# 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,

[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.

[alg.top]
Standard (cubic lattice) CW complex structure on $\mathbb{R}^n$: Lattice of 0-cells, join all by 1-cells, then add 2-cells, 3-cells etc.

Check this rigorously.

[alg.top]
$X$ contractible $\ \nRightarrow$ $X$ deformation retracts to a point

But the converse is true?