Differential Geometry and Differential Equations:the influence of Mirror Symmetry and Physics
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Differential equations  especially those related to integrable systems  have always played a major role in mirror symmetry. Conversely, mirror symmetry and related ideas from physics have had a huge influence on algebraic geometry, symplectic geometry, and topology. The influence on the theory of differential equations and differential geometry is less obvious, as differential equations are generally regarded as a tool. However, throughout history, it is the applications of differential equations which have shaped the subject, by focusing attention on certain equations and particular methods. The case of mirror symmetry is no exception to this. The purpose of this conference is to recognize the benefits "in the other direction" and thus give more prominence to the influence of mirror symmetry and physics on the theory of differential equations and differential geometry.
Place  Date  Speakers (see below for program) 
Waseda University NishiWaseda Campus Building 51 3rd Floor Room 09 (Conference Room 5)

1115 December 2017

Philip Boalch, Universite ParisSud Sebastian Heller, University of Hamburg Kazuki Hiroe, Josai University Shinobu Hosono, Gakushuin University Hiroshi Iriyeh, Ibaraki University Kohei Iwaki, Nagoya University Saburo Kakei, Rikkyo University ChangShou Lin, National Taiwan University Ian McIntosh , University of York Reiko Miyaoka, Tohoku University Atsuhira Nagano, Tokyo University Hajime Nagoya, Kanazawa University Yoshihiro Ohnita, Osaka City University Daisuke Yamakawa, Tokyo University of Science 
Waseda Campus map (Conference takes place at NISHIWASEDA campus)
Note: The elevators in Building 51 are often crowded and slow. Furthermore the largest elevator does not stop at level 3. It is recommended to use the stairs between the main entrance (level 1) and the lecture room (level 3). The stairs can be found at the side of the building (go 10 metres beyond the small metal staircase which is visible from the main entrance, and take the main stairs at the side of the building). If in doubt, just wait for one of the smaller elevators!
Note for speakers: the lecture room has whiteboards and a computer projector (you may connect your own computer or just bring a memory stick).
Tea/coffee breaks: 11:0011:30 (MonFri) and 14:0014:30, 15:3016:00 (MonThu).
PROGRAM OF THE CONFERENCE
Monday 11 December
09:50: opening
10:0011:00 Lin 1
11:3012:30 McIntosh 1
14:3015:30 Hosono
16:0017:00 Kakei
Tuesday 12 December
10:0011:00 Lin 2
11:3012:30 McIntosh 2
14:3015:30 Nagoya
16:0017:00 Nagano
Wednesday 13 December
10:0011:00 Heller 1
11:3012:30 MiyaokaOhnita 1 (Miyaoka)
14:3015:30 Hiroe
16:0017:00 Iriyeh
Thursday 14 December
10:0011:00 Heller 2
11:3012:30 MiyaokaOhnita 2 (Ohnita)
14:3015:30 Yamakawa
16:0017:00 Boalch 1
Friday 15 December
10:0011:00 Iwaki
11:3012:30 Boalch 2
TITLES AND ABSTRACTS
Philip Boalch, Universite ParisSud (1)
TITLE: ABC of quasiHamiltonian geometry
ABSTRACT: QuasiHamiltonian geometry was introduced by AlekseevMalkinMeinrenken
for compact groups in the 1990's as an algebraic way to construct symplectic
moduli spaces of flat connections on Riemann surfaces, complementary to more
analytic approaches (AtiyahBott, Segal, MeinrenkenWoodward), involving loop
groups in the case of surfaces with boundary. In this theory the moduli spaces
are constructed out of two simple pieces C (the conjugacy classes) and D (the
double), via two operations called fusion and reduction. This process can be
viewed as a kind of classical version of a 2d TQFT (producing symplectic manifolds
rather than vector spaces). (It was later shown that D can be derived from certain
examples of C.) I will discuss the world of complex quasiHamiltonian geometry
and some of the examples that occur when one allows connections with "wild''
boundary conditions, corresponding to algebraic connections with irregular singularities.
In brief there are two new classes of simple pieces that occur, called A and
B (the fission spaces and the reduced fission spaces). Lots of new algebraic
symplectic manifolds, such as the wild character varieties, arise by fusing
them together and reducing. I will describe these pieces and draw some pictures
of the types of things that can occur, no doubt underlying some generalisation
of a 2d TQFT.
