\input{"preamble.tex"}

\addbibresource{INIMotivic.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{
\textbf{
    INI KA2W03: Mathematical physics: algebraic cycles, strings and
    amplitudes
  }
    \\ {\normalsize Some sketchy notes} \\
  }







\begin{document}

\date{}
\maketitle


\newpage

% Note: addsec only in KomaScript
\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{preface}{%
\section{Preface}\label{preface}}

These are some (extremely sketchy) notes I've taken while attending the
\href{https://www.newton.ac.uk/event/ka2w03/}{last part of the program
on K-theory, algebraic cycles and motivic homotopy theory at the Isaac
Newton Institute (INI) in Cambridge}. I don't expect these will be even
remotely useful to anyone else, but perhaps this will have some value to
those who just want some vague idea of what topics were discussed in
talks. If you find yourself perplexed by anything here, the fault is
almost surely entirely my own, since I was likely perplexed myself when
I wrote it in the first place. Moreover, if anything appears incorrect,
that too is surely my fault, either due to my own misunderstandings,
typesetting errors, or just not hearing things correctly. The moral of
the story: abandon all hope, all ye who continue reading!

\hypertarget{talk-1-tyler-kelly-open-saito-givental-invariants-and-mirror-symmetry-tuesday-july-19}{%
\section{Talk 1: Tyler Kelly, Open Saito-Givental Invariants and Mirror
Symmetry (Tuesday, July
19)}\label{talk-1-tyler-kelly-open-saito-givental-invariants-and-mirror-symmetry-tuesday-july-19}}

\begin{remark}

Realizations of cycles and motives play a role in the physics of
amplitudes in string theory and the study of Feynman integrals. Today's
focus: mirror symmetry, with a focus on certain integrals over cycles.
Goal of mirror symmetry: link

\begin{itemize}
\tightlist
\item
  Enumerative data for a symplectic manifold, algebraic variety, or
  complex function, or
\item
  Periods of a so-called mirror
\end{itemize}

The ur-example is a quintic threefold.

\end{remark}

\begin{remark}

How this example works: count degree \(d\) rational curves in
\(X \subset {\mathbb{P}}^4\) a quintic threefold. The mirror is a
crepant resolution of
\begin{align*}
Z(\sum x_i^5 - 5\psi \prod x_i ) \subseteq {\mathbb{P}}^4/(C_5)^3
,\end{align*}
a 1-parameter deformation of the quintic. Compute as some integral
\(\int_\gamma \Omega_\psi\).

\end{remark}

\begin{question}

How to construct the mirror? And how to distill the enumerative
information?

\end{question}

\begin{remark}

Today's goal: answer these questions for Landau-Ginzberg (LG) models
\((X,G,W)\) where \(X\) is a quasi-affine variety
\(G\curvearrowright X\) is a finite group, \(W = ?\). These give mirrors
to Fanos, and noncommutative symplectic deformations of complete
intersections in toric varieties.

\end{remark}

\begin{remark}

Example LG model:
\(({\mathbb{C}}^n, \prod \mu_{r_i}, \sum x_i^{r_i}) \rightleftharpoons({\mathbb{C}}^n, 1, \sum x_i^{r_i})\).
Answers to these questions for LG models: the mirror is built via a
combinatorial construction, which annoyingly isn't geometric. For
enumerative data: oscillatory integrals associated to the RHS using
Saito-Givental theory the encapsulate FJRW invariants. Issue:
deformation must be built perturbatively term-by-term.

\end{remark}

\hypertarget{saito-givental-theory-in-dimension-1}{%
\subsection{Saito-Givental Theory in dimension
1}\label{saito-givental-theory-in-dimension-1}}

\begin{remark}

Start with \(W:{\mathbb{C}}^n\to {\mathbb{C}}\) given by
\(\sum x_i^{r_i}\). Build a twisted de Rham complex where
\(\delta = d + h^{-1}dW \vee({-})\) to get
\(0 \to \Omega^0 \xrightarrow{\delta} \cdots\), then take
hypercohomology. For e.g.~\(W = x^r\), a basis can be computed
explicitly for \(H^1\). One can cook up a perfect pairing between a
certain relative homology \(H_n(X, \Re(W/H)\ll 0, {\mathbb{C}})\) and
this hypercohomology, roughly
\begin{align*}
(\psi, \omega) \mapsto \int_\xi e^{W\over h} \omega
\end{align*}
where \(\Re(W/h) \ll 0\) so that the exponential term is small. Now take
deformations to distill enumerative info from this periods. First
approximation: take the versal deformation
\begin{align*}
W^s = \sum x_i^{r_i} + \sum s_{a_1,\cdots, a_n} \prod x_i^{a_i}
.\end{align*}
Integrate to get a power series whose coefficients are
\(\psi_k \in {\mathbb{C}}{\left[\left[ s_{a_1,\cdots, a_n} \right]\right] }\).
Compute this using a dual basis and integration by parts.

Can try \(W = x^4 + s_2 s^2 + s_1 x + s_0\). Problem: lack of linearity!
But this can be fixed to obtain flat coordinates and a primitive form
\(\omega\).

\end{remark}

\begin{remark}

Theorem: there is a deformed potential \(W^t\) which has primitive form
\(\omega = \,dx\) and flat coordinates \(t_0, \cdots, t_{r-2}\) for the
Frobenius manifold construction via Saito-Givental theory for the LG
model \(W = x^r\). This is a generating function for open enumerative
invariants.

\end{remark}

\begin{remark}

Strategy: start with \(X\), where we want to understand its enumerative
geometry. Build the mirror \(\tilde X\) using some type of \emph{open}
enumerative geometry, along with a deformation. Look at integrals of
forms over pieces of \(\tilde X\), and that should recover the
enumerative counts.

\end{remark}

\begin{remark}

Build a moduli of orbidisks \({\mathcal{M}}\) by taking configurations
of marked points on \({\mathbb{P}}^1\), where points off of
\({\mathbb{R}}\) come in pairs of complex conjugates. The local picture
at marked points: \([{\mathbb{C}}/\mu_r]\). Compactify by considering
degenerations of configurations -- internal nodes, boundary nodes,
contracted boundaries:

\includegraphics{figures/2022-07-19_10-37-48.png}

The compactified moduli space is a real manifold with corners.

\end{remark}

\begin{remark}

Take the universal line bundle \({\mathcal{L}}\to {\mathcal{M}}\) and
define the Witten bundle \(R^1(\pi_* {\mathcal{L}})^*\). One needs to
specify a canonical multisection \(s_{\mathrm{can}}\) to control what
happens on the boundary in order to integrate over \({\mathcal{M}}\).
Theorem: in dimension one, the open \(r{\hbox{-}}\)spin invariants exist
and don't depend on choice of this multisection, enjoy topological
recursion, and there are closed form solutions of certain invariants.
Parts of this theorem fail in dimension 2, although analogous statements
can be made.

\end{remark}

\begin{theorem}[Gross-K-Tessler]

Open FJRW invariants exist for
\(({\mathbb{C}}^2, \mu_r \times \mu_s, x^r+y^2)\) and do depend on the
choice of \(s_{\mathrm{can}}\).

\end{theorem}

\hypertarget{dimension-2}{%
\subsection{Dimension 2}\label{dimension-2}}

\begin{remark}

Proceed as before, but run into issues with extra terms involving Gamma
functions in \(h^0\), as opposed to terms attached to \(h^{-1}\)
previously. A solution: insert a hypergeometric function to cancel it,
as Li-Li-Saito-Shen do.

What's the picture?

\includegraphics{figures/2022-07-19_10-56-16.png}

There are some cases with mild singularities (ADE and elliptic type),
outside of these things become unmanageable. Key foothold in the proof:
no wall-crossing to deal with, which is advantageous. Proof uses
topological recursion: compute something in two different ways, set them
equal, and play the two formulae off of each other.

\end{remark}

\hypertarget{talk-2-man-wai-cheung-cluster-dualities-and-superpotentials-for-grassmannian-tuesday-july-19}{%
\section{Talk 2: Man Wai Cheung, Cluster dualities and superpotentials
for Grassmannian (Tuesday, July
19)}\label{talk-2-man-wai-cheung-cluster-dualities-and-superpotentials-for-grassmannian-tuesday-july-19}}

\begin{remark}

Goal: Marsh-Rietsch potential write down a potential for Grassmannians.
Gross-Hacking-Keel-Kontsevich do as well, we'll construct a map \(P\)
going from the former to the latter. Makes use of cluster structure on
\({\operatorname{Gr}}\), cluster varieties, and a GHKK construction.

\end{remark}

\begin{remark}

Cluster algebras: start with a seed
\(s = (\left\{{A_1,\cdots, A_n}\right\}, B = (b_{ij}))\) where \(A_i\)
are variables and \(B = (b_{ij})\) is a skew-symmetrizable matrix. Start
with an initial seed, allow ``mutations'' taking \(s\to s'\) where the
new \(A_i\) are functions of the old, and take all such possible
iterations to generate the algebra.

\end{remark}

\begin{example}[?]

Example:
\(s = \qty{\left\{{A_1, A_2}\right\}, { \begin{bmatrix}  {0} & {-1} \\  {1} & {0} \end{bmatrix} }}\).
Mutate at \(1\) to get \(A_1 A_1' = 1 + A_2\); write
\(A_3 \coloneqq A_1'\) to get \(A_3 = {1+A_2\over A_1}\). The next seed
is \((\left\{{A_3, A_2}\right\}, B')\).

\end{example}

\begin{remark}

Can express cluster structures in terms of triangulations of
\((n+3){\hbox{-}}\)gons:

\includegraphics{figures/2022-07-19_11-30-13.png}

This flip corresponds to the mutation relation
\(A_{24} A_{13} = A_{12} A_{34} + A_{14} A_{23}\). Since we can not flip
the boundary, some variables are frozen -- namely those corresponding to
adjacent nodes, so \(A_{12}, A_{23}, A_{34}, A_{41}\).

\end{remark}

\begin{remark}

The homogeneous coordinate ring of
\({\mathbb{C}}[{\operatorname{Gr}}_k(n)]\) carries a cluster structure.
On the geometry side, cluster varieties are log Calabi-Yau whose rings
of regular functions carry cluster structures. For seeds, variables
correspond to algebraic tori \(({\mathbb{C}}^{\times})^n\), and
mutations are birational maps between tori and defines a gluing between
them.

\end{remark}

\begin{remark}

Formally defining a cluster variety: start with a lattice \(N\) and
skew-symmetric form \(\left\{{{-}, {-}}\right\}\) on \(N\), and integers
\(d_i\), and a seed given by a basis for \(N\). Take a dual basis \(M\)
and scale to \(M^\circ\) by taking a basis \({1\over d_i} e_i^*\). Take
the corresponding tori \(A, X\) and glue by mutations to get the
corresponding cluster variety. The form gives a cluster ensemble lattice
map \(p^*: N\to M^\circ\) where \(n \mapsto \left\{{n, {-}}\right\}\).
Since these commute with mutation, and produce a map of varieties
\(A\to X\).

\end{remark}

\begin{remark}

There is a Langlands dual given by sending \(B\mapsto B^{-t}\). A
conjecture (Bardwellcot-Evans-C-Hong-Lin) of mirror symmetry:
\(A \rightleftharpoons X^L\) and \(X\rightleftharpoons A^L\). For today,
the exchange matrices are skew-symmetric, so the expectation is that
\(A \rightleftharpoons X\) are mirrors.

\end{remark}

\begin{remark}

Cluster varieties can be described by scattering diagrams: defining
\(N_{\mathbb{R}}, M_{\mathbb{R}}\), this is a collection of walls with
finiteness and consistency conditions. A wall is a pair \((d, f_d)\)
where \(d \subseteq M_{\mathbb{R}}\) is the support of walls, a convex
rational polyhedral cone of codimension 1.

\includegraphics{figures/2022-07-19_11-51-51.png}

\end{remark}

\begin{remark}

Crossing walls:
\(p_\gamma: z^m \mapsto z^m f_d^{\pm {\left\langle {n_0},~{m} \right\rangle} }\)
where \(n_0\) is a primitive normal to a wall and \(\gamma\) is a path
between chambers. A motivating example for theta functions:
\(({\mathbb{C}^{\times}})^2\) satisfies
\(H^0(({\mathbb{C}^{\times}})^2; {\mathcal{O}}) = \bigoplus _{m_1, m_2\in {\mathbb{Z}}} {\mathbb{C}}z_1^{m_1} z_2^{m_2}\).
To each nonzero point in \(M^\circ\) assign a theta function
\(\theta_m\), defined from a collection of broken lines with initial
slope \(m\) and endpoint \(Q\) in the positive chamber. Identify points
in the scattering diagram with tropical points of Langlands dual cluster
varieties.

