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

Nishi-Waseda Campus

Building 51

3rd Floor

Room 09

(Conference Room 5)

 

 

11-15

December

2017

 

 

 

Philip Boalch, Universite Paris-Sud

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

Chang-Shou 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 NISHI-WASEDA campus)

Map showing building 51

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: 11:00-11:30 (Mon-Fri) and 14:00-14:30, 15:30-16:00 (Mon-Thu).


PROGRAM OF THE CONFERENCE


Monday 11 December

09:50: opening

10:00-11:00 Lin 1

11:30-12:30 McIntosh 1

14:30-15:30 Hosono

16:00-17:00 Kakei

Tuesday 12 December

10:00-11:00 Lin 2

11:30-12:30 McIntosh 2

14:30-15:30 Nagoya

16:00-17:00 Nagano

Wednesday 13 December

10:00-11:00 Heller 1

11:30-12:30 Miyaoka-Ohnita 1 (Miyaoka)

14:30-15:30 Hiroe

16:00-17:00 Iriyeh

Thursday 14 December

10:00-11:00 Heller 2

11:30-12:30 Miyaoka-Ohnita 2 (Ohnita)

14:30-15:30 Yamakawa

16:00-17:00 Boalch 1

Friday 15 December

10:00-11:00 Iwaki

11:30-12:30 Boalch 2


TITLES AND ABSTRACTS

Philip Boalch, Universite Paris-Sud (1)
TITLE: ABC of quasi-Hamiltonian geometry
ABSTRACT: Quasi-Hamiltonian geometry was introduced by Alekseev-Malkin-Meinrenken 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 (Atiyah-Bott, Segal, Meinrenken-Woodward), 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 quasi-Hamiltonian 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 Paris-Sud (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 Deligne-Malgrange-Sibuya 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 Jimbo-Miwa-Ueno 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 Gauss-Manin connection (i.e. a nonlinear analogue of Picard-Fuchs 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 self-duality equations
ABSTRACT: Solutions of Hitchin’s self-duality equations correspond to special real sections in the Deligne-Hitchin moduli space – twistor lines. A question posed by Simpson in 1995 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would allow for complex analytic procedure to obtain solutions of the self-duality equations. The purpose of my talks is to explain the construction of counter examples given by certain (branched) Willmore surfaces in 3-space (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 self-duality theory. The talks are based on joint work with L. Heller.

SLIDES-1 SLIDES-2

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 & Komatsu-Malgrange irregularity,
(2) blow up & Fourier transform,
(3) knot structure & Stokes phenomenon,
(4) cohomology of curves & de Rham cohomology of differential equations,
and so on.

SLIDES

Shinobu Hosono, Gakushuin University
TITLE: Movable vs monodromy nilpotent cones in mirror symmetry of Calabi-Yau complete intersections
ABSTRACT: I will show some interesting examples of Calabi-Yau complete intersections described by Gorenstein cones, which exhibit nice correspondences between birational geometries and the degenerations of the mirror family of Calabi-Yau 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 n-dimensional 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 Artstein-Avidan, Karasev and Ostrover. The talk is based on a joint work with Masataka Shibata.

SLIDES

Kohei Iwaki, Nagoya University
TITLE: Stokes structure of equivariant CP^1 and wall-crossing
ABSTRACT: I'll investigate the quantum differential equation for the equivariant CP1 model (= the Gauss-Manin 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 2d-4d wall-crossing formula proposed by Gaiotto-Moore-Neitzke. 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 ultra-discrete 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.

Chang-Shou 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 non-trivial 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 W-invariants 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.

SLIDES

Ian McIntosh , University of York
TITLE: Minimal Lagrangian surfaces in the complex projective and complex hyperbolic planes.
ABSTRACT: The Gauss-Codazzi-Ricci equations for a minimal Lagrangian surface in a two complex-dimensional manifold of non-zero 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.

SLIDES

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 non-displaceability in most cases. We introduce the isoparametric geometry and the Floer homology on Lagrangian intersections.

SLIDES

Atsuhira Nagano, Tokyo University
TITLE: Differential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties
ABSTRACT: The work on Calabi-Yau 3-folds by Candelas et al. (1991) is a pioneering work for mirror symmetry. Their Calabi-Yau 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.

SLIDES

Hajime Nagoya, Kanazawa University
TITLE: CFT approach to q-difference 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 q-analog of their result, we give series expansion formulas for tau functions of q-difference 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 non-displaceability in most cases. We introduce the isoparametric geometry and the Floer homology on Lagrangian intersections.

SLIDES

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 Riemann-Hilbert-Birkhoff 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 Grant-in-Aid for Scientific Research (A)25247005 (PI: Martin Guest)


LINKS AND RELATED ACTIVITIES:

Minicourses by Alexander Its (IUPUI), Elizabeth Its (IUPUI) on Riemann-Hilbert Problems

Dates: 20-22 November 2017 Place: Waseda University, Nishi-Waseda Campus further details

Conference on Mirror Symmetry

Dates: 11-15 December 2017 Place: Kyoto University further details