# 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 \( \pi_1({\mathbb{R}}^3\setminus K) \) 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.
:::

::: {.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 \( {\mathbb{Z}}[i] \).
-   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 \( L_p \coloneqq{\mathbb{Z}}[\zeta_p] \) is a UFD, since \( { \operatorname{cl}} (L_p) \neq 1 \) for \( p=23 \) (shown by Kummer) and factorization fails.
-   Kummer fixes this proof assuming a weaker condition: \( p\notdivides { \operatorname{cl}} (L_p) \), 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 \( G_p \).
:::

::: {.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 \( G^{\operatorname{ab}}\curvearrowright[G, G]^{\operatorname{ab}} \) and taking a characteristic polynomial. Analogies:

-   \( G_p \) is similar to \( \pi_1(S^3\setminus K) \),
-   the linking number is similar to the quadratic residue \( { \left( {p} \over {q} \right)  } \).
-   Symmetry of linking numbers is analogous to quadratic reciprocity.
-   \( \operatorname{Spec}{\mathbb{Z}}/p{\mathbb{Z}}\hookrightarrow\operatorname{Spec}{\mathbb{Z}} \) is like a knot in a 3-manifold.
-   \( \operatorname{Spec}{\mathbb{Z}} \) is like \( S^3 \), with primes corresponding to embedded knots.

Analogies (partially) due to Mazur, guided by results of Artin and Tate.
:::

::: {.remark}
Defining \( G_p \): take all rings over \( k \) with some finite rank \( r \) which admit *exactly* \( r \) embeddings in \(  { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }  \), take their union, take ring automorphisms. Constructing such rings: find polynomials \( p \) with \( {\operatorname{disc}}_p = p^k \) for some \( k \). Problem: \( G_p \) is very mysterious! Probably finitely generated but infinite, the only element we can produce is complex conjugation, \( z\mapsto \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu \). Tate (62) shows that \( G_p \) cohomologically looks like a knot group, currently the strongest formal analogy.
:::

::: {.warnings}
Fixing a specific prime \( p \), it is not analogous to any specific not, and \( {\mathbb{Z}} \) is not analogous to any specific manifold. Instead, there is some unknown refinement which recovers properties of both.
:::

::: {.remark}
Idea: compare statistical properties of random knots vs random primes.
:::

::: {.question}
If \( {\mathbb{Z}}\sim M^3 \) and \( p\sim K_p \hookrightarrow M^3 \), what object is analogous to \( K^c \)?
:::

# Talk 2: Symplectic L-functions and their topological analogues

> Abstract: The topology of the symplectic group enters into many different areas of mathematics. After discussing a couple of "classical" manifestations of this, I will explain a new one, in the theory of L-functions, as well as a purely topological analogue of the statement. I am not going to assume any familiarity with the theory of L-functions for the talk. Joint work with Amina Abdurrahman.

## Symplectic Spaces

::: {.remark}
Note: for \( L \) a field,
\[
{ {L}_{\scriptscriptstyle \square} } \coloneqq\left\{{a^2 {~\mathrel{\Big\vert}~}a\in L}\right\}
.\]
:::

::: {.remark}
Recall that any symplectic vector space is isomorphic to \( \qty{{\mathbb{R}}^{2g}, J\coloneqq{
\begin{bmatrix}
  {0} & {\operatorname{id}_g} 
\\
  {- \operatorname{id}_g} & {0}
\end{bmatrix}
}} \). Write
\[
{\operatorname{SP}}(V) = \left\{{g: V\to V{~\mathrel{\Big\vert}~}{\left\langle {gx},~{gy} \right\rangle} = {\left\langle {x},~{y} \right\rangle}}\right\} = \left\{{A\in \operatorname{GL}_{2g}({\mathbb{R}}) {~\mathrel{\Big\vert}~}A^tJA = J}\right\}
.\]
Note
\[
{\operatorname{SP}}_2({\mathbb{R}}) = \left\{{M\in \operatorname{GL}_2({\mathbb{R}}) {~\mathrel{\Big\vert}~}\operatorname{det}M = 1}\right\}
,\]
and \( {\operatorname{SP}}_{2g}({\mathbb{R}}) \) is connected but \( {\operatorname{SP}}_{2g}({\mathbb{R}}) \simeq{\mathbb{R}}^2\setminus\left\{{0}\right\} \). Let \( p: G\to {\operatorname{SP}}_{2g}({\mathbb{R}}) \) be the universal cover; since the base is a topological group, picking any \( p^{-1}(1) \) yields an essentially unique group structure on \( G \) by analytically continuing the group law. In fact \( G \) is a Lie group and \( \ker p \cong {\mathbb{Z}}\in Z(G) \) is central, so there is a SES
\[
{\mathbb{Z}}\to G\to {\operatorname{SP}}_{2g}({\mathbb{R}})
.\]
Note that there are not faithful finite-dimensional reps of \( G \), but there are infinite-dimensional reps important to quantization in physics.
:::

