Doctoral Forum of Mathematics
between Fudan and Kyoto Universities
Date: December 8-12, 2012
Place: Faculty of Science Bldg. No. 3 Room 110
Invited Speakers
Shu Kawaguchi (Kyoto University) Zhi Lu (Fudan University)
Jiangang Ying (Fudan University) Mitsuhiro Shishikura (Kyoto University)
Hisashi Okamoto (RIMS)
Yongqian Zhang (Fudan University)
Engui Fan (Fudan University)
Koji Fujiwara (Kyoto University)

8th (Sat.)

9th (Sun.)

10th (Mon.)

11th (Tue.)

12th (Wed.)

Jiangang Ying

Hisashi Okamoto

Engui Fan

Mitsuhiro Shishikura

Yongqian Zhang

Koji Fujiwara

Shuxia Wang

9:30 - 10:30

Shu Kawaguchi

11:00 - 12:00

Zhi Lu13:30 - 14:00

Mamoru Okamoto

Weimin Peng

Yihang Hao

Yasuhiro Ishitsuka

14:10 - 14:40

Long Hu

Kenta Ishimoto

Kenta Tottori

Jinsong Xu

15:00 - 15:30

Masaki Wada

Caixuan Ren

Pengpeng Xie

Kazuto Ota

15:40 - 16:10

Jiangyan Pu

Miho Hatanaka

Takayuki Okuda

Zhenguo Feng

16:30 - 17:00

Naoya Umezaki

Fei Yang

Chong Zhao

Takamori Kato

Ke Wang
Kanae Akaiwa

November 8. Sat.
A.M.
9:30 - 10:30
11:00 - 12:00

P.M.
13:30 - 14:00
14:10 - 14:40

15:00 - 15:30
15:40 - 16:10
16:30 - 17:00

Chair: Professors
Shu Kawaguchi (Kyoto University)
Degrees of iterations of triangular automorphisms over Q-algebras
Zhi Lu (Fudan University)
Moment-angle complexes and Halperin-Carlsson conjecture
Chair: Students
Mamoru Okamoto (Kyoto University)
Well-posedness for the Chern-Simons-Dirac system in two dimensions
Long Hu (Fudan University)
Exact boundary controllability and synchronization for a coupled systemof 1-D Wave equations
Masaki Wada (Tohoku University)
Perturbation of Dirichlet forms and stability of fundamental solutions
Jiangyan Pu (Fudan University)
Robust consumption portfolio optimization with recursive utility
Naoya Umezaki (University of Tokyo)
On the Grothendieck’s monodromy theorem

November 9. Sun.
A.M.
9:30 - 10:00
10:10 - 10:40
11:00 - 11:30

P.M.
13:30 - 14:00
14:10 - 14:40
15:00 - 15:30

Chair: Students
Shuxia Wang (Peking University)
Small energy scattering for the Klein-Gordon-Zakharov system
Ke Wang (Fudan University)
Boundary feedback stabilization for isothermal Euler equations with friction
Kanae Akaiwa (Kyoto University)
On a matrix tridiagonalization technique based on the quotient-diﬀerence formula
Chair: Students
Weimin Peng (Fudan University)
Global solutions to incompressible Navier-Stokes equations with gravity
Kenta Ishimoto (RIMS)
Stokes dynamics of swimming microorganisms
Caixuan Ren (Fudan University)
Regularization by projection for a backward problem of the time-fractional
diﬀusion equation1

15:40 - 16:10

16:30 - 17:00

Miho Hatanaka (Osaka City University)
The uniqueness of decompositions of a (quasi)toric manifold into productsof real 2 or 4 dimensional (quasi)toric manifolds
Fei Yang (Fudan University)
Rational maps whose Julia sets are Cantor circles

November 10. Mon.
A.M.
9:30 - 10:30
11:00 - 12:00

Chair: Professors
Jiangang Ying (Fudan University)
On Kac’s scattering length formula
Mitsuhiro Shishikura (Kyoto University)
Renormalization in complex dynamics

