| Dec.16（Tue） | | |
| 9:30-10:30
| Ulrich Pinkall (TU Berlin)
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"Conformal equivalence of triangulated surfaces"
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Abstract: We present a new algorithm for conformal mesh parameterization.
It is based on a precise notion of discrete conformal equivalence for triangle
meshes which mimics the notion of conformal equivalence for smooth surfaces.
The problem of finding a flat mesh that is discretely conformally
equivalent to a given mesh can be solved efficiently by minimizing a convex energy
function, whose Hessian turns out to be the well known cot-Laplace operator.
Surprisingly, it turns out that the theory is closely related to the geometry of
polyhedra in hyperbolic space.
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| Dec.16（Tue） | | |
| 10:45-11:45
| Masaaki Umehara (Osaka University)
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"Surfaces with singularities and Osserman-type Ineqalities"
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Abstract: PDF
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File of Talk
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| Dec.16（Tue） | | |
| 12:00-13:00
| Josef Dorfmeister (TU M\"unchen)
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"Loop Groups and Surfaces with Symmetries"
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Abstract: In this talk we will present applications of the loop group method to
the construction of surfaces with symmetries.
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Starting from the well known procedure in the case of CMC surfaces in
R^3, we will discuss the case of CMC surfaces in H^3 with mean curvature H
satisfying 0 <= H < 1, the case of pseudospherical surfaces and, time
permitting, a surface associated with the quantum cohomology of CP^1.
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| Dec.16（Tue） | | |
| 14:30-15:30
| Bertrand Eynard (SPT, Saclay, France)
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" Random matrix methods in enumerative geometry "
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Abstract:
We will introduce some newly discovered invariants of spectral curves.
Given a plane curve S={y(x)} in C^2, called the "spectral curve", we can associate to it a sequence of invariants F_g(S). The F_g's are invariants under symplectic transformations of the spectral curve.
We will give the definitions, and main properties of those F_g's, and present some applications in enumerative geometry, such as Witten-Kontsevich theory, volumes of moduli spaces,
and Gromov Witten theory.
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File of Talk
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| Dec.16（Tue） | | |
| 15:45-16:45
| Kefeng Liu (UCLA, USA)
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" Recent results on the geometry and topology of moduli spaces "
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Abstract: I will present several recent results on the geometry and
topology of moduli spaces of Riemann surfaces, including the proof
of the Faber intersection number conjecture, the goodness of the
Weil-Petersson metric, the Ricci, the perturbed Ricci and the Kahler-Einstein
metric, the Nakano negativity of the Weil-Petersson metric and various
corollaries such as the Gauss-Bonnet type
theorems, vanishing theorems of cohomology groups, rigidity theorems and
Chern number inequalities and the Mumford stability of the logarithmic cotangent
bundles of the moduli spaces. Finally I will discuss the properties of the period map from the Teichmuller space of polarized Calabi-Yau n-folds to the classifying
space of variation of Hodge structures as well as the local and global
geometry of the Teichm\"uller and moduli spaces of polarized Calabi-Yau manifolds.
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File of Talk
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| Dec.16（Tue） | | |
| 17:00-18:00
| Hiraku Nakajima (Kyoto Univ., Japan)
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" Instanton counting (survey) "
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Abstract: Nekrasov defined the instanton partition function by
an equivariant integration of $1$ over moduli spaces of instantons on
$\mathbf R^4$. I will survey its relations to various subjects:
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1) Its leading part is the Seiberg-Witten prepotential defined via a
period integral of hyperelliptic curves.
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2) (Geometric engineering) The full partition function, setting one of
variables 0, is the generating function of Gromov-Witten invariants of
a certain local Calabi-Yau 3-fold so that the leading part corresponds
to the genus 0 part.
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3) The partition function has a natural deformation integrating Chern
classes of natural vector bundles over moduli spaces. They are conjecturally
related to Poincare polynomials of link homology groups a la Khovanov.
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File of Talk
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| Dec.16（Tue） | | |
| 18:15-20:15
| Banquet（懇親会 学情10階 研究者交流室）
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| Dec.17（Wed） | | |
| 9:00 - 10:00 | Yasuyuki Nagatomo（Kyushu Univ.）
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" Harmonic maps into Grassmannian manifolds "
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Abstract: PDF
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| Dec.17 (Wed) | | |
| 10:15-11:15 | John C. Wood (Univ. of Leeds, UK) |
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"A completely explicit formula for harmonic spheres in the unitary group"
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Abstract:
We report on joint work with B.A. Simoes and M.J. Ferreira (Lisbon) which gives a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to a unitary group, and so all harmonic maps from the two-sphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using only combinations of projections and avoiding the usual dbar-problems or loop group factorizations. The formula is obtained using only techniques in the theory of harmonic maps.