\end{remark}

\begin{remark}

Cluster structures for \({\operatorname{Gr}}\): have a distinguished
anticanonical divisor \(D_{\mathrm{ac} } = \sum _{i=1}^n D_i\) where
\(D_k = D_{[i+1, i+k]} = \left\{{p_{[i+1, i+k]} = 0}\right\}\). Let
\({\operatorname{Gr}}_k({\mathbb{C}}^n)^\circ = {\operatorname{Gr}}_k({\mathbb{C}}^n) / D\),
the positroid\ldots{} something. Compactify using boundary divisors
\(D_k = D_{[i+1, i+(n-k)]}\). Each irreducible component yields a term
in the potential of the mirror (something about LG models). Can somehow
get valuations on the coordinate ring and write the potential directly.

\end{remark}

\begin{remark}

Marsh-Rietsch define an isomorphism from the Jacobi ring of
\(({\operatorname{Gr}}_k({\mathbb{C}}^n)^\circ, W)\) to the equivariant
quantum cohomology ring of
\({\operatorname{Gr}}_{n-k}({\mathbb{C}}^n)\). Rietsch-Williams: there
is a Newton-Okounkov body on one side \(\Delta(D)\), identifies points
in this as tropical points on functions extending to the boundary
of\ldots{} a potential \(\Gamma\)? Use theta reciprocity to get
\(\Delta(D) \cong \Gamma\) via \(p^*\).

\end{remark}

\hypertarget{talk-3-dani-kaufman-cluster-varieties-amplitude-symbols-and-mutation-invariants-tuesday-july-19}{%
\section{Talk 3: Dani Kaufman, Cluster varieties, amplitude symbols, and
mutation invariants (Tuesday, July
19)}\label{talk-3-dani-kaufman-cluster-varieties-amplitude-symbols-and-mutation-invariants-tuesday-july-19}}

\hypertarget{cluster-varieties}{%
\subsection{Cluster Varieties}\label{cluster-varieties}}

\begin{remark}

Recall: cluster varieties are varieties with a toric atlas generated by
some extra combinatorial data. Encodes unexpected coordinate functions,
and automorphisms of the cluster structure encode exotic automorphisms
of the variety. Cluster structure for the Grassmannian: uses Plucker
coordinates. New coordinate functions appear as part of a \emph{symbol
alphabet} for scattering amplitudes for \(N = 4\) SYM (super Yang-Mills)
theory.

\end{remark}

\begin{remark}

Idea: the same function can be obtained from several different
constructions, classifying mutation invariants shows that these must
produce the same functions. Examples of such special functions: traces
of monodromy, cluster character, Diophantine equations of Markov
numbers, theta functions, Skein modules elements of closed loops, etc.

\end{remark}

\begin{remark}

Cluster varieties: varieties built out of seed tori glued along
birational morphisms called mutations.

\includegraphics{figures/2022-07-19_15-26-10.png}

Geometric type cluster varieties can be tracked with quivers -- a quiver
is used as the initial seed and generated a cluster variety after taking
mutations. Cluster ensembles associated with a quiver \(Q\): a pair of
cluster varieties \((A, X)\) and a map \(p:A\to X\) defined on
coordinates on the same seed by \(p^*(x_i) = \prod_j a_j^{e_{ij}}\).
Introduced by Fock, Goncharov toward an algebraic theory of higher
Teichmüller spaces. Example: \({\operatorname{Gr}}_p(p+q)\) and
\({\mathrm{Conf}}_n({\mathbb{P}}^{k-1})\) are an \((A, X)\) pair.
Interestingly, the immutable seeds for \({\operatorname{Gr}}_3(7)\) look
like an \(E_6\) diagram! There are ``phase transitions'' for values of
\(p,q\) where there is a jump from finitely many clusters to infinitely
many.

\end{remark}

\hypertarget{amplitudes}{%
\subsection{Amplitudes}\label{amplitudes}}

\begin{remark}

Scattering amplitudes in the planar limit \(N=4\) SYM live in
\({\mathrm{Conf}}+n({\mathbb{P}}^3)\), and the amplitudes for up to two
loops can be written in terms of polylogarithms. The arguments of these
polylogarithms come from the \(X\) cluster variety structure on this
configuration space, and constitute the \emph{symbol} of the amplitude
which encodes its singularity and branch cut structure. Goal: understand
what happens as cluster ensembles transition from finite type to
infinite type.

\end{remark}

\hypertarget{cluster-modular-groups}{%
\subsection{Cluster modular groups}\label{cluster-modular-groups}}

\begin{remark}

Idea: automorphism group \(\Gamma\) of mutation structures, so similar
to a mapping class group. Can define an action of this on the rational
function fields of these varieties. Mutation invariant functions: those
invariant under some subgroup of \(\Gamma\).

\end{remark}

\begin{remark}

Example: take the Markov quiver, which is associated to the punctured
torus and the affine root system of type \(A_1^{(1, 1)}\) and
\(\Gamma = {\operatorname{PSL}}_2({\mathbb{Z}})\). Similarly a
4-punctured sphere is of type elliptic \(D_4^{(1, 1)}\). See Somos
sequences.

\end{remark}

\hypertarget{surface-cluster-ensembles}{%
\subsection{Surface cluster ensembles}\label{surface-cluster-ensembles}}

\begin{remark}

Look at \(S\) associated to a marked surface and take
\(G = {\operatorname{PSL}}_2({\mathbb{R}})\), then \(A\) corresponds to
Penner's decorated Teichmüller spaces and \(X\) with Teichmüller space
with Fock coordinates parameterizing hyperbolic somethings with an
``opening''.

\end{remark}

\begin{remark}

Main theorem of their thesis: classification of invariants for the
action of Dehn twists on surface cluster ensembles. The invariant ring
\({\mathbb{R}}(X)^{\left\langle{\gamma}\right\rangle}\) is generated by
traces of monodromy operators of excised closed curves and invariant
\(X\) coordinates for an excising triangulation of \(\delta\) a simple
closed curve, where \(S\) is a marked surface and \(\gamma\in \Gamma_S\)
is a Dehn twist about \(\delta\). For \(A\), it's generated by curves
that don't intersect \(\delta\).

\end{remark}

\begin{remark}

There's some way to obtain a quiver from a triangulation of a surface.
More generally, each cluster modular group has an element that looks
like a Dehn twist (a \emph{cluster Dehn twist}), even when the cluster
doesn't come from a surface at all.

\end{remark}

\begin{question}

Some questions:

\begin{itemize}
\item
  What can we say about cluster invariants from higher Teichmüller
  spaces? Are these related to higher laminations, tensor webs?
\item
  Are the square root symbols related to polylogarithm relations?
\item
  The collection of invariants for a Dehn twist behave like cluster
  variables for a limit cluster ensemble. How is this limit related to
  compactifications of Teichmüller space?
\end{itemize}

\end{question}

\hypertarget{talk-4-goncalo-tabuada-noncommutative-generalized-riemann-hypothesis-tuesday-july-19}{%
\section{Talk 4: Goncalo Tabuada, Noncommutative generalized Riemann
hypothesis (Tuesday, July
19)}\label{talk-4-goncalo-tabuada-noncommutative-generalized-riemann-hypothesis-tuesday-july-19}}

\hypertarget{grh}{%
\subsection{GRH}\label{grh}}

\begin{remark}

Recall \(\zeta(s) = \sum_{n\geq 1} {1\over n^s}\), which admits an Euler
product expansion \(\prod _{p\text{ prime}} {1\over 1-p^{-s}}\), which
converges for \(\Re(s) > 1\) and has a simple pole at \(s=1\).

\end{remark}

\begin{conjecture}[Riemann, 1859]

\begin{align*}
\zeta(s) = 0 \implies \Re(s) = {1\over 2}
.\end{align*}

\end{conjecture}

\begin{remark}

Formulating GRH: let

\begin{itemize}
\tightlist
\item
  \(K\) be a global field (e.g.~\(K={\mathbb{Q}}\)), so for
  \(\operatorname{ch}(K) = 0\) a finite extension of \({\mathbb{Q}}\)
  and for \(\operatorname{ch}(K) > 0\) an extension of
  \({\mathbb{F}}_q(t)\).
\item
  \(X\) be a smooth proper \(K{\hbox{-}}\)scheme.
\item
  \(0\leq w \leq 2\dim(X)\)
\item
  \(\Sigma_K\) the infinite set of nonarchimedean places of \(K\)
\item
  For \(v\in \Sigma_k\),

  \begin{itemize}
  \tightlist
  \item
    \(K_V\) the completion of \(K\) at \(v\)
    (e.g.~\({ {\mathbb{Q}}_p }\))
  \item
    \({\mathcal{O}}_v\) the valuation ring of \(K_V\)
    (e.g.~\({ {\mathbb{Z}}_{\widehat{p}} }\))
  \item
    \(\kappa_v\) the residue field of \({\mathcal{O}}_v\)
    (e.g.~\({\mathbb{Z}}/p{\mathbb{Z}}\))
  \item
    \(p_v\) the characteristic of \(\kappa_v\) (e.g.~\(p\))
  \item
    \(N_v\) the cardinality of \(\kappa_v\) (e.g.~\(p\))
  \end{itemize}
\end{itemize}

There is a SES
\begin{align*}
1\to I_v \to G_{K_v} \to G_{\kappa_v} \cong \widehat{{\mathbb{Z}}}\to 1
\end{align*}
where \(I_v\) is the inertia group. Let the Frobenius be \(\pi_v\),
defined as the inverse of
\(\lambda \mapsto \lambda^{N_v}\in \widehat{{\mathbb{Z}}}\). Choose
primes \(\ell_v\) and field embeddings
\(\iota_v: {\mathbb{Q}}_{\ell_v} \to {\mathbb{C}}\).

\end{remark}

\begin{definition}[Serre]

\begin{align*}
L_w(X; s) = \prod_{v\in \Sigma_k} L_{w, v}(X, s), \\
L_{w, v}(X; s) \coloneqq{
1
\over
\operatorname{det}\qty{
\operatorname{id}- N_v^{-s} (\pi_v \otimes{\mathbb{C}}) \mathrel{\Big|}
H_\text{ét}^w(X { \underset{\scriptscriptstyle {k} }{\times} } K_v { \underset{\scriptscriptstyle {K_v} }{\times} } \mkern 1.5mu\overline{\mkern-1.5muK_v\mkern-1.5mu}\mkern 1.5mu; {\mathbb{Q}}_{\ell_v} )^{I_v} \otimes{\mathbb{C}}
}
}
.\end{align*}

\end{definition}

\begin{theorem}[Serre]

\(L_w(X; s)\) is a meromorphic function on the half-plane
\(\Re(s) > {w\over 2} + 1\). Some conditions:

\begin{itemize}
\tightlist
\item
  \(L_w(X; s)\) admits a meromorphic continuation to \({\mathbb{C}}\)
\item
  \(L_w(X; s)\) has no poles in the critical strip
  \({w\over 2}<\Re(s) < {w\over 2} + 1\).
\end{itemize}

\end{theorem}

\begin{conjecture}[GRH]

\(L_w(X; s) = 0\) and
\({w\over 2} < \Re(s) < {w\over 2} + 1 \implies \Re(s) = {w+1\over 2}.\)

\end{conjecture}

\begin{example}[?]

For \(K = {\mathbb{Q}}, X = \operatorname{Spec}{\mathbb{Q}}, w=0\) then
\(L_0(X; s) = \zeta(s)\).

For \(K\) a number field and \(X = \operatorname{Spec}K\),
\(L_0(X; s) = \sum_{I{~\trianglelefteq~}{\mathcal{O}}_K} {1\over N(I)^s}\)
is the Dedekind-zeta function.

\end{example}

\begin{remark}

Status of GRH:

\begin{itemize}
\item
  For \(\operatorname{ch}k = 0\), no cases are known. The case of a
  point \(\operatorname{Spec}{\mathbb{Q}}\) is already classical RH!
\item
  \(\operatorname{ch}k > 0\) follows from Grothendieck-Deligne.
\end{itemize}

\end{remark}

\hypertarget{noncommutative-geometry}{%
\subsection{Noncommutative geometry}\label{noncommutative-geometry}}

\begin{remark}

Recall that a DG-category is a category enriched over complexes of
\(K{\hbox{-}}\)modules.

\end{remark}

\begin{example}[?]

Examples: DG-algebras and \(\mathsf{Perf}_{{\mathsf{dg}}}(X)\) for
\(X\in {\mathsf{Sch}}_{/ {K}}\). By Beilinson, \(X = {\mathbb{P}}^1\),
\(\mathsf{Perf}_{{\mathsf{dg}}}({\mathbb{P}}^1)\) is Morita-equivalent
to
\({ \begin{bmatrix}  {K} & {K \oplus K} \\  {0} & {K} \end{bmatrix} }\).