## Surfaces

::: {.remark}
A theorem of Meyer on surfaces: let \( \Sigma \) be a (compact closed oriented smooth) surface and \( \rho: \pi_1 \Sigma\to {\operatorname{SP}}_{2g}{\mathbb{R}} \); this corresponds to a local system, so form the twisted cohomology \( H^1(\Sigma; \rho) \). Using the cup product and symplectic pairing, one can produce a *symmetric* pairing
\[
L: H^1(\Sigma;\rho){ {}^{ \scriptstyle\otimes_{{\mathbb{R}}}^{2} }  }\to {\mathbb{R}}
.\]
There is an isomorphism \( (H^1, L)\cong ({\mathbb{R}}^{p+q}, \operatorname{diag}_{p}(-1) \oplus \operatorname{diag}_q(1)) \), so \( \operatorname{sig}(\Sigma, \rho) \coloneqq\operatorname{sig}L \coloneqq p-q \) is an interesting invariant.
:::

::: {.remark}
Recall that 4-manifolds \( M \) also admit a signature on \( H^2(M; {\mathbb{R}}) \). Let \( M\to\Sigma \) be surface bundle over a surface -- if \( M \) is a product, \( \operatorname{sig}M = 0 \). Chern-Hirzebruch-Serre: there is a monodromy morphism \( \pi_1\Sigma\to {\operatorname{SP}}_{2g}{\mathbb{R}} \) which is trivial if \( \operatorname{sig}M = 0 \). In fact \( \operatorname{sig}(M) = \operatorname{sig}(\Sigma, \rho) \), so this situation naturally arises for fibered 4-manifolds.
:::

