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\title{
\rule{\linewidth}{1pt} \\
\textbf{
    2022 Cantrell Lectures: Akshay Vankatesh
  }
    \\ {\normalsize Lectures by Akshay Venkatesh. University of Georgia,
Spring 2022} \\
  \rule{\linewidth}{2pt}
}
\titlehead{
    \begin{center}
  \includegraphics[width=\linewidth,height=0.45\textheight,keepaspectratio]{figures/cover.png}
  \end{center}
       \begin{minipage}{.35\linewidth}
    \begin{flushleft}
      \vspace{2em}
      {\fontsize{6pt}{2pt} \textit{Notes: Todo } } \\
    \end{flushleft}
    \end{minipage}
    \hfill
    \begin{minipage}{.65\linewidth}
    \end{minipage}
  }







\begin{document}

\date{}
\maketitle
\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
  \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2022-05-04 }
\end{flushleft}


\newpage

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\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{talk-1-number-theory-and-3-dimensional-geometry}{%
\section{Talk 1: Number Theory and 3-Dimensional
Geometry}\label{talk-1-number-theory-and-3-dimensional-geometry}}

\begin{quote}
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.
\end{quote}

\begin{remark}

\begin{itemize}
\tightlist
\item
  60s, Mazur's \emph{Remarks on the Alexander polynomial}: primes are
  analogous to knots.
\item
  Impetus: Thomson 1867, \emph{On vortex atoms}.
\item
  Idea: topological distinction of knots may explain distinct atoms, and
  links for molecules.
\item
  P. G. Tait (and Kirkman? Around 1880) tabulates many knots.
\item
  Rigorous tools to distinguish: Wirtinger, Dehn, Alexander in late
  1800s/early 1990s.
\item
  Dehn (1910), Wirtinger (1905): a presentation of the knot group
  \(\pi_1({\mathbb{R}}^3\setminus K)\) in terms of generators and
  relations.
\item
  Alexander (1927): \emph{Topological invariants of knots and links},
  observes difficulties distinguishing group presentations, introduces
  the simpler Alexander polynomial.

  \begin{itemize}
  \tightlist
  \item
    E.g. for the trefoil, \(p(x) = 1 - x + x^2\).
  \item
    He gives a simple algorithm computing it as the determinant of a
    matrix, and shows how it can be derived from the knot group.
  \item
    Succeeded in distinguishing many tabulated knots (but not all).
  \item
    See SnapPy! You can fly around the hyperbolic space of a knot
    complement, and compute a large number of invariants.
  \end{itemize}
\end{itemize}

\end{remark}

\begin{remark}

\begin{itemize}
\tightlist
\item
  NT: consider solving \(x^2+y^2=z^2\).

  \begin{itemize}
  \tightlist
  \item
    Factor \(x^2+y^2 = (x-iy)(x+iy)\) and write
    \(x\pm iy = (m\pm in)^2\) to obtain
    \(x^2=m^2-n^2, y=2mn, z = m^2+n^2\).
  \item
    This yields \emph{all} solutions, using unique factorization in
    \({\mathbb{Z}}[i]\).
  \end{itemize}
\item
  Consider \(x^p+y^p=z^p\) (Fermat)

  \begin{itemize}
  \tightlist
  \item
    To solve: factor \(z^p = \prod_k x+\zeta_p^k y\).
  \end{itemize}
\item
  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.
\item
  Kummer fixes this proof assuming a weaker condition:
  \(p\notdivides { \operatorname{cl}} (L_p)\), i.e.~this is
  \textbf{regular}.

  \begin{itemize}
  \tightlist
  \item
    This fails for \(p=37\).
  \item
    Kummer proves FLT fails for regular primes.
  \end{itemize}
\item
  Kummer (1850) shows \(p\) is irregular iff \(p\) divides on of the
  first \(p-3\) coefficients of \(x\over e^x-1\).
\item
  Iwasawa (59, 62) gives a polynomial \(P_p\) whose degree quantifies
  the irregularity of \(p\); \(p\) is irregularity iff \(P_p = 1\) is
  constant.

  \begin{itemize}
  \tightlist
  \item
    This definition uses a Galois group \(G_p\).
  \end{itemize}
\end{itemize}

\end{remark}

\begin{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:

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

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

\end{remark}

\begin{remark}

Defining \(G_p\): take all rings over \(k\) with some finite rank \(r\)
which admit \emph{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.

\end{remark}

\begin{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.

\end{warnings}

\begin{remark}

Idea: compare statistical properties of random knots vs random primes.

