## Number Fields and 3-manifolds

An informal learning seminar on the analogy between number fields and 3-manifolds, organized by Autumn Kent and Melanie Matchett Wood

Tuesdays, 11:15am-Noon Van Vleck B203

## Schedule

April 2: Autumn and Melanie describe the basic objects, some of the big questions about them, on both sides of the analogy, e.g. see Section 2.2 of
Analogies between group actions on 3-manifolds and number fields by Sikora
or
Section 14 of Hakenness and b_1 by Reznikov.

April 9: Basic objects, continued

April 16: Splitting, inertia groups, and ramification

April 23: Quadratic Reciprocity

April 30: no meeting

May 7: Chebotarev density theorem

## References

See these nice notes by Li and Sia.
Knots and Primes, book by Morishita (Melanie has from library, you can borrow)

A list of some other references

## Possible future topics

Quadratic reciprocity and symmetry of the linking number (see Li-Sia notes above).

Genus theory and topological analog (classification of the
homology classes of the homology group by linking numbers?) (Ch 6 of Knots and Primes book)

Structure of link groups and Galois groups with restricted ramification
Paper by Morishita

Alexander Polynomial and Iwasawa Theory (Ch 9 of Knots and Primes book)

Class field theory and analog for knots Paper by Niibo

``Knots which behave like the prime numbers'' by McMullen on Chebotarev density theorem for 3-manifolds.