SLIDES Related material: http://arxiv.org/abs/1512.08091,
https://arxiv.org/abs/1703.10376, Oberwolfach Report 2015
with further references
Philip Boalch, Universite ParisSud (2)
TITLE: Irregular isomonodromy
ABSTRACT: By describing the Stokes data of an irregular connection on a curve
in terms of a Stokes local system (as opposed the DeligneMalgrangeSibuya notion
of Stokes filtrations) there is a very simple way (similar to that used in the
tame case) to describe the notion of "monodromy preserving deformations''
of such connections in the general case, with no restriction on the leading
term of the connections. This amounts to an extension of the viewpoint of Kimio
Ueno's master thesis and the subsequent paper of JimboMiwaUeno from around
1980. Further, inspired by Simpson's geometric viewpoint for compact curves,
one sees that, in the tame case, isomonodromy is purely geometric and can be
viewed as a nonabelian GaussManin connection (i.e. a nonlinear analogue of
PicardFuchs equations), attached to any smooth family of complex curves. This
viewpoint may be extended to the irregular case by defining the intrinsic notion
of "irregular curves'' (or "wild Riemann surfaces''), so that an isomonodromy
connection is attached to any admissible family of irregular curves.
Sebastian Heller, University of Hamburg
TITLE: Higher solutions of Hitchin’s selfduality equations
ABSTRACT: Solutions of Hitchin’s selfduality equations correspond to
special real sections in the DeligneHitchin moduli space – twistor lines.
A question posed by Simpson in 1995 asks whether all real sections give rise
to global solutions of the selfduality equations. An affirmative answer would
allow for complex analytic procedure to obtain solutions of the selfduality
equations. The purpose of my talks is to explain the construction of counter
examples given by certain (branched) Willmore surfaces in 3space (with monodromy)
via the generalized Whitham flow. Though these higher solutions do not give
rise to global solutions of the self duality equations on the whole Riemann
surface M, they are solutions on an open dense subset of it. This suggest a
deeper connection between Willmore surfaces, i.e., rank 4 harmonic maps theory,
with the rank 2 selfduality theory. The talks are based on joint work with
L. Heller.
Kazuki Hiroe, Josai University
TITLE: Invariants of differential equations and algebraic curves
ABSTRACT: From algebraic differential equations on Riemann surfaces, we can
define algebraic curves which we will call spectral curves. I will explain that
structures of singularities of these algebraic curves have a close connection
with those of irregular singularities of differential equations. In particular,
I will make comparisons between
(1) Milnor number & KomatsuMalgrange irregularity,
(2) blow up & Fourier transform,
(3) knot structure & Stokes phenomenon,
(4) cohomology of curves & de Rham cohomology of differential equations,
and so on.
Shinobu Hosono, Gakushuin University
TITLE: Movable vs monodromy nilpotent cones in mirror symmetry of CalabiYau
complete intersections
ABSTRACT: I will show some interesting examples of CalabiYau complete intersections
described by Gorenstein cones, which exhibit nice correspondences between birational
geometries and the degenerations of the mirror family of CalabiYau manifolds.
In the examples, I will show that the monodromy nilpotent cones (defined near
the boundary points) are glued together to make a larger cone which we identify
with the movable cone in birational geometry. This talk is based on a recent
paper with H. Takagi (arXiv:1707.08728).
Hiroshi Iriyeh, Ibaraki University
TITLE: On Mahler's conjecture  a symplectic aspect
ABSTRACT: Mahler's conjecture states that for any centrally symmetric convex
body K in the ndimensional Euclidean space, the product of the volume of K
and that of the polar body is greater than or equal to 4^n/n!. The two dimensional
case was solved by Mahler in 1939. In spite of the simpleness of the problem,
this conjecture is still open for n>2. In this talk, we first announce the
solution of this conjecture in the three dimensional case. However, the case
where n>3 is still difficult. In the latter part of this talk, we provide
an approach to this problem from a symplectic geometrical viewpoint based on
a recent work by ArtsteinAvidan, Karasev and Ostrover. The talk is based on
a joint work with Masataka Shibata.
Kohei Iwaki, Nagoya University
TITLE: Stokes structure of equivariant CP^1 and wallcrossing
ABSTRACT: I'll investigate the quantum differential equation for the equivariant
CP1 model (= the GaussManin system for the mirror oscillatory integral) via
the exact WKB method. In particular, I’ll show a computation of the total
Stokes matrix for this equation, and examine the 2d4d wallcrossing formula
proposed by GaiottoMooreNeitzke. I’ll also mention about an equivariant
version of the Dubrovin’s conjecture. My talk is based on a joint work
with H. Fuji (Kagawa), M. Manabe (Max Planck) and I. Satake (Kagawa).
Related material: https://arxiv.org/abs/1708.09365
Saburo Kakei, Rikkyo University
TITLE: Schutzenberger’s Jeu de taquin as a ultradiscrete Toda system
ABSTRACT: Jeu de taquin is a combinatorial operation on skew Young tableaux
that has intimate connections with representation theory. We propose difference
equations that describe jeu de taquin, and investigate their relation to the
discrete Toda equation.
ChangShou Lin, National Taiwan University
TITLE: SU(3) Toda system on flat tori
ABSTRACT: We consider the system
Delta u+2e^{u}e^{v}=4¥pi n_1 delta _0
Delta v+2e^{v}e^{u}=4¥pi n_2 delta _0
in E_tau = C/(Z + Z tau) with tau >0, where delta _0 is the Dirac measure
at the lattice points of E_tau and n_i is a nonnegative integer. It has been
proved that if n_1,n_2 are not congruent mod 3 then an a priori estimate holds,
hence there is a solution in this case.