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We interpret our construction in terms of Segal's Grassmannian model and an explicit factorization of the algebraic loop group, and show how it specializes to give all harmonic maps into a Grassmannian.
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We then describe joint work with M. Svensson (Odense) which extends this interpretation to give explicit formulae for any factorization, in particular, getting formulae for all
harmonic two-spheres into the symplectic and orthogonal groups.
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| Dec.17（Wed) | | |
| 11:30-12:30 | Sumio Yamada（Tohoku Univ.） |
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" Weil-Petersson geometry of Teichm\"uller-Coxeter Complex "
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Abstract:
Teichmuller space is a moduli space of conformal structures on a
topological surface of higher genus.
Recently there has been much progress in understanding the geometry
of the space via the L^2 (Weil-Petersson) deformation theory of
hyperbolic metrics defined on the surface.
In this talk, we will introduce a new construction of a
WP-geodesically complete simplicial complex where the simplex is a copy
of Teichmuller space. This construction suggests a further analogy between
the theory of non-compact symmetric space (and Tits buildings) and that of
Teichmuller space.
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| Dec.17（Wed) | | |
| 14:00-14:45 |
Tim Hoffmann (TU M\"unchen & Kyushu Univ.)
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"The Steiner formula, curvature and discrete surfaces"
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Abstract:
The aim of discrete differential geometry is to find "structure
preserving" discretizations of classical objects and notions form
differential geometry. By preserving the rich structure the smooth
diffenrential geometry provides, these discretizations often can be
derived in many different ways since they resemble their smooth
counterparts in many different aspects.
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In case of surfaces of constant mean curvature, one of the many ways
is through Steiner's formula. The Steiner formula gives the area of a
parallel surface in terms of the distance and the original surface's
area and curvatures. For quadrilateral meshes with planar faces, which
serve as discretizations of conjugate nets, Schief defined curvatures
for these polyhedral surfaces by looking at the area of parallel
meshes which in turn leads to discretizations of surfaces with
constant curvature. However, it depends on a choice for the Gauss map
of the discrete conjugate net. This was carried on by Bobenko,
Pottmann, Schief et al and dualizability and curvature formulas for
discrete surfaces were found in terms of mixed area. The well known
discretizations of surfaces of constant mean curvature can be derived
from this approach.
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| Dec.17（Wed) | | |
| 14:55-15:40 |
Shoichi Fujimori (Fukuoka University of Education）
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" Triply periodic minimal surfaces bounded by vertical symmetry planes "
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Abstract:
We give a uniform and elementary treatment of many classical and new
triply periodic minimal surfaces in Euclidean three-space, based on
a Schwarz-Christoffel formula for periodic polygons in the plane.
Our surfaces share the property that vertical symmetry planes cut
them into simply connected pieces.
This is joint work with Matthias Weber.
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File of Talk
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| Dec.17（Wed) | | |
| 15:50-16:35 |
Katrin Leschke（Univ. of Leicester, UK）
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"Hamiltonian stationary Lagrangian tori in C^2 revisited"
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Abstract: Helein and Romon gave a complete description of HSL tori in
C^2 in terms of Fourier polynomials. In recent work with Romon we showed that
this description is due to the fact that the spectral curve of the associated
harmonic map into the the 2-sphere (the left normal of the HSL torus) has spectral
genus zero. To explain this link I will discuss the relation between this spectral
curve and the multiplier spectral curve of the HSL torus.
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| Dec.17（Wed) | | |
| 16:45-17:30 |
Bennet Palmer (Idaho State Univ., USA)
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" Anisotropic surface energies "
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Abstract: An anisotropic surface energy assigns an energy to a surface which
depends on the direction of the surface at each point. Such an energy
is used to model the shape of an interfaces of anisotropic media.
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We will discuss a capillary (free boundary) problem involving
anisotropic surface energies which includes wetting and line tension.