::: {.remark}
Recall \( \pi_1 \Sigma_g = \left\langle{a_i,b_i {~\mathrel{\Big\vert}~}\prod_i [a_i, b_i] = e}\right\rangle \). Consider trying to form a lift:

```{=tex}
\begin{tikzcd}
    && G \\
    \\
    {\pi_1\Sigma} && {{\operatorname{SP}}_{2g}({\mathbb{R}})}
    \arrow[from=3-1, to=3-3]
    \arrow[from=1-3, to=3-3]
    \arrow[dashed, from=3-1, to=1-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJcXHBpXzFcXFNpZ21hIl0sWzIsMiwiXFxTUF97Mmd9KFxcUlIpIl0sWzIsMCwiRyJdLFswLDFdLFsyLDFdLFswLDIsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

Note \( m\coloneqq\prod [\tilde\rho a_i, \tilde\rho b_i] \in \ker p \cong {\mathbb{Z}} \), and it turns out that
\[
\operatorname{sig}(\Sigma, \rho) = p-q = 4m
.\]
The fact that \( \ker p \in Z(G) \) was essential in making sure this doesn't depend on choices. Note that this only determines \( m \) up to sign!
:::

::: {.remark}
Toward generalizing, the above central extension determines a class \( b\in H^2({\operatorname{SP}}_{2g}({\mathbb{R}}); {\mathbb{Z}}) \) and \( p^* b\in H^2( \Sigma; {\mathbb{Z}}) \). Using the pairing against the fundamental class \( [\Sigma] \) yields
\[
\operatorname{sig}(\Sigma, \rho) = 4 \int_\Sigma p^* b
.\]

When replacing \( {\mathbb{R}} \) with \( L \), replace \( {\mathbb{Z}} \) by \( W_L \), the Witt group of quadratic forms over \( L \).
:::

# 3-manifolds

::: {.remark}
Consider now varying the situation in a 1-dimensional family -- take \( \Sigma\to M\to S^1 \) a surface bundle over a circle with fibers \( \Sigma_t \), each yielding \( \rho_t: \pi_1\Sigma_t \to {\operatorname{SP}}_{2g}({\mathbb{R}}) \) for \( t\in S^1 \). Each \( t \) yields a quadratic vector space \( H^1(\Sigma_t,\rho_t) \) of signature \( (p, q) \). Monodromy yields an element \( m\in \mathop{\mathrm{Aut}}H^1(\Sigma_0, \rho_0) \subseteq {\operatorname{O}}_{p, q} \). If \( p,q>0 \) then \( {\sharp}\pi_0{\operatorname{O}}_{p, q} = 4 \) -- which connected component does this land in? How to separate the components: there is a determinant map
\[
\operatorname{det}: {\operatorname{O}}_{p, q} \to \left\{{\pm 1}\right\}
.\]
There is a *spinor norm*
\[
{\mathrm{spinornorm}}: {\operatorname{O}}_{p, q} &\to \left\{{\pm 1}\right\} = {\mathbb{R}}^{\times}/{ {{\mathbb{R}}}_{\scriptscriptstyle \square} }^{\times}\\
\mathrm{reflection}_v &\mapsto {\left\langle {v},~{v} \right\rangle}
.\]
This works with \( {\mathbb{R}} \) replaced by \( L \), using the fact that \( {\operatorname{O}}_{p, q} \) is generated by reflections. For \( (V, L) \) a symmetric space, one gets
\[
\operatorname{det}: {\operatorname{O}}(V) &\to \left\{{\pm 1}\right\}
\]
and
\[
{\mathrm{spinornorm}}: {\operatorname{O}}(V) &\to L^{\times}/{ {L}_{\scriptscriptstyle \square} }^{\times}
.\]
:::

::: {.theorem title="?"}
It turns out that

-   \( m\in {\operatorname{O}}(H^1(\Sigma_0, \rho_0)) \),
-   \( \operatorname{det}m = 1 \), and
-   \( {\mathrm{spinornorm}}(m) = \int_M p^* c\in L^{\times}/{ {L}_{\scriptscriptstyle \square} }^{\times} \).

Note that there is a Soulé etale Chern class
\[
m\in H^3({\operatorname{SP}}_{2g}(L), L^{\times}/{ {L}_{\scriptscriptstyle \square} }^{\times})
.\]
Pulling this back and integrating yields
\[
p^* c\in H^3(M; L^{\times}/{ {L}_{\scriptscriptstyle \square} }^{\times}) \xrightarrow{\int_M} L^{\times}/{ {L}_{\scriptscriptstyle \square} }^{\times}
.\]
:::

::: {.proof title="?"}
Re-interplay \( {\mathrm{spinornorm}}(m) \) as a quantity \( {\mathrm{RT}}(M, \rho) \) where \( {\mathrm{RT}} \) is the *Reidemeister torsion*, which makes sense for any \( (M, \rho) \) (not just those fibered over \( S^1 \)). This is a bordism invariant, i.e. if \( M\sim N \) are bordant then \( {\mathrm{RT}}(M,\rho) = {\mathrm{RT}}(N,\rho') \). Thus there exists some formula of the desired type, where \( c \) is unknown -- the trick is to compute enough examples to determine \( c \).
:::

## Symplectic \( L{\hbox{-}} \)functions {#symplectic-lhbox-functions}

::: {.remark}
For \( X\in{\mathsf{sm}}\mathop{\mathrm{proj}}{\mathsf{Var}}_{/ {{\mathbb{F}}_q}}  \) and \( \rho: \pi_1 X\to {\operatorname{SP}}_{2g} k \) for \( k \) finite containing \( \sqrt{q} \), there is an associated \( L{\hbox{-}} \)function \( L(X, \rho; T)\in k(T) \) where \( T\approx q^{-s} \). There is a functional equation relating \( T\rightleftharpoons{1\over qT} \). Evaluate at the center of symmetry \( T = {1\over\sqrt q} \) to define \( L(X, \rho) \coloneqq L(X, \rho; 1/\sqrt q) \).
:::

::: {.theorem title="?"}
Suppose \( L(X, \rho) \neq 0 \); then modulo squares
\[
L(X, \rho) = \int_X \rho^* c \qquad \in k^{\times}/{ {k}_{\scriptscriptstyle \square} }^{\times}
,\]
which is a spinor norm of Frobenius (see Zassenhaus). This requires some conditions, e.g. \( \rho\mathrel{\Big|}_{\pi_1 X} \) is surjective, congruences on \( {\sharp}k \) and \( q\operatorname{mod}8 \), and \( \gcd(q,{\sharp}{\operatorname{SP}}_{2g}(k)) = 1 \) (which may not be necessary).
:::

::: {.remark}
Interpretation: \( L(X,\rho) = ab^2 \) where \( a \) is simple and \( b \) is complicated. BSD gives a conjectural formula for this which includes a lot of squares. So there is a cohomological obstruction to the existence of a square root of \( L(X, \rho) \).
:::

::: {.remark}
On the proof: try to pass validity from one known example \( (X,\rho) \) to other examples \( (X', \rho') \) with \( X_{/ {{\mathbb{F}}_q}} , X'_{/ {{\mathbb{F}}_q'}}  \) and images in \( {\operatorname{SP}}_{2g}(k), {\operatorname{SP}}_{2g}(k') \) respectively.

-   Pass from \( q \) to \( q' \) using a moduli space of pairs \( (X,\rho) \) where a topological theorem controls the fiber over \( {\mathbb{C}} \).
-   Pass from \( k \) to \( k' \) using *compatible local systems* from NT.
:::

::: {.remark}
Final comparison:

-   For 2-manifolds, \( \operatorname{sig}(\Sigma, \rho) = \int_\Sigma p^* b \).
-   For 3-manifolds, \( {\mathrm{RT}}(M,\rho) = \int_M p^* c \).
-   Symplectic \( L{\hbox{-}} \)functions: ?
:::

# Talk 3: Relative Langlands Duality

> Abstract: If we are given a compact Lie group G acting on a space X, a powerful tool in "approximately" decomposing the G-action on functions on X is the orbit method. I will describe this method and how it sometimes refines to an exact algebraic statement which involves a "dual" group G\^ and dual space X\^. This is part of a joint work with David Ben-Zvi and Yiannis Sakellaridis about duality in the relative Langlands program. I will do my best to make the talk comprehensible without any familiarity with the framework of the Langlands program.