\end{example}

\begin{definition}[?]

Let \(\mathsf{A}\) be a DG-category, and say \(\mathsf{A}\) is a
\textbf{geometric NC scheme} if there exists a smooth proper
\(X\in {\mathsf{Sch}}_{/ {K}}\) such that there is an admissible
embedding \(A \hookrightarrow\mathsf{Perf}_{\mathsf{dg}}(X)\).

\end{definition}

\begin{example}[?]

\(A\) a finite dimensional \(k{\hbox{-}}\)algebra with finite global
dimension where \(A/K\) is \(K{\hbox{-}}\) separable defines a geometric
NC scheme. For the quiver \(Q \coloneqq(\cdot \to \cdot)\),
\(KQ = { \begin{bmatrix}  {K} & {K} \\  {0} & {K} \end{bmatrix} }\) is
not Morita-equivalent to \(\mathsf{Perf}_{\mathsf{dg}}(Y)\) for any
smooth proper \(Y\). So there exist NC schemes which are not themselves
schemes.

\end{example}

\begin{remark}

Can GRH be formulated in the NC setting? Attempt to define NC
\(\ell{\hbox{-}}\)adic cohomology: let

\begin{itemize}
\tightlist
\item
  \(K\) be a field
\item
  \(\ell \neq \operatorname{ch}(K)\) a prime
\item
  \(A\) a geometric NC scheme
\end{itemize}

Define the étale \({\mathsf{K}}{\hbox{-}}\)theory of \(A\) as
\begin{align*}
{\mathsf{K}}^\text{ét}(A)_{\widehat{\ell}} \coloneqq\mathop{\mathrm{hocolim}}_{n\geq 0} L_{{\operatorname{KU}}} {\mathsf{K}}(A; {\mathbb{Z}}/\ell^n)
.\end{align*}
Define
\begin{align*}
\begin{cases}
H_{\text{even}, \ell}(A) & \pi_0 {\mathsf{K}}^\text{ét}( (A\otimes_K \mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu)_{\widehat{\ell}} ) \otimes_{\mathbb{Z}}{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{\ell} } \right] }
\\
H_{\text{odd}, \ell}(A) & \pi_1 {\mathsf{K}}^\text{ét}( (A\otimes_K \mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu)_{\widehat{\ell}} ) \otimes_{\mathbb{Z}}{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{\ell} } \right] }.
\end{cases}
\end{align*}
These are \({ {\mathbb{Q}}_\ell }{\hbox{-}}\)linear
\(G_{K}{\hbox{-}}\)modules, and can be used to define NC
\(L{\hbox{-}}\)functions.

\end{remark}

\begin{definition}[Noncommutative $L\dash$functions]

Let \(K\) be a global field, \(A\) a NC geometric \(K{\hbox{-}}\)scheme.
Define
\begin{align*}
L_\text{even}(A, s) \coloneqq\prod_{v\in \Sigma_K} L_{\text{even}, v}(A; s) \\
L_{\text{even}, v}(A; s) = {1 \over 
\operatorname{det}\qty{
\operatorname{id}- N_v^{-s} (\pi_v \otimes{\mathbb{C}}) \mathrel{\Big|}
H_{\text{even}, \ell_v} (A\otimes_K K_v)^I_V \otimes{\mathbb{C}}
}
}
,\end{align*}
and \(L_{\text{odd}, v}\) similarly.

\end{definition}

\begin{theorem}[?]

\(L_{\text{even}}(A; s)\) is a meromorphic function on \(\Re(s) > 1/2\),
and odd for \(\Re(s) > 3/2\).

\end{theorem}

\begin{conjecture}[NC generalized RH]

The even zeros in the critical strip are on \(\Re(s) = {1\over 2}\), and
the odd ones on \(\Re(s) = 1\).

\end{conjecture}

\begin{theorem}[Tob?]

Let \(X\) be a smooth proper \(K{\hbox{-}}\)scheme. Then
\(R_\text{even}(\mathsf{Perf}_{\mathsf{dg}}(X)) \iff \left\{{R_w(X)}\right\}_{w \text{ even}}\),
and similarly for odd, where \(R\) denotes the various forms of GRH.

\end{theorem}

\hypertarget{applications-to-commutative-geometry}{%
\subsection{Applications to commutative
geometry}\label{applications-to-commutative-geometry}}

\begin{theorem}[?]

Let \(K\) be a global field with \(\operatorname{ch}(K) = 0\) and let
\(X, Y\) be smooth proper \(K{\hbox{-}}\)schemes and assume condition
(C2). Suppose \(\mathsf{Perf}(X) \cong \mathsf{Perf}(Y)\) is a
Fourier-Mukai equivalence, then
\(\left\{{R_w(X)}\right\}_w \iff \left\{{R_w(Y)}\right\}_{w}\). So this
version of GRH is invariant under derived equivalence.

\end{theorem}

\begin{theorem}[?]

Suppose \(X\) is homologically projectively dual (HPD) to \(Y\), with
\(X\to {\mathbb{P}}(V)\) and \(Y\to {\mathbb{P}}(V {}^{ \vee })\). For
\(L \subseteq V {}^{ \vee }\), write
\(X_L \coloneqq X{ \underset{\scriptscriptstyle {{\mathbb{P}}(V)} }{\times} } {\mathbb{P}}(L^\perp)\)
and
\(Y_L = Y{ \underset{\scriptscriptstyle {{\mathbb{P}}(V {}^{ \vee })} }{\times} } {\mathbb{P}}(L)\).

\end{theorem}

\begin{remark}

See

\begin{itemize}
\tightlist
\item
  Veronese-Clifford duality
\item
  Grassmannian-Pfaffian duality
\item
  Spinor duality?
\item
  Determinantal duality?
\end{itemize}

\end{remark}

\begin{theorem}[?]

Let \(V_1 \cdots V_n\) be simple \(A/J{\hbox{-}}\)modules and consider
\(\mathop{\mathrm{End}}(V_i)\) which are division
\(K{\hbox{-}}\)algebras. Then
\(K \hookrightarrow K_i \hookrightarrow\mathop{\mathrm{End}}(V_i)\)
where the \(K_i\) are finite separable extensions, and
\begin{align*}
\left\{{R_0(\operatorname{Spec}K_i)}\right\}_{1\leq i\leq m} \implies R_\text{even}(A)
.\end{align*}

\end{theorem}

\begin{remark}

This doesn't exactly help prove GRH, but does help establish properties
of various conjectures. See paper of Thomason for paper on étale
\({\mathsf{K}}{\hbox{-}}\)theory.

Some analogs: \(H_\mathrm{dR}\mapsto HP\),
\(H_{{\mathrm{crys}}} \mapsto TP\) (topological periodic cyclic
homology) which comes equipped with a Frobenius and can be used to
define zeta functions.

\end{remark}

\hypertarget{herbert-gangl-the-aomoto-polylogarithm-via-iterated-integrals-wednesday-july-20}{%
\section{Herbert Gangl, The Aomoto polylogarithm via iterated integrals
(Wednesday, July
20)}\label{herbert-gangl-the-aomoto-polylogarithm-via-iterated-integrals-wednesday-july-20}}

\begin{remark}

The plan:

\begin{itemize}
\tightlist
\item
  Iterated integrals and formal versions, leading to the ``algebraic
  fingerprint'' (i.e.~symbols).
\item
  Aomoto polylogarithms
\item
  A new choice of coordinates on configuration spaces
\item
  Relating Aomoto polylogarithms to iterated integrals via the symbol
\end{itemize}

\end{remark}

\hypertarget{classical-polylogarithms}{%
\subsection{Classical Polylogarithms}\label{classical-polylogarithms}}

\begin{remark}

Recall
\begin{align*}
\mathrm{Li}_n(z) = \sum_{k\geq 1} {z^k\over k^n }
.\end{align*}
Here \(n\) is a \emph{weight}, and the \emph{depth} is 1 (the number of
variables). More generally, for example, \(\mathrm{Li}_{m, n}(x, y)\)
has weight \(m+n\) and depth 2. This can be written as an iterated
integral:
\begin{align*}
\mathrm{II}(a_0; a_1,\cdots, a_n: a_{n+1}) = \int_{\Delta_\gamma} {dt_1 \over t_1-a_1} \wedge \cdots\wedge {dt_n \over t_n - a_n}
\end{align*}
where
\(\gamma: [0, 1]\to {\mathbb{C}}\setminus\left\{{ a_1,\cdots, a_n }\right\}\)
with \(\gamma(0) = a_0, \gamma(1) = a_{n-1}\) and \(\Delta_\gamma\) is a
certain simplex. Some properties:

\begin{itemize}
\tightlist
\item
  Form an algebra under the shuffle product
\item
  Path composition
\item
  Path reversal
\item
  An associated differential equation
\end{itemize}

\end{remark}

\begin{remark}

One can encode this formally as an algebra over a field \(F\) with
generators \(I(a_0, \cdots, a_{n+1})\) and relations coming from the
above properties. A new feature: a ``semicircle coproduct'' due to
Goncharov, whose terms are indexed by polygons involving any subset of
the \(a_i\) when arranged along a semicircle:

\includegraphics{figures/2022-07-20_09-14-37.png}

In general, the symbol for \(I(\cdots)\) will take values in
\((F^{\times}){ {}^{ \scriptstyle\otimes_{?}^{n} } }\).

\end{remark}

\hypertarget{aomoto-polylogarithms}{%
\subsection{Aomoto Polylogarithms}\label{aomoto-polylogarithms}}

\begin{remark}

Idea: attach a period to any pair of simplices \((L, M)\) in
\({\mathbb{P}}^n({\mathbb{C}})\) determined by coordinates
\((\ell_0,\cdots, \ell_n; m_0, \cdots, m_n)\).

\begin{itemize}
\tightlist
\item
  To \(L\), attach \(\omega_L\) a differential form,
  \(\operatorname{dLog}\qty{z_1\over z_0} \wedge \cdots\wedge \operatorname{dLog}\qty{z_n\over z_0}\).
\item
  To \(M\) attach an integral cycle.
\end{itemize}

This gives an assignment \((L, M) \mapsto \int_{\Delta_M} \omega_L\),
which generalizes the classical polylogarithm. Some properties:

\begin{itemize}
\tightlist
\item
  Degeneracies go to zero (?)
\item
  Skew-symmetry:
  \((\sigma L, M) = (L, \sigma M) = \operatorname{sign}(\sigma)(L, M)\)
  for any \(\sigma\in S_n\).
\item
  Invariance for \(g\in \operatorname{PGL}_{n+1}({\mathbb{C}})\) in the
  sense that \((gL, gM) = (L, M)\).
\item
  Additivity, a type of scissors congruence property:
  \(\sum_{i=0}^{n+1} (-1)^i (\ell_0, \cdots, \widehat{\ell}_i, \cdots, \ell_{n+1}; M ) = 0\).
\end{itemize}

BGSV ('89) formalized this in an \(F{\hbox{-}}\)algebra with generators
\((L, M)\) and relations coming from these new properties, and found a
good candidate for the category of mixed Tate motives over a field. An
issue is that the coproduct is only defined on generic such pairs.

\end{remark}

\begin{example}[?]

Consider \(n=1\) and \((L, M)\) with \((\ell_0, \ell_1, m_0, m_1)\),
then
\begin{align*}
A_1(L, M) = \int_{m_0}^{m_1} \operatorname{dLog}\qty{z_1 - \ell_1 \over z_0 - \ell_0}
.\end{align*}

One can extract a cross-ratio from this which lives in \(F^{\times}\).

\end{example}

\begin{remark}

A surprise to BGSV: a certain expression can be written as a combination
of Steinberg elements \(a\otimes(1-a)\) which is related to motivic
cohomology and the Bloch-Suslin complex. Goncharov also writes
\(A_3(L, M)\) in terms of polylogarithms, and attaches a symbol to
general \(A_n(L, M)\) in terms of Plucker coordinates.

\end{remark}

\begin{theorem}[G-Charlton-Radchenko]

The symbol of \(A_n(\mathbf{v})\) can be written as a sum of iterated
integrals.

\end{theorem}

\hypertarget{adriana-salerno-and-ursula-whitcher-diagonal-pencils-and-hasse-witt-invariants-wednesday-july-20}{%
\section{Adriana Salerno and Ursula Whitcher, Diagonal pencils and
Hasse-Witt invariants (Wednesday, July
20)}\label{adriana-salerno-and-ursula-whitcher-diagonal-pencils-and-hasse-witt-invariants-wednesday-july-20}}

\begin{remark}

Calabi-Yau threefolds generalize elliptic curves, and K3s are somewhere
in-between. Idea: use mirror symmetry to prove arithmetic properties of
higher dimensional varieties, generalizing things we know about elliptic
curves.