November 11. Tue.
A.M.
9:30 - 10:30
11:00 - 12:00

P.M.
13:30 - 14:00

14:10 - 14:40

15:00 - 15:30
15:40 - 16:10

16:30 - 17:00

Chair: Professors
Hisashi Okamoto (RIMS)
Water-waves and drift of particles
Yongqian Zhang (Fudan University)
Global solution for 2-D steady supersonic ﬂow over a bending wall
Chair: Students
Yihang Hao (Fudan University)
The existence and blow-up criterion of liquid crystals system in critical
Besov space
Kenta Tottori (Tohoku University)
A Monge-Ampère foliation associated with a geodesic in the space ofKähler metrics
Pengpeng Xie (Fudan University)
Sharp perturbation bounds for the total least squares problem
Takayuki Okuda (University of Tokyo)
Classiﬁcation of semisimple symmetric spaces with properly discontinuous
actions of surface groups
Chong Zhao (Fudan University)p-essential normality of quasi-homogeneous Arveson’s submodules2November 12. Wed
A.M.
9:30 - 10:30
11:00 - 12:00

P.M.
13:30 - 14:00
14:10 - 14:40
15:00 - 15:30
15:40 - 16:10
16:30 - 17:00

Chair: Professors
Engui Fan (Fudan University)
Super extension of Bell polynomials with applications to superymmetric equations
Koji Fujiwara (Kyoto University)
Quasi-homomorphisms
Chair: Students
Yasuhiro Ishitsuka (Kyoto University)
Complete intersections of two quadrics and Galois cohomology
Jinsong Xu (Fudan University)
A characterization of birationality of the fourth pluricanonical map
Kazuto Ota (Tohoku University)
A generalization of the theory of Coleman power series
Zhenguo Feng (Fudan University)
Tricomi problem for a quasilinear Lavrentiev-Bitsadze equation of mixed type
Takamori Kato (Kyoto University)
Well-posedness of the ﬁfth order KdV equation with periodic boundary condition3

Degrees of iterations of triangular automorphisms over Q-algebras
Shu Kawaguchi, Kyoto University
Abstract: For a birational map f on projective space deﬁned over C, behavior of the degrees of
the iterations of f (forwardly and backwardly) has been quite studied. In this talk, we would liketo study such behavior when C is replaced by a Q-algebra in general. We consider a simple case,where f is an aﬃne space triangular automorphism deﬁned over a Q-algebra.
Moment-angle complexes and Halperin-Carlsson conjecture
Zhi Lu, Fudan University
Abstract: We give an algebra–combinatorics formula of the Möbius transform for an abstract
simplicial complex K on [m] = {1, ..., m} in terms of the Betti numbers of the Stanley–Reisner facering of K. Furthermore, we employ a way of compressing K to estimate the lower bound of the sumof those Betti numbers by using this formula. As an application, associating with the moment-angle
complex ZK (resp. real moment-angle complex RZK ) of K, we show that the Halperin–Carlsson
conjecture holds for ZK (resp. RZK ) under the restriction of the natural T m -action on ZK (resp.
(Z2 )m -action on RZK ). (Joint work with Xiangyu Cao)
Well-posedness for the Chern-Simons-Dirac system in two dimensions
Mamoru Okamoto, Kyoto University
Abstract: In this talk, we consider the Cauchy problem for the Chern-Simons-Dirac system in
R1+2 . Using the gauge invariance, we reduce the Chern-Simons-Dirac system to a Dirac equation.
We discover the null structure of this Dirac equation. The null structure plays a crucial role in the
proof because it improves the product estimates for wave Sobolev spaces. Using the null structure
estimates, we show that the Cauchy problem of this Dirac equation is local in time well-posed in
H s with s > 1/4. For the proof, we employ the Fourier restriction norm methods.

