|Title and Abstract of Talks ：
abstract (in preparation) |
| Franz Pedit (1) |
|Title: Energy quantization for harmonic 2-spheres in non-compact symmetric spaces |
It is well known from results by Uhlenbeck, Chern-Wolfson and Burstall-Rawnsley that harmonic 2-spheres in compact
symmetric spaces have quantized energies. Using the reformulation of the harmonic map equation as a family of flat
connections, we construct an energy preserving duality between harmonic maps from non-compact symmetric spaces
into their compact duals. Applying this construction to the conformal Gauss maps of Willmore 2-spheres in the n-sphere
provides a generalization and unifying approach to existing quantization results in special cases: Bryant for n=3;
Montiel for n=4; and Ejiri for Willmore 2-spheres admitting a dual Willmore surface.
| Franz Pedit (2) |
|Title: Classification of homogeneous Willmore 2-spheres in the n-sphere |
The simplest surfaces in any ambient geometry are those which arise as orbits via a representation into the
symmetry group defining the geometry. For Willmore surfaces in n-space this symmetry group is the Moebius
group. We show that every Moebius homogeneous Willmore 2-sphere is either a round 2-sphere or is Moebius
congruent to a Veronese minimal 2-sphere in a constant curvature 2m-sphere given by the irreducible SO(3)
representation in 2m+1.
| Francis Burstrall |
|Title: Integrable systems in conformal and Lie sphere geometry |
I shall give an overview of some geometric integrable systems that arise in conformal and Lie sphere geometry.
I shall describe isothermic surfaces and Moebius-flat submanifolds
(which include Guichard surfaces and conformally flat hypersurfaces)
in conformal geometry and their transformation theory.
Then I shall show how these are special cases of an integrable class of Lie applicable submanifolds
in a Lie sphere geometry which is, itself, novel in higher codimension.
| Christoph Bohle |
|Title: Differential Geometry and mcKP |
In my talks I want to explain how the multi-component KP hierarchy (mcKP) can be related to differential geometry.
After briefly going through the general scheme behind this relation, I plan to focus on objects of classical differential
geometry (like multi-dimensional conjugate nets...) and applications to mathematical physics.
| Seiichi Udagawa |
|Title: Finite gap solutions for horizontal minimal surfaces of finite type in 5-sphere
| Takashi Kurose |
|Title: Miura transformation and geometry of curves
The Miura transformation, which played an important role in the development of the theory of
integrable systems, relates the Korteweg-de Vries (KdV) equation and the modified KdV equaion.
On the other hand, it is known that various integral systems naturally appear in geometry of
curves. In this talk, we shall review the cases of the KdV equation and of the modified KdV
equaion, and shall discuss geometric counterparts of the Miura transformation.
| Atsushi Fujioka |
Centroaffine surfaces with parallel or recurrent cubic form relative to the induced connection
The cubic form of a hypersurface in the affine space is an important
invariant in affine differential geometry. In this talk, we shall give
some classification results for nondegenerate centroaffine surfaces in
terms of the cubic form and the induced connection.
| Kentaro Saji |
|Title: Geometric foliations of fronts |
In this talk, generic classification
of geometric foliations near singular points
of fronts and frontals will be discussed.
In particular, a normal form of swallowtail
will be introduced and asymptotic curves
and characteristic curves near swallowtail
will be given.
| Masashi Yasumoto |
|Title: Trivalent maximal surfaces in Minkowski space |
In this talk we introduce trivalent maximal surfaces in Minkowski
3-space based on the combination of two ideas below. In particular, we see that such trivalent
maximal surfaces admit associated families and they also have variational properties and certain
Lam investigated two types of trivalent surfaces in Euclidean 3-space with vanishing mean curvature,
which are called trivalent minimal surfaces. The advantage of this discretization is that we can treat
both integrable geometric aspects of such minimal surfaces and their variational properties, which
generalize many previous works.
On the other hand, Yasumoto described quadrilateral surfaces (or, discrete surfaces) in Minkowski
3-space with mean curvature identically 0, which are called discrete maximal surfaces. Unlike
the case of discrete surfaces in Euclidean 3-space with vanishing mean curvature, it is shown that
discrete maximal surfaces generally have singularities, which occurs only in the case of discrete
constant mean curvature surfaces in Lorentzian spaceforms.
This talk is based on joint work with Wai Yeung Lam (Brown University).
| Atsuhira Nagano |
|Title: Toric K3 hypersurfaces and a Shimura variety
In the 19th century, elliptic modular functions appeared in the study of elliptic curves.
Special values of elliptic modular functions generate class fields over imaginary quadratic fields (Kronecker's Jugendtraum).
In this talk, the speaker will present an extension of the classical result by using K3 surfaces given by hypersurfaces
in toric 3-folds. Namely, K3 surfaces are 2-dimensional analogy of elliptic curves and its periods give
the moduli of K3 surfaces. Moreover, canonical models of Shimura varieties give natural extensions of Kronecker's
Jugendtraum. We will see a construction of an explicit model of a Shimura variety via techniques of periods of
toric K3 hypersurfaces.
| Toru Kajigaya |
|Title: On Lagrangian submanifolds with symmetries
Recently, Lagrangian submanifolds in a symplectic manifold M are of particular interests in several contexts,
e.g., the Lagrangian intersection theory, the Hamiltonian volume minimizing problem and the Lagrangian mean curvature flows.