\end{remark}

\begin{remark}

Consider the Legendre family \(X_\psi: y^2 = x(x-1)(x-\psi)\) satisfies
Picard-Fuchs, and over \({\mathbb{F}}_p\) the trace of Frobenius is
\(a_p = 1 + p - {\sharp}X_\psi({\mathbb{F}}_p)\). Igusa noticed that the
Picard-Fuchs equations for the holomorphic form is hypergeometric, and
\(a_p\) satisfies a truncated hypergeometric formula. Some progress on
arithmetic properties of the Fermat quintic.

\end{remark}

\begin{remark}

What is the mirror of smooth quintics in \({\mathbb{P}}^4\)? Start with
the Fermat quintic pencil \(\sum x_i^5 - \psi \prod x_i = 0\), which
admits a \(C_5^3\) action. Taking the quotient and resolving
singularities yields the mirror \(Y\). For \(\psi \in {\mathbb{Z}}\),
the point counts agree:
\({\sharp}X({\mathbb{F}}_q) \cong {\sharp}Y({\mathbb{F}}_q)\operatorname{mod}q\).

\end{remark}

\begin{remark}

Generalized hypergeometric functions:
\begin{align*}
{}_A F_V(\alpha, \beta | z) = \sum_k {\alpha_{1, k} \cdots \alpha_{A, k} \over \beta_{1, k} \cdots \beta_{B, k}} z^k
.\end{align*}

\end{remark}

\begin{remark}

Daqing-Wang: for the Fermat pencil \(X_\psi\) and its Greene-Plesser
mirror \(Y_\psi\), the unit roots of their zeta functions coincide,
yielding the congruence of point counts. Kadir claims something similar
works for generalize Fermat pencils of CYs in a Gorenstein Fano weighted
projective space and its Greene-Plesser mirror. Recent work by (lots of
people) uses BHK mirror symmetry to generalize to invertible pencils in
\({\mathbb{P}}^n\).

\end{remark}

\begin{remark}

Batyrev's construction: mirror families of CYs using reflexive
polytopes. Look at polar polytopes, lattice polytopes \(\Delta\) which
contains \(0\) in its interior. The facet equation can be normalized to
\(\sum a_i x_i = -1\), and there is an associated \textbf{polar
polytope} \(\Delta^\circ\) associated to the coefficients \((a_i)\). The
polytope \(\Delta\) is \textbf{reflexive} if \(\Delta^\circ\) is again a
lattice polytope. Note that there is no classification of reflexive
polytopes in dimensions \(d\geq 5\).

\end{remark}

\begin{remark}

Building hypersurfaces: for each lattice point choose a parameter
\(\alpha\) and define
\(p_\alpha = \sum_{M \cap} \alpha_m \prod x^{\cdots}\). Can use this to
somehow cook up mirror pairs of CYs? Using that CYs can be realized as
hypersurfaces in toric varieties.

\end{remark}

\begin{remark}

Kernel pairs: for \(\Delta, \Gamma\) combinatorially equivalent
polytopes, put vertices as columns of matrix and see if they have equal
kernels. These are resolutions of quotients of a common toric variety,
and the corresponding vertex pencils satisfy the same Picard-Fuchs
equation.

\end{remark}

\begin{remark}

An alternative to counting points for small primes: for \(X\) a smooth
variety, Frobenius induces a \(p{\hbox{-}}\)linear operator
\(H^n(X; {\mathcal{O}}_X){\circlearrowleft}\), and the matrix for this
map in some basis is the \textbf{Hasse-Witt matrix} \(HW_p\). Katz: for
CYs, this determines the point counts mod \(p\)? A theorem of
Huang-Lian-Yau-Yu shows that \(HW_p\) for \(X\) a Calabi-Yau satisfies a
truncated relation and shows that the period integral has a certain
power series expansion. For smooth CYs, we have a good idea of what
\(HW_p\) is.

\end{remark}

\begin{remark}

Idea: for mirror kernel pairs, can sometimes replace a very complicated
equation with a much simpler equation for its mirror, e.g.~if one wants
to do point counts or study the Picard-Fuchs equation. Can classify K3
surfaces by their \(HW_p\).

\end{remark}

\begin{remark}

Picard-Fuchs for rank 19 lattice-polarized K3s can be written as the
symmetric square of a 2nd order linear Fuchsian differential equation.
For hypergeometric functions, can use Clausen's formula.

\end{remark}

\begin{remark}

Let \(\Gamma \subseteq {\operatorname{PSL}}_2({\mathbb{R}})\), and
define a modular curve as
\(\mkern 1.5mu\overline{\mkern-1.5mu\dcosetl{\Gamma}{{\mathbb{H}}}\mkern-1.5mu}\mkern 1.5mu\).
See Atkin-Lehner map. Group I in the classification is related to
Monstrous Moonshine, and the classification shows it's associated to
\(\Gamma_0(3)^+\) where plus means retaining all Atkin-Lehner
involutions.

\end{remark}

\hypertarget{daniel-kaplan-quiver-varieties-and-symplectic-resolutions-of-singularities-wednesday-july-20}{%
\section{Daniel Kaplan, Quiver varieties and symplectic resolutions of
singularities (Wednesday, July
20)}\label{daniel-kaplan-quiver-varieties-and-symplectic-resolutions-of-singularities-wednesday-july-20}}

\begin{remark}

Three main sections:

\begin{itemize}
\tightlist
\item
  Spaces you (yes you!) care about can be realized as quiver varieties,
\item
  These are well-understood with respect to symplectic resolutions,
\item
  (joint work with Travis Schedler) Even more spaces are ``locally''
  quiver varieties.
\end{itemize}

\end{remark}

\begin{example}[?]

Let \(C_2\curvearrowright{\mathbb{C}}^2\) by negation, then
\(C^2\curvearrowright{\mathbb{C}}[{\mathbb{C}}^2] = {\mathbb{C}}[x, y]\)
by \(f(x, y) \mapsto f(-x, -y)\). The invariants are
\(R\coloneqq{\mathbb{C}}[x, y]^{C_2} = {\mathbb{C}}[x^2, y^2, xy] = {\mathbb{C}}[y,v,w]/\left\langle{uv-w^2}\right\rangle\)
which is a type \(A_1\) singularity at the origin. Take a blowup
\(\operatorname{Bl}_0(X) \to X \coloneqq\operatorname{Spec}R\).

\end{example}

\begin{example}[?]

Let's do the same thing but differently: consider the nilpotent cone
\({\mathrm{Nil}}({\mathfrak{sl}}_2) = \left\{{m = { \begin{bmatrix}  {a} & {b} \\  {c} & {d} \end{bmatrix} } {~\mathrel{\Big\vert}~}{\mathrm{tr}}(m) = \operatorname{det}(m) = 0}\right\} = \left\{{{ \begin{bmatrix}  {a} & {b} \\  {c} & {d} \end{bmatrix} } {~\mathrel{\Big\vert}~}a^2-bc=0}\right\}\),
which is the same equation as before. Take pairs
\((m, \ell\in \ker m)\), then the projection map \((m, \ell) \mapsto m\)
is the cotangent bundle
\({\mathbf{T}} {}^{ \vee }{\mathbb{P}}^1\to {\mathrm{Nil}}({\mathfrak{sl}}_2)\).
Now perhaps it's clear how to generalize, e.g.~by taking other lie
algebras.

\end{example}

\hypertarget{quivers}{%
\subsection{Quivers}\label{quivers}}

\begin{remark}

A quiver is the data \(Q = (Q_0, Q_1, s,t: Q_1\to Q_0)\) where \(Q_0\)
is a vertex set and \(Q_1\) is an arrow set, and everything is over a
field \(k={\mathbb{C}}\) in our case. Fix \(d\in {\mathbb{N}}^{Q_0}\), a
tuple \(d = (d_i)_{i\in Q_0}\), then
\begin{align*}
{\mathsf{Rep}}(Q, d) = \bigoplus _{q\in Q_1} \mathop{\mathrm{Hom}}_{{\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}}({\mathbb{C}}^{d_{s(q)} }, {\mathbb{C}}^{d_{t(q)} })
.\end{align*}
There is an action
\(G_d \coloneqq\prod_{i\in Q_0} \operatorname{GL}_{d_i}({\mathbb{C}})\)
by conjugation, which we sometimes quotient by.

\end{remark}

\begin{remark}

The \(G_d\) orbits are too big! Let \(A,B,C\) be matrices and consider
the quiver
\({\mathbb{C}}^2 \xrightarrow{A} {\mathbb{C}}^2{\circlearrowleft}_{B} \xrightarrow{C} {\mathbb{C}}\).
It's hard to find \(G_d{\hbox{-}}\)invariant functions, and in fact
\({\mathbb{C}}[{\mathsf{Rep}}(Q, d)]^{G_d} = {\mathbb{C}}[\left\{{ {\mathrm{tr}}(\alpha), \alpha \text{ a cycle} }\right\} ]\).
Note that \(B\) is a cycle in this case.

\end{remark}

\begin{remark}

There is a process of doubling a quiver: add inverse arrows, and
duplicate every loop:

\includegraphics{figures/2022-07-20_11-34-06.png}

\end{remark}

\begin{remark}

Note
\begin{align*}
{\mathsf{Rep}}(\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu, d) \to {\mathsf{Rep}}(Q, d) \oplus {\mathsf{Rep}}(Q^*, d) = {\mathsf{Rep}}(Q, d) \oplus {\mathsf{Rep}}(Q, d) {}^{ \vee }\to {\mathbf{T}} {}^{ \vee }{\mathsf{Rep}}(Q, d)
,\end{align*}
which yields a cotangent bundle and puts us in the realm of symplectic
geometry. The moment map is
\begin{align*}
\mu_d: {\mathsf{Rep}}(\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu, d) \to {\mathfrak{g}}_d {}^{ \vee }&\leftarrow{\mathfrak{g}}_d \\
{\mathrm{tr}}(X({-})) &\mapsfrom X
.\end{align*}
Write \(\nu = \sum_{q\in Q_1} [q, q^*]\) where the commutator is
\(aa^* - a^*a\), then \(p\in \mu_d^{-1}(0)\) if \(p\) is a module for
the path algebra
\({\mathbb{C}}\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{Q}}\mkern-1.5mu}\mkern 1.5mu/\left\langle{\nu}\right\rangle = \Pi(Q)\).

\end{remark}

\begin{remark}

Let
\(\theta \in {\mathbb{Z}}^{Q_0} = \mathop{\mathrm{Hom}}_{{\mathsf{Alg}}{\mathsf{Grp}}}(\prod \operatorname{GL}_{d_i} ({\mathbb{C}}), {\mathbb{C}^{\times}})\)
where \(\theta \cdot \dim(\rho) = 0\). Say \(\rho' \subseteq \rho\) is
destabilizing if \(\dim(\rho') \cdot \theta> 0\), and \(\rho\) is
\(\theta{\hbox{-}}\)semistable if it does not contain any destabilizing
\(\rho'\). One can then define
\begin{align*}
{\mathcal{M}}(Q, d, \theta) = \mathop{\mathrm{Proj}}\bigoplus {\mathbb{C}}[ \mu_d^{-1}(0) ]^{k\theta ?}
.\end{align*}
This is an important object: the extra \(\theta\) parameter can be sent
to zero, and generically will yield a smooth space.

\end{remark}

\hypertarget{local-to-global-obstructions}{%
\subsection{Local to global
obstructions}\label{local-to-global-obstructions}}

\begin{theorem}[Bellamy-Schedler]

There is a decomposition into symmetric spaces:
\begin{align*}
{\mathcal{M}}(Q, d, \theta) = \prod_{i = 1}^k \operatorname{Sym}^{n_i} {\mathcal{M}}(Q, d_i, \theta)
\end{align*}
where \(d = \sum n_i d_i\) and \(d_i \cdot \theta = 0\) for all \(i\).
Moreover, this has a symplectic resolution iff each factor does iff
\(\gcd(d_i) = 1\) or \((\gcd(d_i), p\qty{d\over \gcd(d_i)} ) = (2, 2)\).

\end{theorem}

\begin{theorem}[K-Schedler]

Let \({\mathcal{M}}(A, d, \theta)\) be the moduli space of
\(\theta{\hbox{-}}\)semisimple \(A{\hbox{-}}\)modules of dimension
\(d\), where
e.g.~\(A = k\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu / \left\langle{R}\right\rangle\).
There is an isomorphism of formal Poisson varieties after completing:
\begin{align*}
\widehat{{\mathcal{M}}}(\Lambda(Q), d, \theta)_m  { \, \xrightarrow{\sim}\, }\widehat{{\mathcal{M}}}(Q', d', \theta)_0
,\end{align*}
where e.g.~\(Q = Q'\) in type \(\tilde A_n\).