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File of Talk
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| Dec.17（Wed) | | |
| 17:40-18:25 |
John Bolton (Univ. of Durham, UK)
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"Minimal 2-spheres with various symmetry properties in the round 4-sphere"
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Abstract: Each minimal 2-sphere in the round 4-sphere has a holomorphic horizontal lift
to the total space of the twistor bundle, which is the projection from complex projective
3-space to the 4-sphere. Symmetry properties of the minimal 2-sphere are reflected in
corresponding properties of the lift, and this is used to construct, and, in some cases,
classify, minimal 2-spheres in the 4-sphere. This is joint work with
Prof. L. Fernandez (CUNY).
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| Dec.18（Thu) | | |
| 9:00-10:00 |
Ryoichi Kobayashi（Nagoya University, Japan）
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"An Interpretation of the Period Condition of
Algebraic Minimal Surfaces from the View Point of
Lemma on Logarithmic Derivative"
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Abstract: Nevanlinna's Lemma on Logarithmic Derivative
is the source of almost all results in the
value distribution theory of entire holomorphic curves
into projective varieties.
On the other hand, the period condition of algebraic
minimal surfaces is extremely difficult to realize
in the attempt of their construction.
The goal of my talk is to give an interpretation of
the period condition of algebraic minimal surfaces
from the view point of Lemma on Logarithmic Derivative
applied to the Gauss map of algebraic minimal surfaces
lifted on the universal cover.
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File of Talk
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| Dec.18（Thu) | | |
| 10:15-11:15 |
Mark Haskins (Imperial College London)
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"A panoply of Special Lagrangian singularities"
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Abstract:
(joint work with Nikolaos Kapouleas)
Special Lagrangian submanifolds are a special type of higher-dimensional minimal
submanifold that occur naturally in Calabi-Yau manifolds. They have been the
focus of much attention from both mathematicians and string theorists because of
their role in Mirror Symmetry. Singularities of special Lagrangians play a very
important part in this story but as yet are poorly understood. Special Lagrangian
cones with an isolated singularity form the local models for the simplest kinds of singular special Lagrangians. In this talk we will discuss recent progress in the
construction of special Lagrangian cones in dimensions three and higher.
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Three-dimensional special Lagrangian cones with cross-section a 2-torus all arise
from algebraically completely integrable systems constructions. Using spectral
curve methods Carberry-McIntosh proved the surprising result that special
Lagrangian 2-torus cones can come in continuous families of arbitrarily large
dimension. We will show that starting in dimension 6 there are infinitely many
topological types of special Lagrangian cone which can come in continuous
families of arbitrarily large dimension.
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The three main ingredients needed to prove this result are
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a. the integrable systems techniques for special Lagrangian 2-torus cones
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b. gluing constructions of infinitely many topological types of special Lagrangian cones in dimensions 3 and higher
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c. a 'twisted product' construction to produce new special Lagrangian cones from a pair of lower dimensional special Lagrangian cones.
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File of Talk
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| Dec.18（Thu) | | |
| 11:30-12:30 |
Hironori Sakai (Tokyo Metropolitan University)
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"Normalization of differential equations associated to orbifold quantum
cohomology"
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Abstract: Starting from the differential equations associated to quantum cohomology,
the original quantum cohomology can be recovered by taking a normalized
trivialization of the D-module. We will discuss this phenomenon for the
orbifold quantum cohomology of a hypersurface in weighted projective space.
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File of Talk
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| Dec.18（Thu) | | |
| 14:00-15:00 |
Armen Sergeev（Steklov Mathematical Institute, Moscow)
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" Quantization of the Universal Teichm\"uller Space "
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Abstract: PDF
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File of Talk
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| Dec.18（Thu) | | |
| 15:15-16:15 |
Andery Domrin（Moscow University)
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" Meromorphic extension of solutions of soliton equations "
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Abstract: We show that every local (in $x$ and $t$) holomorphic solution
$u(x,t)$ of any soliton equation belonging to a large class (including
Kroteweg--de Vries equation, nonlinear Schroedinger equation, their
modifications and hierarchies) admits analytic continuation to a
globally meromorphic function of $x$ (on the whole complex plane) for
every fixed $t$. The proof uses a local version (independent of any
boundary conditions) of the inverse scattering method. In particular,
we give a simple criterion of solubility of the local holomorphic Cauchy
problem for equations under study in terms of the scattering data of
the initial condition.
