Program
Time 
January. 10 
12:0012:50 
Registration 
13:0014:00 
James R. Lee gEigenvalues of the Laplacian, multiflows, and metric uniformizationh 
14:1015:10 
James R. Lee gMultiway spectral partitioning and higherorder Cheeger inequalitiesh 
15:1015:40 
Tea break 
15:4016:40 
Masato Mimura gVerifiable expanderh 
16:5017:50 
Atsushi Kasue
gKuramochi
boundaries of transient networksh 
18:00 
Banquet (5th floor) 
Time 
January. 11 
9:3010:30 
Hiroki Sako
gSequence of finite
metric spaces : amenability and connectivityh 
10:4011:40 
Hiroshi Hirai gSome combinatorial optimization
problems related to metric spaces of nonpositive
curvatureh 
13:0014:00 
Konrad
Polthier gDiscrete Minimal Surfaces based
on CatmullClark Finite Elements Ih 
14:1015:10 
Konrad
Polthier gDiscrete Minimal Surfaces
based on CatmullClark Finite Elements IIh 
15:1016:00 
Tea break & Poster
session 
16:0016:30 
Masashi Yasumoto
gDiscretization of
linear Weingarten surfaces with Weierstrasstype
representationsh 
16:4017:40 
Wayne Rossman
gDiscretization of
general linear Weingarten surfacesh 
Time 
January. 12 
9:3010:30 
Kenji Kajiwara
gIntegrable
Discrete Deformations of Discrete Curvesh 
10:4011:40 
Kenichi
Maruno gDiscretization of integrable
systems and selfadaptive moving mesh schemesh 
11:5012:50 
Shoichi
Fujimori gDegenerate limits
of triply periodic minimal surfaces of genus 3 h 


Jan. 10
James R. Lee (University of Washington)
Title: Eigenvalues of the Laplacian, multiflows, and metric uniformization
Abstract:
The study of eigenvalues of the
Laplace operator on graphs and manifolds has a long and rich history. In
the Riemannian setting, conformal uniformization is a
powerful tool for understanding the spectrum of the LaplaceBeltrami operator
on surfaces. Even though graphs do not have an inherent conformal
structure, we will introduce a form of metric uniformization
that allows one to deform the underlying geometry by variational
methods.
The uniformization
approach allows us to resolve conjectures of Spielman
and Teng about the spectrum of planar graphs and
their generalizations. In particular, one can provide formal guarantees
on the efficacy of widelyused spectral algorithms for
graph partitioning. This answers positively a question of Gromov on whether there is a generalization of conformal
techniques to minorclosed families of graphs that recovers the separator
theorem of Alon, Seymour, and Thomas. One can
show that the theorems for graphs actually imply corresponding results for
surfaces. Thus as a special case, we recover Korevaar's
bounds on the spectrum of compact surfaces.
James R. Lee (University of Washington)
Title: Multiway spectral partitioning and higherorder Cheeger
inequalities
Abstract:
A basic fact in spectral graph theory is
that the number of connected components in an undirected graph is equal to the
multiplicity of the eigenvalue zero in the Laplacian
matrix of the graph. Cheegerfs inequality and
its variants provide a robust version of this fact for multiplicity 2. It
has been conjectured that an analogous characterization holds for higher
multiplicities: There are k small eigenvalues if and only if the vertex
set of the graph can be partitioned into k sets with small boundary.
In the setting of mathematical physics,
Simon and HoeghKrohn (1972) asked whether every
hyperbounded Markov operator has a spectral gap. Much later, Laurent Miclo showed that a positive answer to the first conjecture
yields a positive answer to the second. In joint work with OveisGharan and Trevisan, we
prove Miclo's higherorder Cheeger
conjecture. The proof is based on an eigenfunction
localization method using random partitions of metric spaces. The
argument is constructive and yields an algorithm for multiway spectral
partitioning that appears to work well in practice. Moreover, the study
of higherorder Cheeger inequalities has some
striking applications in computational complexity theory.
Masato Mimura (Tohoku University)
Title: Verifiable expander
Abstract:
We say an infinite connected graph Gamma
is a verifiable expander if any finite connected graph Lambda which ``locally
looks the same as
Gamma" is an expander. That means,
there exists a universal bound from below of the first positive Laplace
eigenvalues of such Lambda. We will
discuss a recent development on this notion.
Atsushi Kasue (Kanazawa
University)
Title: Kuramochi boundaries of transient networks
Abstract:
We study functions of finite Dirichlet sum on a transient network and show that a family
of the Dirichlet forms induced on the boundaries of
finite subnetworks converges in a certain variational sense to the Dirichlet
form induced by the random walk on the Kuramochi
boundary of the network, if the subnetworks increase
and exhaust the whole network.
Jan. 11
Hiroki Sako (Niigata University)
Title: Sequence of finite metric spaces : amenability
and connectivity
Abstract:
I will give a talk on discrete metric
spaces, especially paying attention on disjoint unions of finite metric spaces.
Among them, one having high connectivity is called an expander sequence. The
item attracts not only mathematicians but also computer scientists. The objects
relate to construction of networks and error correcting codes. We will focus on
mathematical structure of uniformly locally finite metric spaces. It has been
already known that a space cannot satisfy both of the following:
i) Coarse amenability,
ii) Containing an
expander sequence.
I would like to
show that every uniformly locally finite metric space has exactly one of i) or a little weaker condition than ii). In the proof, a Banach space of continuous linear operators plays a key
role.