\end{remark}

\begin{question}

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

\end{question}

\hypertarget{talk-2-symplectic-l-functions-and-their-topological-analogues}{%
\section{Talk 2: Symplectic L-functions and their topological
analogues}\label{talk-2-symplectic-l-functions-and-their-topological-analogues}}

\begin{quote}
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.
\end{quote}

\hypertarget{symplectic-spaces}{%
\subsection{Symplectic Spaces}\label{symplectic-spaces}}

\begin{remark}

Note: for \(L\) a field,
\begin{align*}
{ {L}_{\scriptscriptstyle \square} } \coloneqq\left\{{a^2 {~\mathrel{\Big\vert}~}a\in L}\right\}
.\end{align*}

\end{remark}

\begin{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
\begin{align*}
{\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\}
.\end{align*}
Note
\begin{align*}
{\operatorname{SP}}_2({\mathbb{R}}) = \left\{{M\in \operatorname{GL}_2({\mathbb{R}}) {~\mathrel{\Big\vert}~}\operatorname{det}M = 1}\right\}
,\end{align*}
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
\begin{align*}
{\mathbb{Z}}\to G\to {\operatorname{SP}}_{2g}({\mathbb{R}})
.\end{align*}
Note that there are not faithful finite-dimensional reps of \(G\), but
there are infinite-dimensional reps important to quantization in
physics.

\end{remark}

\hypertarget{surfaces}{%
\subsection{Surfaces}\label{surfaces}}

\begin{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 \emph{symmetric} pairing
\begin{align*}
L: H^1(\Sigma;\rho){ {}^{ \scriptstyle\otimes_{{\mathbb{R}}}^{2} }  }\to {\mathbb{R}}
.\end{align*}
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.

\end{remark}

\begin{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.

\end{remark}

\begin{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:

\begin{center}
\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}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwyLCJcXHBpXzFcXFNpZ21hIl0sWzIsMiwiXFxTUF97Mmd9KFxcUlIpIl0sWzIsMCwiRyJdLFswLDFdLFsyLDFdLFswLDIsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

Note
\(m\coloneqq\prod [\tilde\rho a_i, \tilde\rho b_i] \in \ker p \cong {\mathbb{Z}}\),
and it turns out that
\begin{align*}
\operatorname{sig}(\Sigma, \rho) = p-q = 4m
.\end{align*}
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!

\end{remark}

\begin{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
\begin{align*}
\operatorname{sig}(\Sigma, \rho) = 4 \int_\Sigma p^* b
.\end{align*}

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

\end{remark}

\hypertarget{manifolds}{%
\section{3-manifolds}\label{manifolds}}

\begin{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
\begin{align*}
\operatorname{det}: {\operatorname{O}}_{p, q} \to \left\{{\pm 1}\right\}
.\end{align*}
There is a \emph{spinor norm}
\begin{align*}
{\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}
.\end{align*}
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
\begin{align*}
\operatorname{det}: {\operatorname{O}}(V) &\to \left\{{\pm 1}\right\}
\end{align*}
and
\begin{align*}
{\mathrm{spinornorm}}: {\operatorname{O}}(V) &\to L^{\times}/{ {L}_{\scriptscriptstyle \square} }^{\times}
.\end{align*}

\end{remark}

\begin{theorem}[?]

It turns out that

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

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

\end{theorem}

\begin{proof}[?]

Re-interplay \({\mathrm{spinornorm}}(m)\) as a quantity
\({\mathrm{RT}}(M, \rho)\) where \({\mathrm{RT}}\) is the
\emph{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 \(c\).

\end{proof}

\hypertarget{symplectic-lhbox-functions}{%
\subsection{\texorpdfstring{Symplectic
\(L{\hbox{-}}\)functions}{Symplectic L\{\textbackslash hbox\{-\}\}functions}}\label{symplectic-lhbox-functions}}

\begin{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
\(L(X, \rho) \coloneqq L(X, \rho; 1/\sqrt q)\).

\end{remark}

\begin{theorem}[?]

Suppose \(L(X, \rho) \neq 0\); then modulo squares
\begin{align*}
L(X, \rho) = \int_X \rho^* c \qquad \in k^{\times}/{ {k}_{\scriptscriptstyle \square} }^{\times}
,\end{align*}
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).

\end{theorem}

\begin{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 \(L(X, \rho)\).

\end{remark}

\begin{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.

\begin{itemize}
\tightlist
\item
  Pass from \(q\) to \(q'\) using a moduli space of pairs \((X,\rho)\)
  where a topological theorem controls the fiber over \({\mathbb{C}}\).
\item
  Pass from \(k\) to \(k'\) using \emph{compatible local systems} from
  NT.
\end{itemize}

\end{remark}

\begin{remark}

Final comparison:

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

\end{remark}

\hypertarget{talk-3-relative-langlands-duality}{%
\section{Talk 3: Relative Langlands
Duality}\label{talk-3-relative-langlands-duality}}

\begin{quote}
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.
\end{quote}

\addsec{ToDos}
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\addsec{Exercises}
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\addsec{Figures}
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\printbibliography[title=Bibliography]


\end{document}