\end{theorem}

\begin{remark}

Local to global obstructions: since \(X\) is Poisson, it has a
stratification into symplectic leaves,
\(X = {\textstyle\coprod}_i S_i\). There is an action of
\(\pi_1(S_i, x_i)\) on the set of local resolutions
\(\left\{{ \tilde U_{s_i} \xrightarrow{\pi} U_{s_i} }\right\}/W\) where
\(W\) is a Nakajima-Weyl group. This action can be nontrivial, and there
is a compatibility with leaf closures: if
\(\mkern 1.5mu\overline{\mkern-1.5muS_i\mkern-1.5mu}\mkern 1.5mu \supseteq S_j\)
then
\({ \left.{{ \pi_{\mkern 1.5mu\overline{\mkern-1.5muS\mkern-1.5mu}\mkern 1.5mu_i} }} \right|_{{S_j}} } = \pi_{S_j}\).

\end{remark}

\begin{example}[?]

In the situation \(C_2\curvearrowright({\mathbb{C}^{\times}})^2\) there
are four fixed points \((\pm 1, \pm 1)\) yielding simple singularities
with local resolutions. The theorem implies there is a global symplectic
resolution.

\end{example}

\hypertarget{jonathan-rosenberg-twisted-derived-equivalences-and-string-theory-dualities-with-b-field-wednesday-july-20}{%
\section{Jonathan Rosenberg, Twisted derived equivalences and string
theory dualities with B-field (Wednesday, July
20)}\label{jonathan-rosenberg-twisted-derived-equivalences-and-string-theory-dualities-with-b-field-wednesday-july-20}}

\begin{remark}

Type II string theory: spacetime is
\(X = {\mathbb{R}}^{10-2n} \times M\) where \(M\) is compact Kähler with
a Ricci-flat metric and trivial canonical (the Calabi-Yau condition),
not necessarily simply connected. For us: \(M\) is smooth projective,
\(X\) is quasi-projective. The physics is controlled by D-branes (D for
Dirichlet), either complex submanifolds in IIB or isotropic submanifolds
for the symplectic structure in IIA. More generally, IIA branes are
coherent sheaves, and IIA branes are classified by the Fukaya category.
In type IIB, there is interest in
\({ \mathsf{D} }^b {\mathsf{Coh}}(X)\), since complexes can capture
joining branes, separating, collapsing into other configurations, etc.
There is a coarser topological classification given by topological
\({\mathsf{K}}{\hbox{-}}\)theory (Minasian-Moore, Witten). The D-branes
give classes in \({\operatorname{KU}}^0(M)\) in IIB and
\({\operatorname{KU}}^{-1}(M)\) for IIA.

\end{remark}

\begin{remark}

Today: orientifold string theories. Classically, strings moving in time
yield maps \(\Sigma\to X\) where \(\Sigma\) is the string worldsheet and
\(X\) is spacetime. This is a \(\sigma{\hbox{-}}\)model. Orientifold
theory: ask for strings to be equivariant with respect to involutions
\(\iota\) and worldsheet parity operators -- if
\(\iota = \operatorname{id}_X\), this recovers type I string theory
(forgetting string orientation). In this theory, D-brane charges live in
real \({\mathsf{K}}{\hbox{-}}\)theory \({\mathsf{K}}R(X, \iota)\), which
reduces to \({\operatorname{KO}}(X)\) when
\(\iota = \operatorname{id}_X\). 2015: joint work classifies orientifold
string theories associated to (anti)holomorphic involutions on elliptic
curves: there are 10. One case corresponds to Karoubi and Donovan's
twisted \({\mathsf{K}}{\hbox{-}}\)theory.

\end{remark}

\begin{remark}

Can view an elliptic curve with antiholomorphic involution as a smooth
genus one curve, possibly with no real points. Karoubi and Weibel (2003)
show that \({\mathsf{K}}R\) of the complex curve is related to
\({\mathsf{K}}^\mathsf{Alg}\) of the real variety. Suggests that duality
of orientifold string theories may be reflected in a equivalence of
derived categories for \emph{real} (rather than complex) curves.

\end{remark}

\begin{remark}

Derived equivalence: prototypical example is
\({ \mathsf{D} }(X) \cong { \mathsf{D} }(Y)\) for \(X,Y\) smooth
varieties over \(k\) via Mukai duality. Formulated in terms of the
Poincaré sheaf on \(X\times \widehat{X}\) for \(X\) an abelian variety
and \(\widehat{X}\) its dual. Integration over the fiber is really
derived pushforward, so define the transform as
\begin{align*}
\phi({\mathcal{F}}) = {\mathbb{R}}\pi_{2, *} ({\mathcal{L}} \overset{\mathbb{L}}{ \otimes} \pi_{1}^* {\mathcal{F}})
.\end{align*}
AVs are simple examples of CYs?

\end{remark}

\begin{remark}

Elliptic curves are self-dual, so it works here, which suggests it may
also work fiberwise for elliptic fibrations. Căldăraru considers
elliptic fibrations \(f: X\to Z\) with no section. The role of
\(\widehat{X}\) is the relative Jacobian \(\widehat{f}: J\to Z\), the
relative module space of semistable sheaves of rank 1 and degree zero on
the fibers of \(f\). Then \(\widehat{f}\) is an elliptic fibration with
section, but there's no universal sheaf to play the role of the Poincaré
sheaf \({\mathcal{L}}\). Two reasons for this:

\begin{itemize}
\tightlist
\item
  \(J\) may have singularities. Not so serious, can resolve and replace
  \(J\) by a small resolution
\item
  May exist locally but not globally.
\end{itemize}

It turns out that \({\mathcal{L}}\) exists as a twisted sheaf: the
transition functions may only agree up to a 2-cycle, defining a class
\(\alpha\) in the Brauer group \(H_\text{ét}^2(J; {\mathbb{G}}_m)\). A
similar recipe goes through: pull up to
\(X{ \underset{\scriptscriptstyle {Z} }{\times} } J\), tensor by the
\(\alpha^{-1}{\hbox{-}}\)twisted sheaf \({\mathcal{L}}\), and push down
to \(X\).

\end{remark}

\begin{theorem}[JR 2017]

View a complex elliptic curve
\(\mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu\) defined
over
\(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu = {\mathbb{C}}\)
with a free antiholomorphic involution, regarded as \(C({\mathbb{C}})\)
with its \({ \mathsf{Gal}} ({\mathbb{C}}/{\mathbb{R}})\) action for
\(C\) a smooth projective curve with \(g=1\) defined over
\({\mathbb{R}}\) and \(C({\mathbb{R}}) = \emptyset\). Think of
\(C \to \operatorname{Spec}{\mathbb{R}}\) as an elliptic fibration
without a section, and the relative Jacobian as
\(E\to \operatorname{Spec}{\mathbb{R}}\). There is a Mukai equivalence
\({ \mathsf{D} }(C) { \, \xrightarrow{\sim}\, }{ \mathsf{D} }(E, \alpha)\)
where
\(\alpha\in \mathop{\mathrm{Br}}(E)/\mathop{\mathrm{Br}}({\mathbb{R}})\)
is the unique nonzero class.

\end{theorem}

\begin{remark}

This is an AG version of \(T{\hbox{-}}\)duality of orientifold theories,
and \(\alpha\) corresponds to the \(B{\hbox{-}}\)field.

\end{remark}

\hypertarget{generalizations}{%
\subsection{Generalizations}\label{generalizations}}

\begin{remark}

Generalizing from genus 1 curves to K3s: let \(E_1, E_2\) be elliptic
curves over \({\mathbb{C}}\) with real \(j{\hbox{-}}\)invariant, so they
admit antiholomorphic involutions. Cook up a real structure on
\(X = (E_1\times E_2)/\mu_2\) and hence on its blowup \(\widehat{X}\)
which is a Kummer K3. Projection gives an elliptic fibration without a
section. The relative Jacobian gives a dual \(K3\) fibred over
\({\mathbb{P}}^1\) with a canonical section and the same derived
equivalence as in the theorem. Can interpret as duality of orientifold
type IIB string theories on K3s.

\end{remark}

\begin{remark}

Generalizing to genus 1 curves over general fields: let
\(\operatorname{ch}k = 0\) and \(C_{/ {k}}\) be a smooth connected
projective curve with \(g=1\) and let \(E\) be its Jacobian. Then \(J\)
is elliptic over \(k\) and \(C\) is an \(E{\hbox{-}}\)torsor, so defines
a class in
\(\delta(C) \in H^1_\text{ét}(\operatorname{Spec}k, E(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu))\).
Can how a \(d_2\) vanishes in the Leray spectral sequence
\begin{align*}
E_2^{p, q} H^p_\text{ét}(\operatorname{Spec}k; {\mathbb{R}}^q \pi_* {\mathbb{G}}_m) \Rightarrow H_\text{ét}^{p+q}(E; {\mathbb{G}}_m)
.\end{align*}
Use Tsen's theorem to get vanishing. Then
\(\mathop{\mathrm{Br}}(E) \cong \mathop{\mathrm{Br}}(E, \operatorname{Spec}k) \oplus \mathop{\mathrm{Br}}(k)\)
where the first term is the \emph{relative Brauer group}.

\end{remark}

\hypertarget{andrew-harder-hyperelliptic-curves-and-planar-2-loop-feynman-graphs-wednesday-july-20}{%
\section{Andrew Harder, Hyperelliptic curves and planar 2-loop Feynman
graphs (Wednesday, July
20)}\label{andrew-harder-hyperelliptic-curves-and-planar-2-loop-feynman-graphs-wednesday-july-20}}

\hypertarget{introductory}{%
\subsection{Introductory}\label{introductory}}

\begin{remark}

Feynman graphs \(\Gamma\): vertices, edges, and half-edges where (for
us) all vertices has a unique adjacent half-edge. Can attach a Feynman
integral: to any internal edge, attach
\(q_e = \sum \delta_{e, \ell} k_e + \sum \delta_{e, v} p_v\) where
\(\delta \in \left\{{0, \pm 1}\right\}\) and
\(I_\Gamma = c \int_{{\mathbb{R}}^{?}} {dk \over \prod_e (q_e^2 -m_e^2) }\)
for \(m_e \in {\mathbb{C}}\). Can reduce dimension to write
\(G = \sum x_e (q_e^2 - m_e^2)\), look at discriminant locus for a
family of quadrics? Yields a locus \(UF=0\) for two polynomials \(U\)
and \(F\). Somehow get a parametric form
\begin{align*}
I_\Gamma = c\int_\sigma {U^? \over F^?} \Omega_0
\end{align*}
where \(\sigma\) is a cycle given by
\(\left\{{{\left[ {x_1: \cdots : x_E} \right]} {~\mathrel{\Big\vert}~}x_i \in {\mathbb{R}}_{\geq 0}}\right\}\);
this is an integral on the complement of \(Z(UF) = 0\).

\end{remark}

\begin{theorem}[?]

After blowing up \({\mathbb{P}}^{{\left\lvert {E} \right\rvert} - 1}\),
the integral \(I_\Gamma\) is well-defined and a relative period of the
pair \((P\setminus Z(U^bF^a), B)\) where \(P\) is a toric variety and
\(B\) its toric boundary, and \(a,b \in \left\{{0, 1}\right\}\)
depending on the dimension \(D\). Thus the geometry of \(Z(U)\) and
\(Z(F)\) determine \(I_\Gamma\).

\end{theorem}

\begin{remark}

The \(U\) and \(F\) are defined combinatorially from the Feynman graph.
Let \(\Gamma_{\in } = (V, E)\) and let \(S_1\) be the set of all
spanning trees of \(\Gamma_{\in}\). Let \(x^T = \prod_{e\not\in T} x_e\)
for \(T\) any spanning tree, where we have coordinates
\(\left\{{x_e}\right\}_{e\in E}\). This somehow defines \(U\), and \(F\)
is defined by looking at spanning 2-trees (subgraphs with 2 connected
components, no loops, containing all vertices). It turns out \(F\) is
homogeneous, so defines a projective variety.

\end{remark}

\begin{example}[?]

The theta graph yields a smooth elliptic curve in \({\mathbb{P}}^2\).
More generally, connected the two vertices with more edges yields
Calabi-Yau \((n-1){\hbox{-}}\)folds.

\end{example}

\hypertarget{extensions-to-2-loop-graphs}{%
\subsection{Extensions to 2-loop
graphs}\label{extensions-to-2-loop-graphs}}

\begin{remark}

2-loop graphs: two trivalent nodes connected by chains of edges:

\includegraphics{figures/2022-07-20_14-54-37.png}

What does the mixed Hodge structure on cohomology look like? This is an
important step toward understanding periods, and the answer depends
heavily on \(D\). Bloch works out kite and double box graphs.

