# Talk 1: Number Theory and 3-Dimensional Geometry

> Abstract: There is a wonderful analogy between the theory of numbers, and 3-dimensional geometry. 
  For example,  prime numbers behave like knots!   
  I will explain some of the history of this analogy and how it is evolving.

:::{.remark}

- 60s, Mazur's *Remarks on the Alexander polynomial*: primes are analogous to knots.
- Impetus: Thomson 1867, *On vortex atoms*.
- Idea: topological distinction of knots may explain distinct atoms, and links for molecules.
- P. G. Tait (and Kirkman? Around 1880) tabulates many knots.
- Rigorous tools to distinguish: Wirtinger, Dehn, Alexander in late 1800s/early 1990s.
- Dehn (1910), Wirtinger (1905): a presentation of the knot group $\pi_1(\RR^3\sm K)$ in terms of generators and relations.
- Alexander (1927): *Topological invariants of knots and links*, observes difficulties distinguishing group presentations, introduces the simpler Alexander polynomial.
  - E.g. for the trefoil, $p(x) = 1 - x + x^2$.
  - He gives a simple algorithm computing it as the determinant of a matrix, and shows how it can be derived from the knot group.
  - Succeeded in distinguishing many tabulated knots (but not all).
  - See SnapPy! You can fly around the hyperbolic space of a knot complement, and compute a large number of invariants.
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:::{.remark}

- NT: consider solving $x^2+y^2=z^2$. 
  - Factor $x^2+y^2 = (x-iy)(x+iy)$ and write $x\pm iy = (m\pm in)^2$ to obtain $x^2=m^2-n^2, y=2mn, z = m^2+n^2$.
   - This yields *all* solutions, using unique factorization in $\ZZ[i]$.
- Consider $x^p+y^p=z^p$ (Fermat) 
  - To solve: factor $z^p = \prod_k x+\zeta_p^k y$.
- Lame (1847): an incorrect proof that falsely assumes $L_p \da \ZZ[\zeta_p]$ is a UFD, since $\cl(L_p) \neq 1$ for $p=23$ (shown by Kummer) and factorization fails.
- Kummer fixes this proof assuming a weaker condition: $p\notdivides \cl(L_p)$, i.e. this is **regular**.
  - This fails for $p=37$.
  - Kummer proves FLT fails for regular primes.
- Kummer (1850) shows $p$ is irregular iff $p$ divides on of the first $p-3$ coefficients of $x\over e^x-1$.
- Iwasawa (59, 62) gives a polynomial $P_p$ whose degree quantifies the irregularity of $p$; $p$ is irregularity iff $P_p = 1$ is constant.
  - This definition uses a Galois group $G_p$.
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:::{.remark}
Mazur (63) observes similarity between the Alexander polynomial and Iwasawa's polynomial.
Both are pure group theory, and the construction arises from action of $G^\ab\actson [G, G]^\ab$ and taking a characteristic polynomial.
Analogies: 

- $G_p$ is similar to $\pi_1(S^3\sm K)$, 
- the linking number is similar to the quadratic residue $\qsymb{p}{q}$.
- Symmetry of linking numbers is analogous to quadratic reciprocity.
- $\spec \ZZ/p\ZZ \embeds \spec \ZZ$ is like a knot in a 3-manifold.
- $\spec \ZZ$ is like $S^3$, with primes corresponding to embedded knots.

Analogies (partially) due to Mazur, guided by results of Artin and Tate.
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:::{.remark}
Defining $G_p$: take all rings over $k$ with some finite rank $r$ which admit *exactly* $r$ embeddings in $\kbar$, take their union, take ring automorphisms.
Constructing such rings: find polynomials $p$ with $\disc_p = p^k$ for some $k$.
Problem: $G_p$ is very mysterious!
Probably finitely generated but infinite, the only element we can produce is complex conjugation, $z\mapsto \bar{z}$.
Tate (62) shows that $G_p$ cohomologically looks like a knot group, currently the strongest formal analogy.
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:::{.warnings}
Fixing a specific prime $p$, it is not analogous to any specific not, and $\ZZ$ is not analogous to any specific manifold.
Instead, there is some unknown refinement which recovers properties of both.
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:::{.remark}
Idea: compare statistical properties of random knots vs random primes.
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:::{.question}
If $\ZZ\sim M^3$ and $p\sim K_p \embeds M^3$, what object is analogous to $K^c$?
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