Exact boundary controllability and synchronization for a coupled system of 1-D wave
equationsLong Hu, Fudan University
Abstract: In this talk, we ﬁrstly present a uniﬁed constructive method to establish the exact
boundary controllability for a kind of coupled system of 1-D wave equations with Dirichlet type,
Neumann type and Coupled third type boundary conditions in the framework of semi-global classical solutions. Based on this, we then introduce exact synchronization for a coupled system of 1-D
wave equations with boundary conditions of various types and we show that the synchronization4

can be realized by means of less boundary controls. The latter part of this lecture is a recent joint
work with Prof. Tatsien Li and Prof. Bopeng Rao.

Perturbation of Dirichlet forms and stability of fundamental solutions
Masaki Wada, Tohoku University
Abstract: Let (E, F) be a symmetric regular Dirichlet form as follows:∫∫(u(y) − u(x))(v(y) − v(x))J(x, y)dxdy
E(u, v) =Rd ×Rdκ1
κ2≤ J(x, y) ≤(0 < κ1 ≤ κ2 )dd|x − y| ϕ(|x − y|)|x − y| ϕ(|x − y|)ϕ(r) = rα exp(m(r − 1) ∨ 0) (0 < α < 2, m ≥ 0)
It is proved that the associated fundamental solution p(t, x, y) admits the two-sided estimates byChen, Kumagai et al. Let µ be a positive Radon smooth measure in a certain class and consider∫
the perturbed form E µ (u, u) = E(u, u) − Rd u2 (x)dµ. Denote the fundamental solution associatedwith E µ by pµ (t, x, y). We call it the stability of fundamental solution that pµ (t, x, y) has the sametwo-sided estimates as p(t, x, y) up to positive constants. In this talk, we establish a necessary andsuﬃcient condition on µ for the stability.

Robust consumption portfolio optimization with recursive utility
Jiangyan Pu, Fudan University
Abstract: This talk examines a continuous time intertemporal consumption and portfolio choiceproblem for an investor with recursive preferences. The investor worries about model misspeciﬁcation and seeks robust decision rules. We formulate the HJB equation and provide explicit solutions.
We also compare our robust solutions to that with non-robustness.

On the Grothendieck’s monodromy theorem
Naoya Umezaki, University of Tokyo
Abstract: Grothendieck showed that the action of Galois group on the etale cohomology, which
we call monodromy, can be unipotent by a ﬁnite base extension.
In this talk, we introduce the monodromy in above sense and give my result, which says that thedegree of base extension can be controlled only by certain invariants of the varieties.

Small energy scattering for the Klein-Gordon-Zakharov system
Shuxia Wang, Peking University
5

Abstract: In this talk, I will introduce the radial-improved Strichartz estimates for Klein-Gordon
and wave equations; and then prove the small energy scattering of Klein-Gordon-Zakharov system
in Sobolev spaces by using normal form reduction and the Strichartz estimates given above.

Boundary feedback stabilization for isothermal Euler equations with friction
Ke Wang, Fudan University
Abstract: A second-order quasilinear hyperbolic equation is derived that is equivalent to theisothermal Euler equations with friction. For the corresponding mixed initial-boundary value problem, we consider non-stationary solutions locally around a stationary state on a ﬁnite time intervaland discuss the well-posedness of this kind of problem. We introduce a strict H 2 -Lyapunov function
and show that the boundary feedback constant can be chosen such that the H 2 -Lyapunov function
and hence also the H 2 -norm of the diﬀerence between the non-stationary and the stationary state
decays exponentially with time. An exponential estimate for the C 1 -norm is also established. Then,we generalize this result to the gas ﬂows in a planar tree-like network.

On a matrix tridiagonalization technique based on the quotient-diﬀerence formula
Kanae Akaiwa, Kyoto University
Abstract: The quotient-diﬀerence (qd) algorithm was proposed by H. Rutishauser in 1954 and isusually known as be applicable to computing matrix eigenvalues with a tridiagonalization techniqueby the Householder transformation. In this talk, we mainly describe that dense matrix can be
reduced into tridiagonal one through using the recursion formula of the qd algorithm instead of
the Householder transformation. We also prove that the reduced matrix has the same eigenvalues,
except for multiplicity, as original one from the viewpoint of the Jordan canonical form. Some
examples of tridiagonalization are numerically given.

