- 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,

- 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]

has characteristic polynomial .

[alg] [cat.theory]

Given an additive category we can form the Grothendieck group . If further is a tensor category then is also a ring, the Grothendieck ring.

If is semisimple then any equality in corresponds to an isomorphism in .

[vect.bund]

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

Given local trivializations of a vector bundle on open sets whose intersection is non-empty, these local trivializations define a *clutching function* (or ), since on the intersection the composite restricted to each is a linear isomorphism.

Given vector bundles , define .

*Fibers.*The obvious thing we want for the direct sum is to let the fiber over each be . So for in the total space the projection map is .*We need local triviality.*Define an open cover and local maps on each that satisfy the properties of local trivializations (except the homeomorphism bit, because we haven’t put a topology on the total space yet).*Topologize.*Finally, topologize the total space so that the local trivializations on are in fact local homeomorphisms from trivial bundles on to the preimage .

[alg]

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

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

[tricks]

1. For a polynomial with roots ,

2.