\input{"preamble.tex"} \addbibresource{Cantrell2022.bib} \let\Begin\begin \let\End\end \newcommand\wrapenv[1]{#1} \makeatletter \def\ScaleWidthIfNeeded{% \ifdim\Gin@nat@width>\linewidth \linewidth \else \Gin@nat@width \fi } \def\ScaleHeightIfNeeded{% \ifdim\Gin@nat@height>0.9\textheight 0.9\textheight \else \Gin@nat@width \fi } \makeatother \setkeys{Gin}{width=\ScaleWidthIfNeeded,height=\ScaleHeightIfNeeded,keepaspectratio}% \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 % Note: addsec only in KomaScript \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} \listoftodos[List of Todos] \cleardoublepage % Hook into amsthm environments to list them. \addsec{Definitions} \renewcommand{\listtheoremname}{} \listoftheorems[ignoreall,show={definition}, numwidth=3.5em] \cleardoublepage \addsec{Theorems} \renewcommand{\listtheoremname}{} \listoftheorems[ignoreall,show={theorem,proposition}, numwidth=3.5em] \cleardoublepage \addsec{Exercises} \renewcommand{\listtheoremname}{} \listoftheorems[ignoreall,show={exercise}, numwidth=3.5em] \cleardoublepage \addsec{Figures} \listoffigures \cleardoublepage \printbibliography[title=Bibliography] \end{document}