Global solutions to oncompressible Navier-Stokes equations with gravity
Weimin Peng, Fudan University
Abstract: In this talk, we consider a special class of initial data to the 3D incompressible NavierStokes equations with gravity. We show that, under such conditions, the incompressible NavierStokes equations with gravity are globally wellposed. The important features of the initial data isthat the velocity ﬁelds minus gravity term are almost parallel to the corresponding vorticity ﬁelds
in a very large space domain.

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Stokes dynamics of swimming microorganisms
Kenta Ishimoto, RIMS
Abstract: The ﬂow around a swimming microorganism obeys the inertialess Stokes equations
with time-depending boundary conditions. Due to the linearity of the governing equations, there
exists a constraint on motion of the swimmer, which is known as the scallop theorem. In this talk,the theorem and its breakdown due to small inertia will be argued. A numerical approach to sucha swimmer will be also discussed.

Regularization by projection for a backward problem of the time-fractional diﬀusion
equation
Caixuan Ren, Fudan University
Abstract: Time-fractional diﬀusion equations, which frequently appear in the diﬀusion process,describe the continuous time random walk phenomena. In this talk we investigate a backward timefractional diﬀusion equation where one wants to extract the initial temperature distribution fromthe observation data provided along the ﬁnal time t = T. Based upon the eigenfunction expansion,an analytical solution is deduced. However, such an approach does not depend continuously on theobservation data. Hence, we propose a regularization by projection method where the truncatedlevel plays the role of the regularization parameter. Under appropriate regularity assumptions of theexact solution, a uniform error estimate with an optimal convergence rate between the reconstructed
solution and the exact one is obtained both for a priori and a posteriori parameter choice strategies.
Finally, numerical examples are presented to illustrate the validity and eﬀectiveness of the proposed
method.

The uniqueness of decompositions of a (quasi)toric manifold into products of real 2
or 4 dimensional (quasi)toric manifolds
Miho Hatanaka, Osaka City University
Abstract: A quasitoric manifold is a closed smooth even domensional manifold with an eﬀective
smooth action of compact torus, such that it is locally equivariantly diﬀeomorphic to a representation space of compact torus and the orbit space is a simple convex polytope. The uniquenessof a direct decomposition of a closed smooth manifold does not hold in general up to diﬀeomorphism. However, it holds for decompositions of (quasi)toric manifolds into products of real 2 or 4
dimensional (quasi)toric manifolds up to diﬀeomorphism, which I will talk about.

Rational maps whose Julia sets are Cantor circles
Fei Yang, Fudan University
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Abstract: We give a family of rational maps whose Julia sets are Cantor circles and show thatevery rational map whose Julia set is a Cantor set of circles must be topologically conjugate to
one element of this family on their corresponding Julia sets. In particular, we give the formulas ofsome rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to
any McMullen maps on their Julia sets. Moreover, a series of non-hyperbolic rational maps whose
Julia sets are Cantor circles are constructed.

On Kac’s scattering length formula
Jiangang Ying, Fudan University
Abstract: The scattering length formula was formulated and proved in special cases by Kac etal. It was discussed by a series of authors, including M. Taylor, H. Tamura, Y. Takahashi. The
formula was proved by M. Takeda in symmetric case and by P. He assuming weak duality. In thistalk, we shall describe the background of this formula and prove it in the general framework ofright Markov processes.