\end{remark}

\begin{theorem}[DHNV]

For \((a,1,1)\) hypersurfaces, identifies when certain homology (or some
of its weight-filtered pieces) are Hodge-Tate under conditions on \(a\)
and \(D\).

\end{theorem}

\begin{remark}

Note that a Tate HS has \(h^{i, j} = 0\) for \(i\neq j\), and mixed Tate
is an iterated extension of such a Hodge structure. This means it's as
simple as it could reasonably be. Idea of proof: blow up a linear
subspace, remove some singular hyperplanes.

Interestingly, for \(D=4\) only the cases \((3,1,1)\) and \((1,1,1)\)
yields a non-Hodge-Tate mixed Hodge structure. These correspond to
multiple polylogarithms.

\end{remark}

\begin{remark}

WIP: can say more in \((a,1,c)\) case when \(a, c> 1\). Note that the
\((a,1,c)\) case can't yield K3s.

See Prym varieties?

\end{remark}

\hypertarget{noriko-yui-modularity-of-certain-calabi-yau-threefolds-over-mathbbq-wednesday-july-20}{%
\section{\texorpdfstring{Noriko Yui, Modularity of certain Calabi-Yau
threefolds over \({\mathbb{Q}}\) (Wednesday, July
20)}{Noriko Yui, Modularity of certain Calabi-Yau threefolds over \{\textbackslash mathbb\{Q\}\} (Wednesday, July 20)}}\label{noriko-yui-modularity-of-certain-calabi-yau-threefolds-over-mathbbq-wednesday-july-20}}

\begin{remark}

Definition: smooth projective variety \(X\) over \({\mathbb{C}}\),
\(\dim_{\mathbb{C}}X = 3\), and

\begin{itemize}
\tightlist
\item
  \(H^i(X; {\mathcal{O}}_X) = 0\) for \(i=1,2\), and
\item
  \(K_X \cong {\mathcal{O}}_X\).
\end{itemize}

By conjugation, \(h^{i, j} = h^{j, i}\), and by Serre duality
\(h^{i, j} = h^{3-i, 3-j}\). Note \(h^{1,1}, h^{2,2} > 0\) since
Calabi-Yau threefolds are Kähler. The outer edge of the Hodge diamond
contains only zeros and ones and has 7 rows, up to 4 columns. A typical
example: quintic threefolds \(\sum x_i^5 = 0\), or elliptic curves are
dimension 1 CYs and K3s are dimension 2 CYs. More generally, CYs can be
constructed as intersection threefolds in weighted projective spaces.
Other examples:

\begin{itemize}
\tightlist
\item
  Double octics \(w^2 = f_8(x,y,z,w)\)
\item
  \(E\times S/\iota\) where \(E\) is an elliptic curve, \(S\) is a K3,
  \(\iota\) is a non-symplectic involution
\item
  Toric CYs
\end{itemize}

\end{remark}

\begin{remark}

Let \(X_{/ {{\mathbb{Q}}}}\) be Calabi-Yau iff
\(X\otimes_{\mathbb{Q}}{\mathbb{C}}\) is Calabi-Yau in the usual sense.
Suppose it has defining equations in
\({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{m} } \right] }\) for some
\(m\), pick a \emph{good prime} \(p\) coprime to \(m\) so that \(X\) has
good reduction: \(X_p\) is smooth over
\(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}_p\mkern-1.5mu}\mkern 1.5mu\).
Define the zeta function
\(Z_p(X, T) \in {\mathbb{Q}}{\left[\left[ T \right]\right] }\) as usual,
then the Weil conjectures hold for \(X\). Since \(P_1 = P_5 = 1\),
\begin{align*}
Z_p(X, T) = {P^3(T) \over (1-T) P^2(T) P^2(pT) (1-p^3 T)}
.\end{align*}
There is a compatible system of \(\ell{\hbox{-}}\)adic Galois
representations on étale cohomology. Define an \(L{\hbox{-}}\)function
\(L_i(X, s) = \prod_{p\neq \ell \text{ good}} P^i(p^{-s})^{-1}\times (\cdots)\),
where the last factor accounts for when \(\ell=p\). Can compute
\({\sharp}X({\mathbb{F}}_p)\) using the Lefschetz trace formula for
\(\operatorname{Frob}_p\curvearrowright H_\text{ét}(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu; { {\mathbb{Q}}_\ell })\).
Here \({\sharp}X({\mathbb{F}}_p) = 1 + p^3 + (1+p ) t_2(p) - t_3(p)\)
where \(t_2, t_3\) are bounded above by \(h^{1, 1}, \beta_?\) by RH.

\end{remark}

\begin{remark}

The modularity question: are there global functions or automorphic forms
that determine \(L(X, s)\)? This would be the Langlands philosophy.

Say \(X\) is rigid if \(h^{2, 1} = 0\) so \(\beta_3 = 2\) and the Hodge
diamond only has two unknowns, \(h^{1, 1}, h^{2,2}\). These resemble
elliptic curves in terms of the characteristic polynomial.

\end{remark}

\begin{theorem}[?]

Every rigid Calabi-Yau threefold is modular: there is some modular form
\(f\) of weight \(4\) on some \(\Gamma_0(N)\) such that
\(L(f, s) = L(X, s)\).

\end{theorem}

\begin{remark}

There are 14 one-parameter families of CYs defined over \({\mathbb{Q}}\)
of hypergeometric type, and at conifold points each family as a rigid
motive and corresponds to a special cusp form of weight 4 on some
\(\Gamma_0(N)\) where only \(2,3,5\divides N\). The next simplest case
is when \(h^{2, 1} = h^{1, 2} = 1\) and not zero, so \(\beta_3 = 4\).
Constructing such CYs: Nygaard and van Geemen take a certain complete
intersection whose singular locus has 16 ordinary double points and 4
plane conics intersecting in a square; take its resolution. Van Geemen
and Werner/Consani and Scholten construct another using the affine
equation of a skew pentagon, yielding a quintic Calabi-Yau threefold
defined over \({\mathbb{Q}}\) with bad primes \(2,3,5\).

\end{remark}

\begin{theorem}[?]

For \(X\) the Calabi-Yau coming from a complete intersection as above,
\begin{align*}
L(X, s) = L(\chi^3, s) \cdot L(\chi_{(-1)}, s)
,\end{align*}
where \(\chi\) is a Hecke character for \({\mathbb{Q}}(i)\)?

\end{theorem}

\begin{remark}

Note that something similar happens for elliptic curves with CM. Idea of
proof: split \(H^3\) into two rank 2 motives to decompose the L
function.

\end{remark}

\begin{remark}

Since \(X\) is covered by some \(E^3\) where \(E: y^2 = 1+x^4\) which
has CM by \({\mathbb{Q}}(i)\), so \(L(X, s) \divides L(E^3, s)\) and we
can use this to define when a Calabi-Yau has CM in terms of the cover.
General expectation: if \(X\) has CM for an imaginary quadratic number
field \(K\), then \(L\) splits into factors involving Hecke characters
of \(K\). To avoid Hecke characters, define \emph{real} multiplication
instead: if \(X\) is Calabi-Yau with \(h^{p, q} = 1\) for \(p+q=3\) and
\(\beta_3 = 4\), set \(K = {\mathbb{Q}}(\sqrt d)\) and say \(X\) has RM
with \(K\) if

\begin{itemize}
\tightlist
\item
  There is an algebraic correspondence \(\Psi:X\to X\) defined over
  \(K\).
\item
  There is an automorphism of \(H^3\) preserving the Hodge structure,
\item
  There is an automorphism of \(H^3\) preserving Galois representations.
\end{itemize}

\end{remark}

\begin{remark}

See Hilbert modular forms, Adrianov \(L{\hbox{-}}\)functions.

\end{remark}

\hypertarget{johannes-walcher-rationality-of-mums-and-2-functions-thursday-july-21}{%
\section{Johannes Walcher, Rationality of MUMs and 2-functions
(Thursday, July
21)}\label{johannes-walcher-rationality-of-mums-and-2-functions-thursday-july-21}}

\hypertarget{rational-2-functions-are-abelian}{%
\subsection{Rational 2-functions are
abelian}\label{rational-2-functions-are-abelian}}

\begin{remark}

Motivations: mirror symmetry links enumerative geometry (A model) to
variations of Hodge structure (B model). On the B side, arithmetic
properties naturally arise. Do they have an interpretation on the A
side?

Classical mirror symmetry involves maximal unipotent monodromy (MUM) in
the moduli space of CYs. Johannes entered this area by looking at
extensions of algebraic cycles.

\end{remark}

\begin{remark}

2-functions: for \(s \in {\mathbb{Z}}_{>0}\), and
\(s{\hbox{-}}\)function is a power series
\(W\in z{\mathbb{Q}}{\left[\left[ z \right]\right] }\) of the form
\(\sum_d n_d \mathrm{Li}_s(z^d)\). Motivating examples: \(g=0\)
prepotential expanded around MUM in flat coordinates. Equates generating
functions for GW invariants and that for BPS invariants,
\(\sum a_dq^d = \sum n_d \mathrm{Li}_3(q^d)\).

A new feature: framing, where
\(Y \in 1 + z{\mathbb{Z}}{\left[\left[ z \right]\right] }\) defined an
automorphism of \(z{\mathbb{Z}}{\left[\left[ z \right]\right] }\). See
Lagrange inversion for finding coefficients of power series. Theorem: if
\(W\) is a rational 3-function, then all of its framings are
3-functions. Relates to some classical theorems, e.g.~the Wolstenholme
theorem and the Jacobsthal-Kazandzidis congruence.

\end{remark}

\begin{remark}

Over a number field, the \(s{\hbox{-}}\)functions form a finite rank
free module. Muller shows that there can't be a natural choice of basis
for \(K\) an arbitrary number field -- if
\(W\in zK{\left[\left[ z \right]\right] }\) is rational 2-function, so
\(\exp(\delta W) \in K(z)\), then \(K/{\mathbb{Q}}\) must be an abelian
extension.

\end{remark}

\hypertarget{mum}{%
\subsection{MUM}\label{mum}}

\begin{remark}

For \(\pi: Y\to B\) a smooth family of Calabi-Yau threefolds of mirror
quintic type, there is a \({\mathbb{Z}}{\mathrm{VHS} }\) (pure, weight
3). By analog with \(p{\hbox{-}}\)adic Hodge theory, Kontsevich et al
have a theorem that lets you conclude integrality from rationality in
some special cases? Look at monodromy of Picard-Fuchs equations. See
Bogner's ``strange operator''.

\end{remark}

\begin{remark}

Early 2000s: Almkvist-Zudilin look at Calabi-Yau differential operators,
which are 4th order operators with MUM points, duality, integral
fundamental periods, etc so that it could come from the integral
cohomology of a CY.

See Beauville-Apery families of elliptic curves, Hadamard product.

\end{remark}

\begin{remark}

Idea: take a family \(Y\to B\) and \(C\hookrightarrow Y\) some 2-cycles
(curves) which is also a nice family over \(B\), this gives an extension
of VHSs. Somehow classified by intermediate Jacobian? By Griffith's
transversality, the extension is fully determined by a piece in a
certain \(\operatorname{Ext} ^1\) group. Extraction of integers appears
to depend on finding a good basis for 2-functions. Speculation: analytic
2-functions with coefficients in a number field arise as solution to
inhomogeneous linear ODEs with algebraic coefficients, and unravelling
the enumerative interpretation (e.g.~in in open GW/BPS theory) will
involve natural such bases. Maybe \(K\) is the trace field of a
hyperbolic 3-manifold?

\end{remark}

\hypertarget{matt-kerr-k_2-and-quantum-curves-thursday-july-21}{%
\section{\texorpdfstring{Matt Kerr, \(K_2\) and quantum curves
(Thursday, July
21)}{Matt Kerr, K\_2 and quantum curves (Thursday, July 21)}}\label{matt-kerr-k_2-and-quantum-curves-thursday-july-21}}

\begin{quote}
Holy moly, Matt has a lot of equations and writes super fast! I was not
able to record even half of what was written.
\end{quote}

\hypertarget{a-miraculous-coincidence}{%
\subsection{A miraculous coincidence}\label{a-miraculous-coincidence}}

\begin{remark}

Suppose \(f\in \operatorname{Mero}({\mathbb{C}}, {\mathbb{C}})\) with no
poles in the horizontal strip
\({\left\lvert {\Im(z)} \right\rvert} < 2\pi\), then
\(e^{\pm 2\pi i {\frac{\partial }{\partial z}\,}}\) is a shift operator
\(f(z) \mapsto f(z+2\pi i)\). Example:
\(e^{2\pi i {\frac{\partial }{\partial z}\,}} z^k = \sum_\ell \qty{ (2\pi i)^\ell \over \ell! } \qty{{\frac{\partial }{\partial z}\,}}^\ell z^k = (z+ 2\pi i)^k\).
Consider
\begin{align*}
\widehat{\phi }\coloneqq e^r + e^{-r} + e^{2\pi {\frac{\partial }{\partial r}\,}} + e^{-2\pi i {\frac{\partial }{\partial r}\,} }\curvearrowright L^2({\mathbb{R}})
.\end{align*}
This is some kind of quantum Hamiltonian for a Fermi gas? Note that this
is unbounded, but its inverse is bounded and self-adjoint with spectrum
that can be enumerated. This suggests a curve of the form
\(x+x^{-1}+y+y^{-1}= \lambda\), is there any AG or Hodge theory arising
in this spectral problem?