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| Dec.18（Thu) | | |
| 16:25-17:10 |
Toshihiro Nogi (Osaka City Univ.)
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"On holomorphic sections of a holomorphic family of Riemann surfaces of
genus two"
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Abstract: (Joint work with Yoichi Imayoshi and Yohei Komori)
We study a holomorphic family of Riemann surfaces of genus two
constructed by Gonzalo Riera.
The main goal of this talk is to estimate the number of holomorphic
sections of this family.
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File of Talk
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| Dec.18（Thu) | | |
| 17:20-18:05 |
Yu Kawakami（Kyushu Univ.& OCAMI）
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"Recent progress in the value distribution of the hyperbolic Gauss map"
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Abstract:
In this talk, I will explain my recent work on the value distribution of
the hyperbolic Gauss map. In particular, I will define ``algebraic''
class of constant mean curvature one (CMC-1) surfaces in the hyperbolic
three-space and give the ramification estimate for the hyperbolic Gauss
map of them.
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File of Talk
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| Dec.19（Fri) | | |
| 9:00-10:00 |
Craig A. Tracy (UC Davis, USA)
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"The Asymmetric Simple Exclusion Process : Integrable Structure and Limit Theorems"
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Abstract:
We consider the asymmetric simple exclusion process (ASEP) on the
integer lattice in the case of step initial condition. Using ideas
from Bethe Ansatz we show that the probability distribution for the
position of an individual particle is given by an integral whose
integrand involves a Fredholm determinant. We use this formula to
derive a limit theorem for ASEP which extends Kurt Johansson's result
for TASEP to ASEP.
This is joint work with Harold Widom.
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File of Talk
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Ref. Japanese Longevity
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| Dec.19（Fri) | | |
| 10:15-11:15 |
Hao Xu (Zhejiang Univ., P.R.China)
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"Intersection numbers on the moduli spaces of stable curves"
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Abstract:
In the first part, we give a survey of algorthms for computing $\psi$ class
intersection numbers, Witten's r-spin intersection numbers and higher
Weil-Petersson volumes of moduli spaces of curves. In particular, our work
on effective recursion formulae of higher Weil-Petersson volumes is motivated
by the work of Prof. Mulase and Safnuk.
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In the second part, we present our proof of the Faber intersection number
conjecture on moduli spaces of curves as well as some vanishing identities
of Gromov-Witten invariants. The latter has been proved recently by X. Liu
and Pandharipande.
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File of Talk
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| Dec.19（Fri) | | |
| 11:30-12:30 |
Motohico Mulase（University of California, Davis, USA)
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"An integrable system approach to the newly discovered topological recursion"
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Abstract:
A recent discovery in topological string theory
predicts that Gromov-Witten invariants of
toric Calabi-Yau threefolds are miraculously
calculated by an effective recursion formula
that uses only classical Riemann surface theory.
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We found that some of such examples can
be explained through a deformation theory of
KP tau-functions. In this talk our new theory,
based on a joint work with Brad Safnuk, will
be reported.
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| Dec.19（Fri) | | |
| 14:00-15:00 |
Iskandar Taimanov (Novosibirsk) |
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" The Moutard transformation and blowing up solutions
of the Novikov-Veselov equation "
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Abstract:
The Moutard transformation is a generalization of the
Darboux transformation he case of two-dimensional
Schrodinger operators. Although the Darboux transformation was
widely used for constructing differential operators with
interesting properties the Moutard transformation until recently
was not used in the spectral theory. In our joint papers with S.P. Tsarev we
constructed the first known examples of two-dimensional Schrodinger operators on
the plane with bounded and fast decaying potentials which have nontrivial
(and even multi-dimensional kernel). Moreover we contsruct
examples of solutions of the Novikov-Veselov equation (one of
two-dimensional generalizations of the Korteweg-de Vries equation)
which blow up in finite time and have smooth fast decaying Cauchy data.
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File of Talk
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| Dec.19（Fri) | | |
| 15:15-16:15 |
Andrey Mironov (Sobolev Inst. of Math.)
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"Spectral data for Hamiltonian-minimal Lagrangian tori in ${\bold C}P^2$"
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Abstract:
We find spectral data that allow to find in explicit form
Hamiltonian-minimal Lagrangian tori in ${\bold C}P^2$ in terms
of theta functions of spectral curves.