Renormalization in complex dynamics
Mitsuhiro Shishikura, Kyoto University
Abstract: One often observes “self-similarities” within the phase space and the parameter spaceof chaotic dynamical systems. While the self-similarity in the phase space is easily understood via
the dynamics itself, the self-similarity in the parameter space is explained by recurrent behavior of
orbits and small scale bifurcations reproducing the large scale bifurcations. The renormalization isa tool to study such phenomenon. The key is to consider the return map along recurrent orbit anddeﬁne it be a new dynamical system, and this construction is considered to be a meta-dynamics
in the space of certain dynamical systems. In one dimensional dynamics, the renormalization
has been the focus of the research for several decades in connection with the conjectures on thedensity of hyperbolicity and the local connectivity. In this talk, we discuss the general ideas ofrenormalization and recent development.

Water-waves and drift of particles
Hisashi Okamoto, RIMS
Abstract: We compute trajectories of ﬂuid particles in a water-wave which propagates with aconstant shape in a constant speed. The Stokes drift, which asserts that ﬂuid particles are pushedforward by a wave, is proved in a new method. Numerical examples with various gravity andsurface tension coeﬃcients are presented.

8Global solution for 2-d steady supersonic ﬂow over a bending Wall
Yongqian Zhang, Fudan University
Abstract:We are concerned with the steady supersonic ﬂow moving over a two dimensional bending wall. The bending wall is assumed to be a small perturbation of a convex one. Under theassumption that the total variation of the tangent of the perturbation is suﬃciently small, theglobal approximate solutions are constructed via wave front tracking scheme, and the convergenceof approximate solutions to the entropy solution is shown. Also the asymptotic behaviour of thesolution is studied.

The existence and blow-up criterion of liquid crystals system in critical Besov space
Yihang Hao, Fudan University
Abstract:We build the existence of strong solution to liquid crystals system both locally andglobally in critical Besov space. According to Serrin’s well-known criterion, we give a generalcriterion for liquid crystals on Besov type. Our criterion has slight diﬀerence, as in our system wewill confront the nonlinear term |∇d|2 d.
A Monge-Ampère foliation associated with a geodesic in the space of Kähler metrics
Kenta Tottori, Tohoku University
Abstract: A geodesic in the space of kähler metrics was deﬁned by Mabuchi, Semmes, Donaldson. In this talk, I want to explain the geodesic. It corresponds to a solution of Homogeneous
Complex Monge-Ampère equation and the solution deﬁnes the so called Monge-Ampère foliation.
In particular, the talk will focus on the foliation.

Sharp perturbation bounds for the total least squares problem
Pengpeng Xie, Fudan University
Abstract: In this talk, new perturbation analysis of the total least squares(TLS) is considered.
The results are much better than the earlier results. We also apply the statistical condition estimate(SCE) to TLS. In the numerical examples, we compare our perturbation bounds and theresulting forward errors with other error estimates given in the literature. As a byproduct, We
show that the condition number derived by Li and Jia (cf. [ Linear Algebra Appl., 435 (2011),
pp. 674–686]) is the same as that obtained by Baboulin and Gratton (cf. [ SIAM J. Matrix Anal.
Appl., 32 (2011), pp. 685–699]).

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Classiﬁcation of semisimple symmetric spaces with properly discontinuous actions ofsurface groups
Takayuki Okuda, University of Tokyo
Abstract: We consider symmetric spaces M := G/H with connected linear simple Lie groups
G. In this talk, we give a classiﬁcation of such symmetric spaces for which there exist properly
discontinuous actions of surface groups as isometries of M. Our method is based on the results ofT. Kobayashi [Math. Ann. ’89] and Y. Benoist [Ann. Math. ’96] together with combinatorialtechniques of nilpotent orbits.

p-essential normality of quasi-homogeneous Arveson’s submodules
Chong Zhao, Fudan University
Abstract: The ﬁrst author and Wang proved that each homogeneous principal submodule of theDrury-Arveson module Hn2 is essentially normal, and hence in dimensions n = 2, 3 each homogeneous submodule of Hn2 is essentially normal. For the Bergman modules L2a (Bn ) on the unit ball,
Douglas and Wang recently proved that every principal submodule is essentially normal.
We develop some new techniques to prove the essential normality of Drury-Arveson’s quasihomogeneous principal submodules, by a combination of the approach of Douglas and Wang. As
a consequence, we prove that each quasi-homogeneous submodule of Hn2 is essentially normal for
dimensions n = 2, 3, and determine the related K-homology.