\end{remark}

\begin{remark}

Consider elliptic curves, pullbacks of the Legendre family
\(y^2 = x(x-1)(x-16t^2)\). This has two periods \(A(t), B(t)\) given by
integrating over the two torus cycles \(\alpha,\beta\), which are
complicated but annihilated by the Picard-Fuchs operator
\(L = D_t^2 - 16t^2 (D_t + 1)^2\). Solve \(L({-}) = c\) a constant using
variation of parameters to get
\begin{align*}
C(t) \approx B(t) \int A(t) {\,dt\over t} - A(t) \int B(t) {\,dt\over t}
.\end{align*}
Can choose constant terms such that the monodromy of \(C\) lies in
\({\mathbb{Z}}(2) \left\langle{A(t), B(t)}\right\rangle\) where
\({\mathbb{Z}}(2) = 2\pi i {\mathbb{Z}}\). Set
\(\nu(t) = {1\over \pi^2} \log^2(-t) + {7\over 6} + { \mathsf{O}} (t\log(t))\)
and define \(t_n\) by \(\nu(t_n) = n+2\). Then
\(-\log(-t_n) = \pi\sqrt{n + {5\over 6}}\), which yields a sequence
matching the exponents in the eigenvalues of \(\widehat{\phi}\).
Something is going on!

\end{remark}

\begin{remark}

Local mirror symmetry conjecture linking inhomogeneous Picard-Fuchs
equations of Hori-Vafa models and enumerative geometry of local CYs: the
\(B\) model mirror of \(K_{{\mathbb{P}}^1\times {\mathbb{P}}^1}\) is the
\(\mathrm{VMHS}\) of \(H^3\) of
\(Y_a = V(x+x^{-1}+ y+y^{-1}+ a + uv) \subseteq ({\mathbb{C}^{\times}})^2\times \times {\mathbb{C}}^2\)
with coordinates \({\left[ {x, y} \right]}\) and
\({\left[ {u, v} \right]}\). See Frobenius duality for differential
operators? The Geneva conjectures connected difference operators and
quantum curves to enumerative geometry of CYs.

\end{remark}

\hypertarget{beilinsons-formula}{%
\subsection{Beilinson's Formula}\label{beilinsons-formula}}

\begin{remark}

Like \(X_{/ {{\mathbb{C}}}}\) smooth projective, and note
\begin{align*}
{\mathsf{K}}_2({\mathbb{C}}(X)) = 
{
{\mathbb{C}}(X)^{\times}\wedge {\mathbb{C}}(X)^{\times}\over 
\left\langle{F \wedge (1-F)}\right\rangle 
} 
\xrightarrow{\text{T}} \bigoplus _{p\in ?} {\mathbb{C}^{\times}}
\end{align*}
where \(T\) is the \emph{tame symbol}. We want to compute the regulator
\begin{align*}
\ker(T) = H_M^2(X, {\mathbb{Z}}(2)) \xrightarrow{R_X} H^2_D(X, {\mathbb{Z}}(2)) \cong H^1(X^{\mathrm{an}}, {\mathbb{C}}/{\mathbb{Z}}(2))
.\end{align*}
Beilinson shows that if \(\left\{{f, g}\right\}\in \ker(T)\) and
\(\gamma\in \pi_1(X\setminus?, x_0)\) then
\begin{align*}
{\left\langle {R_X\left\{{f, g}\right\}},~{[\gamma]} \right\rangle} \equiv \int_\gamma \log(f) {dg\over g} - \log g(x_0) \int_\gamma {df\over f}
.\end{align*}

\end{remark}

\begin{remark}

Consider \(X \xrightarrow{\pi} S\) a family of smooth curves and
\(z\in H_m^2(X, {\mathbb{Z}}(2))\) we can consider their fibers
\(z_2\in H_M^2(X_s, {\mathbb{Z}}(2))\). This maps to
\begin{align*}
R(S) = H^1(X_s, {\mathbb{C}}/{\mathbb{Z}}(2)) \cong \operatorname{Ext} ^1_{\mathrm{MHS}}({\mathbb{Z}}(2), H^1(X_s, {\mathbb{Z}}))
.\end{align*}
By Griffiths transversality,
\(\nabla R = F^1 H^1_{X_{/ {S}} } \otimes\omega_S^1\). If \(z\) lifts to
\(\left\{{F, G}\right\}\) (?) then
\(\nabla R = {dF\over F}\wedge {dG\over G}\).

\end{remark}

\hypertarget{families-of-elliptic-curves}{%
\subsection{Families of elliptic
curves}\label{families-of-elliptic-curves}}

\begin{remark}

Let \(\Delta \subseteq {\mathbb{R}}\times [-1, 1]\) be a reflexive
polytope, e.g.~the polytope for \({\mathbb{P}}^n\), let \(\Delta^\circ\)
be is polar dual. Somehow construct an elliptic surface
\({\mathcal{E}}\) out of this! The symbol
\(\left\{{-x, -y}\right\}\in {\mathsf{K}}_2({\mathbb{C}}({\mathcal{E}}))\)
lifting to
\(z\in H^2_M(\Sigma\setminus{\mathcal{E}}_\infty, {\mathbb{Z}}(2))\) and
take its regulator in \(H^1(E_?, {\mathbb{C}}/{\mathbb{Z}}(2))\).

Idea: regulators of \({\mathsf{K}}_2\) of curves relate to Hori-Vafa
models, so we've reduced to relating regulators to quantum curves:

\includegraphics{figures/2022-07-21_10-53-34.png}

\end{remark}

\hypertarget{quantum-curves}{%
\subsection{Quantum Curves}\label{quantum-curves}}

\begin{remark}

Theorems:

\begin{itemize}
\tightlist
\item
  For all \(a\in V\), there is a
  \(\psi_a\in L^2({\mathbb{R}})\setminus\left\{{0}\right\}\) such that
  \(\widehat{\phi }\psi_a = a\psi _a\).
\item
  Asymptotically, 100\% of \(\sigma(\widehat{\phi})\) belong to \(V\).
\end{itemize}

The first primarily uses Beilinson's formula, the second uses
two-variable Fourier transforms. These establish a link between the
spectrum \(\sigma(\widehat{\phi})\) (quantum curve side) to the
\({\mathbb{Z}}{\hbox{-}}\)locus of truncated HNFs (regulator side).

\end{remark}

\begin{remark}

See Grothendieck period conjecture.

\end{remark}

\hypertarget{chloe-xiaohua-ai-from-feynman-amplitudes-to-multiple-l-values-thursday-july-21}{%
\section{Chloe Xiaohua Ai, From Feynman Amplitudes to multiple L-values
(Thursday, July
21)}\label{chloe-xiaohua-ai-from-feynman-amplitudes-to-multiple-l-values-thursday-july-21}}

\begin{question}

Recall that the multiple zeta function is defined as
\begin{align*}
\zeta(\mathbf{s}) = \sum_{n_1> n_2 > \cdots > 0} n_1^{-s_1} n_2^{-s_2}\cdots n_k^{-s_k}
.\end{align*}
The classical zeta generalizes to Dedekind zeta functions
\(\zeta_K(S) = \sum_{0\neq n\subset {\mathcal{O}}_K} N_{K/{\mathbb{Q}}}(n)^{-s}\),
is there an analog for the multiple zeta function?

\end{question}

\begin{remark}

The Hecke formula: takes automorphic periods to \(L{\hbox{-}}\)functions
by integrating a well-chosen Eisenstein series with respect to a
normalized Haar measure. Can this idea be extended to multiple
variables?

\end{remark}

\begin{remark}

Goncharov's theory on Hodge correlators: complex numbers given by
integrals assigned to complex curves or varieties. Can construct
integrals all arising from a single object: Green functions/currents.
Many \(L{\hbox{-}}\)functions can be interpreted as Hodge correlators:
Rankin-Selberg integral, polylogarithms, multiple Eisenstein-Kronecker
series. Hodge correlators are periods of motivic correlators. The Green
functions in Goncharov's construction work over \({\mathbb{C}}\), what
about number fields?

\end{remark}

\begin{remark}

Plectic conjecture: in the presence of real multiplication by a totally
real number field, motives have canonical functorial additional
structure. See Hilbert modular varieties, Hilbert modular forms. Goal:
define higher plectic Green functions, and define multivariate zeta
functions by integrating them. These new multivariate functions will
depend on some combinatorial data: a graph \(\Gamma\). If \(\Gamma\) is
a special tree, then the generalized multiple zeta values
\(Z_{I, v}(\Gamma, {{\partial}}\Gamma)\) can be expressed as finite
\({\mathbb{Z}}{\hbox{-}}\)linear combinations of classical multiple zeta
values. In the totally real case, \(Z\) be expanded as sum in values of
\({\mathcal{L}}_m\), a generalized \(m{\hbox{-}}\)logarithm which is a
nontrivial iterated integral.

\end{remark}

\hypertarget{lawrence-barrott-the-doran-harder-thompson-conjecture-via-the-gross-siebert-program-thursday-july-21}{%
\section{Lawrence Barrott, The Doran-Harder-Thompson conjecture via the
Gross-Siebert program? (Thursday, July
21)}\label{lawrence-barrott-the-doran-harder-thompson-conjecture-via-the-gross-siebert-program-thursday-july-21}}

\begin{remark}

Goal: understand proportion of period integral via enumerative
invariants on the mirror. For mirror symmetry, start with a family of
CYs \(X \xrightarrow{f} {\mathbb{A}}^1\) with \(X\) smooth and
\(f^{-1}(0) = \displaystyle\bigcup D_i\) an SNC divisor. Locally this is
the map \(f^*: t\to \prod_{i=1}^n x_i\) for \(t\) a coordinate on
\({\mathbb{A}}^1\), and we ask that \(X\) is etale locally isomorphism
to \(\operatorname{Spec}k[x_i, y_i]\) for \(y_i\) some free variables.
Want to encode fibers of this family via period integrals; use the dual
intersection complex \(\Sigma_X\), the subset of the cone
\(\bigoplus {\mathbb{R}}_{\geq 0} D_i\) where \(\mathbf{a}\in \Sigma_X\)
if the \(D_i\) with \(a_i > 0\) all intersect with nonempty
intersection.

\end{remark}

\begin{example}[?]

Take a quartic K3 degenerating to a toric boundary. Resolve 24
singularities via blowups to get a cone over a standard simplex. Sum
coordinates to \(\sum D_i\) to get a map
\(\Sigma_X \to {\mathbb{R}}_{\geq 0}\), then \(\Sigma_X(Z)\) gives a
collection of integer points.

\end{example}

\begin{theorem}[Gross-Siebert]

There is a ring structure on formal monomials \(k[\Sigma_X(Z)]\)
extending the one induced by ``conewise addition''.

\end{theorem}

\begin{remark}

This is a graded ring and the mirror is
\(\widehat{X} = \mathop{\mathrm{Proj}}k[\Sigma_X(Z)]\) where
\(z^pz^q = \sum N_r z^r\) where \(N_r\) is a count of punctured (open)
GW invariants. This also works for pairs \((Y, D)\) of a variety and an
anticanonical divisor \(D\in {\left\lvert {-K_Y} \right\rvert}\). Having
a family with a map to \({\mathbb{A}}^1\) yields a Landau-Ginzburg
model. In this case, \(\mathop{\mathrm{Proj}}\Sigma_{(Y, D)}(Z)\) has a
canonical map to \({\mathbb{A}}^1\), and superpotential
\(\widehat{Y} \xrightarrow{W} {\mathbb{A}}^1\).

\end{remark}

\begin{remark}

Tyurin degenerations: a family \(X \xrightarrow{f} {\mathbb{A}}^1\) CYs
with \(X\) smooth and \(f^{-1}(0) = Y_1 \cup_D Y_2\) glued along a
anticanonical divisor \(D \in {\left\lvert {-K_X} \right\rvert}\).

