteristic polynomial $z^n + a_{n-1}z^{n-1} + \dotsb + a_0$.

# Why the Grothendieck group/ring?

[alg] [cat.theory]
Given an additive category $\mathcal{C}$ we can form the Grothendieck group $\mathcal{K}(\mathcal{C})$. If further $\mathcal{C}$ is a tensor category then $\mathcal{K}(\mathcal{C})$ is also a ring, the Grothendieck ring.

If $\mathcal{C}$ is semisimple then any equality in $\mathcal{K}(\mathcal{C})$ corresponds to an isomorphism in $\mathcal{C}$.

# Scratchpad for Vector bundles & K-theory

[vect.bund]
This is a scratchpad for my summer school course in vector bundles and K-theory.

# Clutching functions

Given local trivializations of a vector bundle on open sets $U, V$ whose intersection is non-empty, these local trivializations define a clutching function $\tilde{h}_{UV} : U \cap V \rightarrow GL_n(\mathbb{R})$ (or $GL_n(\mathbb{C})$), since on the intersection the composite $h_{UV} = h_V^{-1} \circ h_U : (U \cap V) \times \mathbb{R}^n \rightarrow (U \cap V) \times \mathbb{R}^n$ restricted to each $x \in U \cap V$ is a linear isomorphism.

# Direct sum of vector bundles

Given vector bundles $p_1 : E_1 \rightarrow X, p_2 : E_2 \rightarrow X$, define $E_1 \oplus E_2 = \bigcup_{x\in X} (E_1)_x \oplus (E_2)_x$.

1. Fibers. The obvious thing we want for the direct sum is to let the fiber over each $x$ be $(E_1)_x \oplus (E_2)_x$. So for $(e_1, e_2)$ in the total space the projection map is $(p_1 \oplus p_2)(e_1,e_2) = p_1(e_1) = p_2(e_2)$.
2. We need local triviality. Define an open cover $\{U_\alpha\}$ and local maps $h_\alpha$ on each $U_\alpha$ that satisfy the properties of local trivializations (except the homeomorphism bit, because we haven’t put a topology on the total space yet).
3. Topologize. Finally, topologize the total space so that the local trivializations on $U_\alpha$ are in fact local homeomorphisms from trivial bundles on $U_\alpha$ to the preimage $(p_1\oplus p_2)^{-1}(U_\alpha)$.

[alg]
Subgroups of finitely-generated abelian groups are finitely-generated.

Not necessarily true for finitely-generated non-abelian groups.

[tricks]
1. For a polynomial $f(z) = c_n z^n + c_{n-1}z^{n-1} + \dotsb + c_0$ with roots $a_1, \dotsc ,\ a_n$,

$\displaystyle \sum a_i = -\frac{c_{n-1}}{c_n},$
$\displaystyle \prod a_i = (-1)^n \frac{c_0}{c_n}$

2. $\displaystyle{f^\prime (z) = f(z) \left( \frac{1}{z-a_1} + \dotsb + \frac{1}{z-a_n} \right)}$