Hiroshi Hirai (University of Tokyo)
Title: Some combinatorial optimization problems related to
metric spaces of nonpositive curvature.
Abstract:
The maxflow mincut
theorem, due to FordFulkerson, says that in a network with terminal pair s,t, the maximum value of an (s,t)flow is equal to the minimum capacity of an (s,t)cut. This theorem is one of most fundamental
theorems in combinatorial optimization and graph theory. In this talk, we will
discuss some generalizations of this theorem, and explain interesting
connections to nonpositivelycurved metric spaces.
(i) Multicommodity flow problem:
There are several generalizations
of the maxflow mincut theorem to gmultiflowh
settings. For a class of multiflow problems, the dual
problem, which corresponds to gmincuth in the single flow setting, is
formulated as a location problem on a CAT(0) B_2
complex, called a "folder complex". The study of this
formulation has brought significant contributions to several unsolved problems
in multiflow theory.
(ii) Multifacility location problem (0extension problem):
Problems of
locating several facilities on graphs (or metric spaces) have numerous
applications in operation research, machine learning and computer vision. Multifaclity location problem (or 0extension problem) is
an important special case where the cost of a location is given by a
linear function on distances between facilities. The computation complexity of
this problem depends on the underlying graph G. Recently the following
dichotomy theorem was established:
If G is an "orientable modular graph", then the problem is
polynomial time solvable (Hirai SODA13). Otherwise it is NPhard (Karzanov 98).
The idea behind
the proof of the polynomial time solvability is formulating the problem as an
optimization problem on a certain metrized complex
constructed from orientable modular graph G. It
turned out that this complex is an "orthoscheme
complexh introduced by BradyMcCammond in geometric
group theory.
Joint with J. Chalopin, V. Chepoi, and D. Osajda, we showed that several nonpositively curved
complexes are constructed from orientable modular
graphs. Examples include CAT(0)cube complexes and
Euclidean buildings of type C.
Konrad Polthier (Freie
Universität Berlin)
Title: Discrete Minimal Surfaces based on CatmullClark
Finite Elements I & II
Abstract:
We
present a novel concept for discrete minimal surfaces based on CatmullClark finite elements. These surfaces are globally
C1 though determined by a discrete control grid. We will report about the
theoretical setup and provide several examples of these discrete minimal
surfaces. A special advantage of this new discretization scheme is the uniform
integration of discrete differential geometry, finite element analysis and
computer aided design. Joint work with Anna Wawrzinek.
Masashi Yasumoto (Kobe
University)
Title: Discretization of linear Weingarten surfaces with Weierstrasstype
representations
Abstract:
In this talk we discretize linear
Weingarten surfaces in nonEuclidean spaceforms given
by Weierstrasstype representations. In particular, discretizations
of maximal surfaces in LorentzMinkowski 3space and
linear Weingarten surfaces of Bryant and Bianchi type are introduced.
Furthermore, we compare smooth, discrete and semidiscrete surfaces with the
same curvature conditions. This talk is partly based on the joint work with
Wayne Rossman, and will be highly related to the talk
by Wayne Rossman that follows it.
Wayne Rossman (Kobe
University)
Title: Discretization of general linear Weingarten surfaces
Abstract:
Continuing from the previous talk by
Masashi Yasumoto, we will consider how to discretize
a more general class of surfaces, including those that do not have Weierstrasstype representations. We will see how general
discrete linear Weingarten surfaces can be defined using constant conserved
quantities of associated flat connections as a tool. Then, since smooth linear Weingarten
surfaces typically have singularities (such as cuspidal
edges, swallowtails, cuspidal crosscaps, etc), we will examine how to define corresponding
singularities in the discretized case.
Jan. 12
Kenji Kajiwara (Kyushu University)
Title: Integrable Discrete Deformations of Discrete Curves
Abstract:
We
present some results on integrable discrete
deformations of space/plane discrete curves in various settings, including: (i) isoperimetric deformation of plane curves (discrete mKdV equation) (ii) conformal deformation of plane curves
(discrete Burgers equation) (iii) torsionpreserving deformation of space
discrete curves of constant torsion (discrete mKdV
and discrete sineGordon equation) (iv) curvaturepreserving deformation of
space discrete curves of constant curvature (discrete mKdV
and discrete sineGordon equation).
Kenichi Maruno
(Waseda University)
Title: Discretization of integrable systems and
selfadaptive moving mesh schemes
Abstract:
In recent years, we have investigated integrable discretizations of integrable nonlinear partial differential equations (PDEs)
whose solutions possess singularities.
For PDEs in this class, we obtained various discrete integrable
systems with selfadaptive moving mesh which can be powerful tools for
numerical simulations of nonlinear PDEs.
One of keys to construct selfadaptive moving mesh schemes is a
discretization of hodograph transformations which are
transformations between Lagrangian description and Eulerian description.
In a geometric point of view, selfadaptive moving mesh schemes can be
interpreted as Eulerian description of the motion of
discrete curves. In this talk, I
will review recent results of discrete integrable
systems with selfadaptive moving mesh.
Shoichi
Fujimori (Okayama University)
Title: Degenerate limits of triply periodic minimal surfaces of genus 3
Abstract:
We consider limits of triply periodic
minimal surfaces in Euclidean 3space. We prove that some important examples of
singly or doubly periodic minimal surfaces can be obtained as limits of triply
periodic minimal surfaces. This is joint work with Norio Ejiri
and Toshihiro Shoda.