Super extension of Bell polynomials with applications to superymmetric equations
Engui Fan, Fudan University
Abstract: Bell polynomials into super version, which are found to be eﬀective in systematicallyconstructing super bilinear representation, bilinear Backlund transformation, Lax pair and inﬁnite conservation laws of supersymmetric equations. We take supersymmetric KdV equation andsupersymmetric sine-Gordon equation to illustrate this procedure.

Quasi-homomorphisms
Koji Fujiwara, Kyoto University
Abstract: A function f on a group G is a quasi-homomorphism if there exists a constant C suchthat for any two group elements g and h, |f (gh) − f (g) − f (h)| is at most C. If C = 0, then f is ahomomorphism. Quasi-homomorphisms are useful to show some interesting properties of groups,
and I’d like to discuss that. This will be a survey talk.

10Complete intersections of two quadrics and Galois cohomology
Yasuhiro Ishitsuka, Kyoto University
Abstract: There are a classical relation between Jacobians of a hyperelliptic curve over C andlines on intersection of two quadrics over C. In this talk, we consider the relation over the rational
number ﬁeld Q, and some related topics.
A characterization of birationality of the fourth pluricanonical map
Jinsong Xu, Fudan University
Abstract: The pluricanonical systems is an important way of understanding the structure ofalgebraic varieties. Besides the classic work of E.Bombieri on surfaces, there are a lot of progressfor higher dimensional varieties in recent years. Generally, it is diﬃcult to describe the behaviorof the pluricanonical map φm when m small. In this talk, I will give a characterization of nonbirationality of 4-th pluricanonical map of an algebraic threefold of general type. We show that
under the assumption that volume(X) is large, the 4-th pluricanonical map is not birational if andonly if X is ﬁbred by surfaces of type (vol(S), pg (S)) = (1, 2).

A generalization of the theory of Coleman power series
Kazuto Ota, Tohoku University
Abstract: The classical Coleman power series theory says that every norm compatible system of
local units in p-power cyclotomic ﬁelds is interpolated by a power series. There are several generalization of this theory, which play important roles in Iwasawa theory. Recently, Shinichi Kobayashi(Tohoku Univ.) found a new generalization of Coleman theory to formal groups associated to elliptic curves, and he applied it to computation of p-adic height parings. In this talk, we generalize
his theory to higher dimensional formal groups.

Tricomi problem for a quasilinear Lavrentiev-Bitsadze equation of mixed type
Zhenguo Feng, Fudan University
Abstract: In this report, we consider the Tricomi problem of a quasilinear Lavrentiev-Bitsadze
mixed type equation∂ 2u ∂ 2u(sgn uy ) 2 + 2 − 1 = 0,∂x∂ywhose coeﬃcients depend on the ﬁrst order derivative of unknown function. We prove the existenceof solution to this problem by the using hodograph transformation, which can be applied to study
more diﬃcult problems for nonlinear mixed type equations arising in gas dynamics.

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Well-posedness of the ﬁfth order KdV equation with periodic boundary condition
Takamori Kato, Kyoto University
Abstract: In this talk, we consider the well-posedness of the Cauchy problem for the ﬁfth order
KdV equation in the periodic setting. This equation is completely integrable and is derived fromgeneralizing the KdV equation in some sense. The perturbation theory does not work. Here theperturbation means the nonlinear term is regarded as an approximate linear solution. To overcome
this diﬃculty, we focus on the rich symmetry of this equation. Actually we use the algebraic
structure of the nonlinear terms and the fundamental analysis tools so that the strong nonlinear
interactions are canceled and we can get the optimal result in some sense.

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