Clean up

MFO26.tex

@@ -72,30 +72,30 @@
 \subsection{Topology of Kähler spaces: D-modules, perverse sheaves, Hodge modules}
 
 Over the last decade, Saito's theory of Hodge modules has seen spectacular
-applications in birational geometry.  More recent developments connect the
-theory to singularity theory, commutative algebra, and the topology of algebraic
-varieties.  The following topics in this area will be of particular interest to
-our workshop.
+applications in birational geometry.  More recent developments, which are of
+significant importance, connect the theory to singularity theory, commutative
+algebra, and the topology of algebraic varieties.  The following topics in this
+area will particularly interest our workshop.
 
 
-\subsubsection{Singularities and Hodge ideals}
+\subsubsection{Singularities and Hodge Ideals}
 
 In a series of influential papers starting with \cite{MR4081135}, Mustaţă and
 Popa used Hodge modules to refine and generalize well-known invariants of
 singularities, most notably the multiplier ideals used in analysis and algebraic
-geometry. An alternative approach towards similar ends was recently suggested in
-the preprint \cite{arXiv:2309.16763} of Schnell and Yang.  First applications
-pertain to Bernstein--Sato polynomials and their zero sets; these are important
-invariants of singularities originating from commutative algebra that are hard
-to compute.  Schnell and Yang apply their results to conjectures of
+geometry. Schnell and Yang’s recent preprint \cite{arXiv:2309.16763} suggested
+an alternative approach toward similar ends.  The first applications pertain to
+Bernstein--Sato polynomials and their zero sets; these are essential invariants
+of singularities originating from commutative algebra that are hard to compute.
+Schnell and Yang apply their results to conjectures of
 Debarre--Casalaina-Martin--Grushevsky concerning the Riemann--Schottky problem
 and the singularities of Theta divisors of principally polarized Abelian
 varieties. 
 
-Very recently, Park and Popa applied perverse sheaves and D-module theory to
-improve Goresky--MacPherson's classic Lefschetz theorems in the singular setting.
-A program put forward by Friedman--Laza aims at understanding the Hodge
-structures of degenerating Calabi--Yau varieties. 
+Park and Popa recently applied perverse sheaves and D-module theory to improve
+Goresky--MacPherson's classic Lefschetz theorems in the singular setting. A
+program put forward by Friedman--Laza aims at understanding the Hodge structures
+of degenerating Calabi--Yau varieties. 
 
 
 \subsubsection{Lagrangian fibrations}
@@ -103,41 +103,47 @@ structures of degenerating Calabi--Yau varieties.
 A Lagrangian fibration of a hyperkähler manifold $M$ is a proper holomorphic map
 $f : M \to B$ whose generic fibers are Langrangian. 
 
-If $M$ is compact, a well-known conjecture in the field predicts that $B$ is
-projective space.  In case where $B$ is smooth, the conjecture has been
-established more than 16 years ago in a celebrated work of Hwang.  Today, there
-is new insight, as Bakker--Schnell recently found a purely Hodge theoretic proof
-of Hwang's result, \cite{arXiv:2311.08977}.  There is hope that these methods
-give insight into the singular setting, which remains open to date. 
 
-In the non-compact setting, Lagrangian fibrations have been studied in the
-framework of the so called $P=W$ conjecture, which has recently been proven by
-Maulik--Shen and Hausel--Mellit--Minets--Schiffmann, \cite{arXiv:2209.02568,
-arXiv:2209.05429}.  In the same setting, Shen--Yin discovered a remarkable
+\paragraph{Compact Setting}
+
+If $M$ is compact, a well-known conjecture in the field predicts that $B$ is
+projective space.  In the case where $B$ is smooth, Hwang established the
+conjecture more than 16 years ago in a celebrated paper.  There is new insight
+today, as Bakker--Schnell recently found a purely Hodge theoretic proof of
+Hwang's result in \cite{arXiv:2311.08977}.  Hopefully, these methods will give
+insight into the singular setting, which remains open to date.
+
+
+\paragraph{Non-compact Setting}
+
+In the non-compact setting, geometers study Lagrangian fibrations in the
+framework of the ``$P=W$ conjecture,'' which Maulik–Shen and
+Hausel–Mellit–Minets–Schiffmann have recently proved \cite{arXiv:2209.02568,
+arXiv:2209.05429}.  In the same setting, Shen–Yin discovered a remarkable
 symmetry of certain pushforward sheaves and conjectured that more general
-symmetries exist.  These conjectures have recently been established by Schnell,
-\cite{arXiv:2303.05364}, who also proved two conjectures of Maulik--Shen--Yin on
+symmetries exist. Schnell has recently established these conjectures in
+\cite{arXiv:2303.05364} and also proved two conjectures of Maulik–Shen–Yin on
 the behavior of certain perverse sheaves near singular fibers.
 
 
 \subsubsection{Singer--Hopf conjecture and fundamental groups of Kähler manifolds}
 
-The Singer--Hopf conjecture asserts that a closed aspherical manifold of real
+The Singer-Hopf conjecture asserts that a closed aspherical manifold of real
 dimension $2n$ has positive signed Euler characteristic, $(-1)^n \cdot
-\chi(X)\geq 0$. This conjecture goes back to 1931, when Hopf formulated a
-related version for Riemannian manifolds. Recently, Hodge-theoretic refinements
-of this conjecture for Kähler manifolds have been put forward by
-Arapura--Maxim--Wang, \cite{arXiv:2310.14131}. Special cases of these conjecture
-have been proven, but the statement remains open in full generality.
+\chi(X)\geq 0$. This conjecture goes back to 1931 when Hopf formulated a related
+version for Riemannian manifolds. Recently, Arapura–Maxim–Wang suggested
+Hodge-theoretic refinements of this conjecture for Kähler manifolds in
+\cite{arXiv:2310.14131}. While the methods of \cite{arXiv:2310.14131} suffice to
+show particular cases, the statement remains open in full generality. 
 
 In a related direction, Llosa-Isenrich--Py found an application of complex
 geometry and Hodge theory to geometric group theory, settling an old question of
-Brady on finiteness properties of groups, \cite{zbMATH07790946}. As a byproduct,
-the authors also obtain a proof of the classical Singer conjecture in an
-important special case in the realm of Kähler manifolds.
+Brady on the finiteness properties of groups \cite{zbMATH07790946}. As a
+byproduct, the authors also obtain a proof of the classical Singer conjecture in
+an essential particular case in the realm of Kähler manifolds.
 
 Our goal in this workshop is to bring together several experts in geometric
-group theory with experts on Hodge theory, and to explore further potential
+group theory with experts on Hodge theory and to explore further potential
 applications of the methods from one field to problems in the other.