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File of Talk
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| Dec.19（Fri) | | |
| 16:30-17:15 |
Katsuhiro Moriya （Univ. Tsukuba) |
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"Super-conformal surfaces in the Euclidean four space in terms of
null complex holomorphic curves"
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Abstract:
A super-conformal surface in the Euclidean four space is a surface
whose curvature ellipse is a circle.
It is the stereographic projection of a surface with vanishing Willmore
energy in the conformal four sphere.
A surface with vanishing Willmore energy in the conformal four sphere
is a twister projection of
a complex holomorphic curve in the three dimensional complex projective
space.
In this talk, a super-conformal surface in the Euclidean four space is
constructed by a null complex holomorphic curve in
the four dimensional complex Euclidean space.
The use of quaternionic analysis makes our proof shorter than Dajczer
and Tojeiro's.
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File of Talk
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| Dec.19（Fri) | | |
| 17:30-18:15 |
Tetsuya Taniguchi (Kitasato Univ) |
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"Fourier-Mukai transforms and spectral data of harmonic tori
into compact symmetric spaces"
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Abstract: PDF
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File of Talk
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| Dec.20（Sat) | | |
| 9:00-10:00 |
Franz Pedit (Univ. Tuebingen & Univ. Massachusetts) |
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"Global Aspects of Integrable Surface Geometry"
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Abstract:
We will discuss the relevance of the spectral curve construction for
conformally immersed tori to global problems in surface geometry and
indicate how to extend these ideas to conformally immersed surfaces of
arbitrary genus.
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File of Talk
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| Dec.20（Sat) | | |
| 10:15-11:15 |
Martin Kilian (Univ. College Cork, Ireland) |
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"On the Lawson Conjecture"
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Abstract: While there are no compact minimal surfaces in Euclidean
3-space, Lawson showed in 1970 that the curvature of the 3-sphere allows
for embedded compact minimal surfaces of arbitrary genus. In particular,
in collaboration with Hsiang he investigated minimal tori in the
3-sphere, and conjectured that the only embedded minimal torus in the
3-sphere is a torus which possesses a 2-parameter family of isometries,
the so-called Clifford torus. In recent work with M. U. Schmidt, we prove
that Lawson's conjecture indeed holds, and in this talk I will give an
outline of the proof, which uses modern methods from the theory of
integrable systems.
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| Dec.20（Sat) | | |
| 11:30-12:30 |
Martin Ulrich Schmidt（Univ. Mannheim，Germany) |
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"On the Moduli of Alexandrov embedded cmc cylinders in S^3"
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Abstract: PDF
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| Dec.20（Sat) | | |
| 14:00-14:45 |
Emma Carberry （Univ. of Sydney, Australia) |
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"Almost-complex tori in the 6-sphere"
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Abstract: Octonionic multiplication defines a natural almost-complex structure on
$S^6\subset{Im}\O$ and almost-complex curves $M^2\rightarrow S^6$ are rather pleasant
examples of minimal surfaces. In particular, the cone over such a curve is associative
and hence absolutely volume minimising. These almost-complex curves come in two types:
they are either {\it isotropic} (in which case Bryant has shown they can be algebraically
constructed from holomorphic maps) or they are {\it superconformal}. I shall describe a
spectral curve approach to superconformal almost-complex tori; the main point of which is
to also obtain an algebraic characterisation of these surfaces and hence study their
moduli. An interesting feature is that the relevant abelian variety in this case is the
intersection of two Prymians.
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| Dec.20（Sat) | | |
| 15:00-16:00 |
Francis Burstall (Univ. Bath, UK) |
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"Conserved quantities in geometric integrable systems"
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Abstract: Many geometric integrable systems admit integrable specialisations:
examples include constant mean curvature surfaces in 3-dimensional
space forms (specialising both isothermic surfaces and constrained
Willmore surfaces); submanifolds of space-forms with constant
sectional curvatures and flat normal bundle (specialising conformally
flat submanifolds with flat normal bundle and, in two dimensions,
Guichard surfaces) and the special isothermic surfaces studied by
Bianchi and Darboux. I shall describe joint work with Calderbank and
Santos which provides an enlightening characterisation of such
specialisations in terms of the associated loop of flat connections
and parallel sections thereof depending polynomially in the loop
parameter. I shall also describe (if times permits) an application
with Hertrich-Jeromin, Rossman and Santos to discrete constant mean
curvature surfaces.
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