I've gotta choose a master's thesis, and im torn between
>Differential topology (reference books: Milnor topology from a differentiable viewpoint, Bredon Topology and geometry, Differential topology Hirsh)
>manifolds, vector bundles, de Rham cohomology, framed cobordism. Describe exotic spheres with connections to number theory.
OR
>Topology of Vector Fields and Vector Bundles (reference books: Milnor, characteristic classes, Hatcher Algebraic topology, Hatcher Vector Bundles and K-Theory
>poincare-hopf theorem, parallelisable manifolds, tangent bundles of manifolds, vector bundles over spaces, characteristic classes
I want to end up doing a PhD after my masters (although I'm also interested in Algebraic geometry)
James Long
Can anyone help me with the following problem?
The problem is to find the residue field of each point of [math]\operatorname{Spec} {\mathbb{F}_p}\left[ x \right][/math] and the count how many points per residue field.
I know the points are and where f is any monic irreducible polynomial over [math]{\mathbb{F}_p}[/math]. I also know that if f is of degree n then the residue field will be [math]{\mathbb{F}_{{p^n}}}[/math]. So pretty much I just want to know how to count the number of monic irreducible polynomials of degree n for each n.
Denote by [math]\Pi_n[/math] the set of monic irreducible polynomials of degree [math]n[/math] of [math]\mathbb F_p[t][/math]. Notice that, for each [math]n \ge 1[/math], [eqn]t^{p^n}-t = \prod_{\beta \in \mathbb F_{p^n}} (t-\beta) = \prod_{d\, |\, n}\prod_{P \in \Pi_d} P[/eqn] Indeed, the degree of each element in [math]\mathbb F_{p^n}[/math] divides [math]n[/math]. Conversely, if [math]d |n[/math], and [math]P \in \Pi_d[/math], then it splits in [math]\mathbb F_{p^n}[/math] and thus divides [math]t^{p^n}-t[/math]. Since [math]t^{p^n}-t[/math] and its derivative are coprime, we get the desired equality. Now, taking the degrees on each side, we get [eqn]p^n = \sum_{d|n} d|\Pi_d|[/eqn] Using Möbius inversion, we get: [eqn]|\Pi_n| = \frac{1}{n}\sum_{d|n}\mu\left(\frac{n}{d}\right)p^d[/eqn]
Angel White
>finish masters degree >get offered PhD position >ask the prof I'm going to work with for some project ideas >tells me to read some articles on arXiv and recreate the data >do so within 5 months >email her about it >"oh shit lmao lemme get back to you on that sonic the hedgehog" >email her colleague I want to work with for some extra work >same thing happens >so bored that I'm smoking 2 to 3 cigarettes a day >PhD doesn't start for another 3 months
Michael Smith
Find a different Ph.D. position.
Juan Peterson
Do symplectic geometry instead.
Elijah Price
fug mayne, which one is more interesting/rich/more useful to know?
James Brooks
> for some time been haunted by the feeling that mathematics is just a symbol-game.
That's what you get for studying anything else than analysis.