If n_1,n_2 are congruent mod 3 then there are bubbling solutions and the existence
problem becomes very nontrivial from the analytic point of view. On the other
hand, this is an integrable system in the sense that any solution can be expressed
by holomorphic data, which arises from a complex ODE (derived from the Winvariants
of the solution). By applying this
idea, we prove among other things:
Theorem: If n_1 ¥leq n_2, with n_1 odd and n_2 even, then the equation has
no even solution in any E_tau.
This is a joint work with Zhijie Chen.
Ian McIntosh , University of York
TITLE: Minimal Lagrangian surfaces in the complex projective and complex hyperbolic
planes.
ABSTRACT: The GaussCodazziRicci equations for a minimal Lagrangian surface
in a two complexdimensional manifold of nonzero constant holomorphic sectional
curvature are much simpler than one might expect. The symmetry of the Lagrangian
condition means the Gauss and Ricci equations are the same and reduce to a scalar
elliptic p.d.e. governing the metric. The Codazzi equation tells us that the
second fundamental form is determined entirely by a cubic holomorphic differential.
The sign of the holomorphic sectional curvature determines just one sign in
the Gauss equation, but this sign make a huge difference to the methods which
work to describe solutions. For the positive sign (complex projective space)
methods from integrable systems work when the surface is a torus. For the negative
sign (complex hyperbolic space) the study of solutions is closely linked to
Higgs bundles and nonabelian Hodge theory. My aim is to survey some of what
is known about these methods.
Reiko Miyaoka, Tohoku University
TITLE: Approach from hypersurface geometry to the Floer theory on Lagrangian
intersections, I
ABSTRACT: The Gauss images of isoparametric hypersurfaces in the sphere provide
us with lots of examples of Lagrangian submanifolds in the complex hyperquadric.
We discuss the Floer homology of these examples, and show the Hamiltonian nondisplaceability
in most cases. We introduce the isoparametric geometry and the Floer homology
on Lagrangian intersections.
Atsuhira Nagano, Tokyo University
TITLE: Differential equations concerned with mirror symmetry of toric K3 hypersurfaces
with arithmetic properties
ABSTRACT: The work on CalabiYau 3folds by Candelas et al. (1991) is a pioneering
work for mirror symmetry. Their CalabiYau varieties are given by toric hypersurfaces
and their results are closely related to generalized hypergeometric differential
equations. On the other hand, classical Kronecker's Jugendtraum suggests that
hypergeometric functions can be applied to algebraic number theory. In this
talk, the speaker will present arithmetic properties of hypergeometric differential
equations related to mirror symmetry of toric K3 hypersurfaces.
Hajime Nagoya, Kanazawa University
TITLE: CFT approach to qdifference Painlevé equations
ABSTRACT: In conformal field theory(CFT), conformal blocks play central roles.
Gamayun, Iorgov, and Lisovyy found series expansion formulas for Painlevé
tau functions in terms of conformal blocks. Their conjecture was proved by Iorgov,
Lisovyy, and Teschner using CFT. As a qanalog of their result, we give series
expansion formulas for tau functions of qdifference Painlevé equations.
This is based on joint works with M. Jimbo, H. Sakai, and Y. Matsuhira.
Related material: https://academic.oup.com/integrablesystems/article/2/1/xyx009/4159874
Yoshihiro Ohnita, Osaka City University
TITLE: Approach from hypersurface geometry to the Floer theory on Lagrangian
intersections, II
ABSTRACT: The Gauss images of isoparametric hypersurfaces in the sphere provide
us with lots of examples of Lagrangian submanifolds in the complex hyperquadric.
We discuss the Floer homology of these examples, and show the Hamiltonian nondisplaceability
in most cases. We introduce the isoparametric geometry and the Floer homology
on Lagrangian intersections.
Daisuke Yamakawa, Tokyo University of Science
TITLE: Introduction to wild character varieties
ABSTRACT: In this talk I will first review the general theory of meromorphic
connections on compact Riemann surfaces, especially the RiemannHilbertBirkhoff
correspondence between the meromorphic connections and the monodromy/Stokes
data. Then I will give the definition of the (possibly twisted) wild character
varieties.
Organizing Committee: Martin Guest (Waseda), Yasushi Homma (Waseda), Shinobu Hosono (Gakushuin)
Partially supported by JSPS GrantinAid for Scientific Research (A)25247005 (PI: Martin Guest)
LINKS AND RELATED ACTIVITIES:
Minicourses by Alexander Its (IUPUI), Elizabeth Its (IUPUI) on RiemannHilbert Problems
Dates: 2022 November 2017 Place: Waseda University, NishiWaseda Campus further details
Conference on Mirror Symmetry
Dates: 1115 December 2017 Place: Kyoto University further details