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



[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.

Scratchpad for Vector bundles & K-theory

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).

Subgroups of finitely-generated abelian groups are finitely-generated.

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

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.