In particular, homogeneous Lagrangian submanifolds, or more generally, Lagrangian submanifolds with symmetries
give nice examples of Lagrangian submanifolds with several properties (e.g., minimal, special, monotone, Hamiltonian stable etc). In this talk, I explain some known results for Lagrangian submanifolds with symmetries in a symplectic or Kahler manifold by using the notions of Hamitonian actions and moment maps. After that, I introduce the following own results:
(i) A classification result of homogeneous Lagrangian submanifolds in the complex hyperbolic spaces
(a joint work with Takahiro Hashinaga).
(ii) A Hsiang-Lawson type theorem for minimal Lagrangian submanifolds in a Kahler manifold.
| Hideo Takioka |
Infinitely many knots with the trivial $(2,1)$-cable $\Gamma$-polynomial
For coprime integers $p(>0)$ and $q$, the $(p,q)$-cable $\Gamma$-polynomial of a knot is
the $\Gamma$-polynomial of the $(p,q)$-cable knot of the knot, where the $\Gamma$-polynomial
is the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials.
In this talk, we show that there exist infinitely many knots with the trivial $(2,1)$-cable $\Gamma$-polynomial,
that is, the $(2,1)$-cable $\Gamma$-polynomial of the unknot.
Moreover, we see that the knots have the trivial $\Gamma$-polynomial,
the trivial first coefficient HOMFLYPT polynomial,
and the distinct Conway polynomials.
| Kohei Iwaki |
|Title: Exact WKB analysis and related topics
Exact WKB analysis, developed by Voros et.al., is an effective method for global study of
(singularly perturbed) ordinary differential equations defined on a complex domain.
After recalling fundamental facts about exact WKB analysis, I’ll talk about
relationships to other research topics, such as cluster algebras, topological recursion,
integrable systems of Painlev\’e type, etc.
| Hitoshi Yamanaka |
|Title: Invariant function on GKM-representation space
After recalling several notions, such as GKM-representations, GKM-manifolds and representation coverings,
I propose a conjecture concerning the existence of torus-invariant Morse functions on GKM-manifolds. Toward
solving the conjecture, I will examine the Hessian matrix of invariant functions on a GKM-representation spaces
at the origin. The result reveals a Morse-theoretic nature of the GKM-conditon.
| Shintaro Yanagida |
|Title: Turaev's skein algebra for torus and a variant of homological mirror symmetry
|Abstract: Turaev introduced skein algebras for surfaces as quantization of Goldman's
Lie algebras. Morton and Samuelson showed that the skein algebra for torus coincides with a specialization
of (the Drinfeld double of non-extended) Hall algebra of elliptic curves. We introduce a version of Hall algebra
for elliptic curves over a field with one element, and show that the skein algebra of torus is isomorphic to our
Hall algebra without specialization. We shall further explain a speculation on an extension of homological mirror
symmetry for torus.
| Kentaro Mitsui |
|Title: Models of torsors under elliptic curves
Any smooth curve of genus one over a general base field can be given as a torsor (a principal homogeneous space)
under an elliptic curve. In the case of a function field of a curve C, we may define an elliptic surface as a model of
the smooth curve, which is a fibration over the curve C. There are at most finitely many singular fibers. When the
curve C is defined over the field of complex numbers, these fibers were classified by Kodaira. We generalize his
result to the case where the curve C is defined over a general perfect field such as a finite field. We apply this
classification to study rational points on torsors over local fields.
| Shintaro Akamine |
|Title: Behavior of the Gaussian curvature of timelike minimal surfaces with singularities
A timelike minimal surface in the 3-dimensional Lorentz-Minkowski space is a surface with a Lorentzian metric
whose mean curvature vanishes identically. The shape operator of a timelike surface is not always diagonalizable.
The diagonalizability of the shape operator of a timelike minimal surface is determined by the sign of the Gaussian
curvature away from flat points. In this talk we prove that the sign of the Gaussian curvature of any timelike minimal
surface is determined by the degeneracy and the orientations of the two null curves that generate the surface.
Moreover, we also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal
surface. This talk is based on the preprint arXiv:1701.00238.
| Joseph Cho |
|Title: Deformation and singularities of maximal surfaces with planar curvature lines
Minimal surfaces with planar curvature lines in the Euclidean 3-space have been studied since the late
19th century, and have connections to various different subjects of differential geometry. On the other
hand, the classification of maximal surfaces with planar curvature lines in the Minkowski 3-space has
only recently been given. In this talk, we use an alternative method not only to refine the classification
of maximal surfaces with planar curvature lines, but also to show that there exists a deformation consisting
exactly of all such surfaces. Furthermore, we investigate the types of singularities that appear on maximal
surfaces with planar curvature lines.