\end{remark}

\hypertarget{dht-conjecture}{%
\subsection{DHT Conjecture}\label{dht-conjecture}}

\begin{remark}

Let \(A \xrightarrow{f} {\mathbb{A}}^2\) be a family of CYs and suppose

\begin{itemize}
\tightlist
\item
  \(f^{-1}(t, 0) = Y_1 \cup_D Y_2\)
\item
  \(f^{-1}(0, t)\) is an SNC with a maximality condition
\end{itemize}

Here locally \(t_1 = \prod x_i, t_2 = \prod y_i\), and \(X\) is
étale-isomorphic to \(\operatorname{Spec}k[x_i, y_i, z_i]\) where the
\(z_i\) are free variables. For some reason, \(\Sigma_X\) looks like a
Toblerone, and we now get \emph{two} maps to \({\mathbb{A}}^1\).

\includegraphics{figures/2022-07-21_14-15-01.png}

\end{remark}

\begin{remark}

The DHT conjecture: \(\widehat{X}\) is a fibred Calabi-Yau and
\(\widehat{X} \to {\mathbb{P}}^1\) with fibers mirror to \(D\).
Topologically, the two maps to \({\mathbb{A}}^1\) can be glued to get
the map to \({\mathbb{P}}^1\), and here
\(\widehat{Y}_1 \xrightarrow{W_1} {\mathbb{A}}^1, \widehat{Y}^2 \xrightarrow{ W_2} {\mathbb{A}}^1\)
can be glued to get the fibration. Doran-Kostiuk-You construct examples
as complete intersections on toric varieties of the form
\(\Pi\times {\mathbb{P}}^1\) and related the periods of the fibres via
Hadamard products of relative \(I\) functions. Moral: can glue periods
for \(\widehat{Y_1}, \widehat{Y_2}\) to get periods for all of \(X\).

\end{remark}

\begin{remark}

The GS program gives a scattering diagram (pairs of walls and functions)
on the Toblerone which gives gluing instructions for the mirror family.
Scattering diagrams have canonical differential forms, i.e.~canonical
sections of \(\omega{\widehat{X}}\), but
\({ \left.{{\omega_{\widehat{X}}}} \right|_{{\widehat{D}}} } \neq \omega_{\widehat{D}}\)
-- there are correction terms coming from other parts of the scattering
diagram. Ruddat-Seibert give a way to compute period integrals
associated to 1-dimensional balanced subsets of \(\Sigma_X\) and form an
\(n{\hbox{-}}\)cycle \(\beta\) and integrating the canonical
differential from from the scattering diagram, yielding
\(\int_\beta \omega_{\widehat{X}}\). Since \(\beta\) is contained in a
fiber, one gets a relation
\begin{align*}
\int_\beta \omega_{\widehat{D}} (1 + \sum N_k) = \int_\beta { \left.{{\omega_{\widehat{X}}}} \right|_{{\widehat{D}}} }
,\end{align*}
and the \(N_k\) count curves with \(k+1\) marked points lying on the
fiber, all tangent to \(Y_1, Y_2\) (and possible negative or punctured).

\end{remark}

\begin{remark}

An ideal theorem: any purple curve (DZG: forgot to mark which was
purple!) can be formed by gluing components on
\(\Sigma_{X_1, D}, \Sigma_{X_2, D}\):

\includegraphics{figures/2022-07-21_14-27-38.png}

\end{remark}

\hypertarget{amanda-francis-hanging-gardens-and-divisors-from-adinkras-thursday-july-21}{%
\section{Amanda Francis, Hanging gardens and divisors from Adinkras
(Thursday, July
21)}\label{amanda-francis-hanging-gardens-and-divisors-from-adinkras-thursday-july-21}}

\begin{remark}

Physics motivations: standard model has 5 bosons and 12 fermions, and
supersymmetry switches and matches these in pairs. Form an algebra from
supersymmetry operators \(Q_i\) transforming bosons to fermions, where
\(Q_i \phi = \pm (iH)^s \psi\), where \(H\) is a Hamiltonian. Visualize
these with Adinkras (etymology: Ghanaian art): a bipartite graph
(bosons/fermions), \(Q_i\) operators represented by colored edges,

\end{remark}

\begin{remark}

Chromotopology: bipartite, \(N{\hbox{-}}\)regular with a consistent edge
coloring (each vertex has \(N\) incident edges, one of each color), and
other conditions.

\end{remark}

\begin{remark}

\end{remark}

\begin{remark}

Somehow get an algebraic (Riemann) surface \(X\) from this. Match up
with dessin d'enfants?

If a height function \(h\) on a triangular mesh sends adjacent vertices
to distinct values, this is a discrete Morse function. So we can
triangulate these graphs by adding new vertices and new fractional
heights to get Morse functions. Build divisors from critical points of
\(h\) by defining multiplicities \(m_v\); this yields a Morse divisor
\(D_h = \sum_{v\in M} -m_v [v]\).

\end{remark}

\begin{remark}

The Jacobian of \(X\) carries interesting data. Neat fact: the Belyi map
\(X\to {\mathbb{P}}^1\) factors through a beach ball \(B_N\), and the
map \(X\to B_N\) is less ramified. See Carvacho-Hidalgo-Quispe: \(N=5\)
hypercube curve is the generalized Fermat curve, a complete intersection
\(X \hookrightarrow{\mathbb{P}}^4\). Remainder of talk: discussion of
counting height functions on \(N{\hbox{-}}\)hypercubes. The number blows
up very quickly!

\end{remark}

\hypertarget{elana-kalashnikov-unwinding-the-gelfandcetlin-toric-degeneration-on-the-mirror-thursday-july-21}{%
\section{Elana Kalashnikov, Unwinding the Gelfand---Cetlin toric
degeneration on the mirror (Thursday, July
21)}\label{elana-kalashnikov-unwinding-the-gelfandcetlin-toric-degeneration-on-the-mirror-thursday-july-21}}

\begin{remark}

Batyrev mirror symmetry: anticanonical CYs have mirrors which are Fanos
in toric varieties, where \(Z \subseteq X_p\) and
\(Z {}^{ \vee }\subseteq X_{p {}^{ \vee }}\) for \(p\) a polytope and
\(p {}^{ \vee }\) its dual polytope. How to extend from mirror
hypersurfaces in a Calabi-Yau to mirrors in a Grassmannian?
Gelfand-Cetlin degenerate \({\operatorname{Gr}}_n(m)\leadsto X_p\) a
toric variety and proceed as before (?), how do we leave the toric
world? It turns out that there is a Grassmannian on the mirror, and a
natural generalization of the Greene-Plesser quintic mirror.

\end{remark}

\begin{remark}

Generalizing Greene-Plesser mirrors: mirror pairs of the form
\(Z \subset {\mathbb{P}}^{n-1} \rightleftharpoons Z {}^{ \vee }\subseteq {\mathbb{P}}^{n-1}/ H\)
for \(H\) a finite group constructed as follows. Write
\({\mathbb{P}}^{n-1} = {\mathbb{C}}^n { \mathbin{/\mkern-6mu/}}{\mathbb{C}^{\times}}\)
as a GIT quotient. Think of this as
\(\operatorname{Mat}_{1\times n}({\mathbb{C}}) { \mathbin{/\mkern-6mu/}}\operatorname{GL}_1({\mathbb{C}})\).
Define
\(\tilde H_n = \left\{{(\zeta_1, \cdots, \zeta_n) {~\mathrel{\Big\vert}~}\zeta_i^n=1, \prod \zeta_i = 1}\right\}\),
then
\(\tilde H_n \curvearrowright\operatorname{Mat}_{1\times n}({\mathbb{C}})\),
and finally
\(H_n = \tilde H_n / \tilde H_n \cap\operatorname{GL}_1({\mathbb{C}})\).
Replacing \(\operatorname{GL}_1({\mathbb{C}})\) with
\(\operatorname{GL}_r({\mathbb{C}})\) yields
\({\operatorname{Gr}}_n(r) = \operatorname{Mat}_{r\times n}({\mathbb{C}}) { \mathbin{/\mkern-6mu/}}\operatorname{GL}_r({\mathbb{C}})\),
and one defines \(H_{n, r}\).

\end{remark}

\begin{theorem}[?]

The CYs \(Z \subseteq {\operatorname{Gr}}_n(r)\) and
\(\tilde Z \in {\operatorname{Gr}}_n(r)/ H_{n, r}\) are related.

\end{theorem}

\begin{remark}

Take a toric degeneration
\(Z \subseteq {\operatorname{Gr}}_n(r)\leadsto Z_p \subseteq X_p\) a
Fano toric, apply BB mirror symmetry to get
\(Z_p {}^{ \vee }\subseteq X_{p {}^{ \vee }}\), take a ``geometric
transition'' (blow up, change the stability conditions, variation of
GIT) to get \(\tilde Z_p \subseteq \tilde X_p\), and smooth to get
\(\tilde Z\).

\includegraphics{figures/2022-07-21_16-18-56.png}

\end{remark}

\begin{remark}

Degenerate toric variety to one associated to a certain reflexive
polytope coming from a quiver moduli spaces
\({\mathcal{M}}_\theta(LQ, (1,1,\cdots, 1))\) which is a Fano where
\(\theta\) is a stability condition. Here \(LQ\) is a ladder quiver.
There are correspondences between arrows in \(LQ\) and coordinates on
\(X_{P_{n, r}}\), paths \(0\to 1\) and partitions
\(\lambda \vdash r\times (n-r)\), subsets of \([n]\) of size \(r\), and
sections of a very ample line bundle \({\mathcal{O}}(D)\) where
\(S_\lambda = \prod_{a\in \lambda} x_a\) yielding
\(X_{P_{n, r}} \hookrightarrow{\mathbb{P}}^{{n\choose r} - 1}\) with
Plucker coordinates \(p_\lambda\). What equations cut out
\(X_{P_{n, r}}\) in this projective space? Can define operations
\(\sigma \wedge \lambda, \sigma \vee \lambda\) for two ``incomparable''
grid paths (pairs of paths which cross) through an \(r\times (n-r)\)
grid by taking maximal or minimal parts of each, and these generate the
correct equations. This yields the toric degeneration described as step
1 in the image above.

\end{remark}

\begin{remark}

Step 2: use a new version of polytope duality. Let
\(P \subseteq N_{\mathbb{R}}\) be a reflexive polytope, let \(Q\) denote
the \textbf{primitive dual} of \(P\). For fixed \(P, Q\), BB mirror
symmetry looks like \(X_P\rightleftharpoons X_Q/G\) and
\(X_Q \rightleftharpoons X_P/G\), where
\(G = M/\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu \cong \mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu {}^{ \vee }/ N\).
See Gelfand-Cetlin polytope, toric variation of GIT. What geometric
transition means in this context:

\includegraphics{figures/2022-07-21_16-37-56.png}

\end{remark}

\hypertarget{toric-un-degenerating}{%
\subsection{Toric un-degenerating}\label{toric-un-degenerating}}

\begin{remark}

We get \(H_{n, r}\) from mirror symmetry and \(G\) from the
Gelfand-Cetlin polytope, and it turns out that \(H_{n, r} \leq G\) is
the maximal subgroup for which \({\operatorname{Gr}}_n(r)\) is an
equivariant subvariety. Constructing the toric degeneration as an
equivariant family: let
\(Z \subseteq X\coloneqq{\mathbb{P}}^{{n\choose r} - 1} \times {\mathbb{C}}^k\)
be the family cut out by Plucker relations with extra parameters tracked
by the \({\mathbb{C}}^k\) part. Extend the action of \(G\) to
\({\mathbb{C}}^k\) so that \(Z\) is \(G{\hbox{-}}\)equivariant. Then
\(Z/G \subseteq X/G \to {\mathbb{C}}^k/G\) is a family with special
fiber \(X_{P_{n, r}}/G\) and general fiber
\({\operatorname{Gr}}_n(r)/H_{n, r}\), so \(X_{P_{n, r}}/ G\) smooths to
the Grassmannian analogy of Greene-Plesser. This more or less comes from
the orbit-stabilizer theorem and carefully tracking the stabilizer.

\end{remark}

\begin{theorem}[?]

The family of hypersurfaces
\(\sum_\lambda p_\lambda^n + \psi \prod_{\lambda } p_\lambda = 0\) in
\(\operatorname{Bl}{\operatorname{Gr}}_n(r)/H_{n, r}\) is a Calabi-Yau
compactification of a Laurent polynomial mirror for
\({\operatorname{Gr}}_n(r)\). This is a candidate mirror to
\(Z \subseteq {\operatorname{Gr}}_n(r)\).

\end{theorem}



\newpage
\printbibliography[title=Bibliography]


\end{document}
