The East Asia Core Doctorial Forum on Mathematics, Kyoto, 2014

Abstracts
Talks by professors take place in Room 110.
For parallel sessions, talks written in the ﬁrst line take place in Room 110
and those written in the second line take place in Room 127.

Tuesday, 21 January
9:30–10:30
Jian Zhou (Tsinghua University)
Boson-fermion correspondence and Witten-Kontsevich tau-function
Keywords:
Abstract: Tau-functions and boson-fermion correspondence play important roles in Kyoto school’s approach
to integrable hierarchies. An amazing conjecture of Witten proved by Kontsevich states that the generating
function of intersection numbers on Deligne-Mumford moduli spaces is a tau-function of the KdV hierarchy.
In this talk we will explain some recent result on understanding the Witten-Kontsevich tau-function from thepoint of view of boson-fermion correspondence.

10:40–11:40
Takashi Sakajo (Kyoto University)
Mathematical theory of potential ﬂows in multiply connected domains
Keywords: Takashi Sakajo
Abstract: We deal with the incompressible and inviscid ﬂows in two-dimensional domains whose velocity ﬁeld,
say u(t, x) = (u(t, x, y), v(t, x, y)) at time t and at the position x = (x, y), is a solution of 2D Euler equations.
Suppose further that the vector ﬁeld is irrotational, namely it satisﬁes vx − uy = 0. Then the velocity ﬁeld
u(t, x) is represented by a holomorphic function F (z) = u−iv, called the complex velocity, in which the complex
z-plane is identiﬁed with the two-dimensional space as z ∼ x + iy. When the complex velocity has the primitive
W (z), i.e. W ′ (z) = F (z), its imaginary part ψ = Im[W (z)], which is known as the stream function, gives riseto the Hamiltonian of the velocity ﬁeld, i.e. u = ∂y ψ and v = −∂x ψ.
When these ﬂows are considered in multiply connected domains, there arise many interesting problems.
Mathematically, in the construction of the complex velocity in these domains, a transcendental function called
the Schottky-Klein prime function, which is an extension of the elliptic functions to an M -punctured disk,
plays an important role. In order to consider the multiply connected domains with arbitrary boundary shapes,
one needs to construct conformal mappings between multiply connected domains mathematically as well as
computationally. The other problems are a topological classiﬁcation of structurally stable Hamiltonian vectorﬁelds on multiply connected domains, and a description of all possible transitions between them. On the other
hand, from the application point of view, the potential ﬂow theory is available to construct mathematical models
that contribute to the theoretical understanding of bioﬂuids such as insect ﬂights and schooling of ﬁsh as wellas environmental ﬂows in rivers and coastal domains.
In the presentation, I will introduce this theory in multiply connected domains and show its recent researchdevelopments with many examples.

13:40–14:10
Hansol Hong (Seoul National University)
Localized mirror functors and HMS for P1a,b,c (HMS = Homological mirror symmetry)
Keywords: Lagrangian Floer theory, matrix factorizations, Homological mirror symmetry1Abstract: We will begin with a brief introduction to the homological mirror symmetry (HMS for short)conjecture. For a Kähler manifold X, HMS predicts that there is a Kähler manifold Y whose “complex geometry”
is equivalent to the “symplectic geometry” of X, and vice versa. Symplectic side of HMS is formulated by acertain (A∞ -)category called the Fukaya category of X, and for the Landau-Ginzburg mirror (Y, W : Y → C)
of X, the complex counterpart is given by the matrix factorization category of W .
I will next explain the natural construction of mirror functors between these two categories associated withimmersed Lagrangians L in X. The construction of the functor depends on the Floer theory of this immersed
Lagrangian, in particular, the holomorphic discs one of whose boundary condition is L.
We ﬁnally apply the construction to the orbifold projective lines P1a,b,c together with a speciﬁc Lagrangianin P1a,b,c which generates the Fukaya category. Here, P1a,b,c is an orbifold sphere which has three singular points
with Za , Zb and Zc singularities, respectively. Our goal is to prove HMS for P1a,b,c when a1 + 1b + 1c ≤ 1.

Xumin Gu (Fudan University)
Partial Regularity of Solutions to the Four-Dimensional Navier-Stokes Equations
Keywords: 4D Navier-Stokes equations, partial regularity, Campanato’s characterization
Abstract: In this talk, we consider suitable weak solutions of incompressible Navier-Stokes equations in four
spatial dimensions. We prove that the two-dimensional time-space Hausdorﬀ measure of the set of singularpoints is equal to zero.
We prove our results in three steps. In the ﬁrst step, we give some estimates of the scaling invariant
quantities. In the second step, we establish a weak decay estimate of certain scaling invariant quantities basedon the estimate we proved in the ﬁrst step by using an iteration method. In the last step, we improve the decay
estimate by a bootstrap argument, and apply parabolic regularity to get a good estimate of the L3/2 -mean
oscillations of u, which yields the Hölder continuity of u according to Campanato’s characterization of Hölder
continuous functions.

14:20–14:50
Maho Yokota (Kyoto University)
Noether problem for some easy groups
Keywords: Noether problem, rationality, Galois theory
Abstract: Let K be a ﬁeld and G a ﬁnite group acting on a rational function ﬁeld K(xg : g ∈ G) by xg ·h = xgh
for any g, h ∈ G. Noether problem asks whether the invariant ﬁeld K(G) := K(xg : g ∈ G)G is rational over K
or not. This problem is simple, however the answer depends greatly on the property of G and K, and known
results are far from general answer.
It is known that K(G) is rational over K if G has a faithful representation of at most degree three and K is
algebraic closed. I found if the order of G is odd, the assumption about faithful representation can be weakened.
Namely, we only need the following assumptions; K is algebraic closed and G has a faithful representation whichis the direct sum of representations of at most degree three. In this talk, I talk about introduction to Noether
problem at ﬁrst and about outline of the proof of my theorem.

Jaehoon Kang (Seoul National University)
Tangential limits for harmonic functions with respect to ϕ(∆): stable and beyond
Keywords: Bernstein function, subordinate Brownian motion, Poisson kernel, harmonic function, tangential
limits, Lp -Hölder space
Abstract: We study the boundary behavior of harmonic functions with respect to the non-local operator.
The classical Fatou theorem, which was showed by Fatou in 1906, states that if f ∈ Lp (Rd−1 ) for p ∈ [1, ∞],the Poisson integral of f , deﬁned on the upper half space Rd+ has a non-tangential limit for almost every
boundary point ξ ∈ Rd−1 . After the classical Fatou theorem, there have been a lot of studies on the boundary
behavior of harmonic functions such as, for general operator and general open set, Fatou theorem, relative Fatou
theorem (the existence of non-tangential boundary limit of ratio of two harmonic functions) and the existenceof tangential limits. In this talk, we discuss tangential limits for regular harmonic functions with respect to thenon-local operator ϕ(∆) := −ϕ(−∆) in the C 1,1 open set D in Rd , where ϕ is the complete Bernstein function
and d ≥ 2. For p ∈ (1, ∞] and β > 1/p, suppose that the exterior function f is local Lp -Hölder continuous oforder β on Dc . Then, we obtain that the regular harmonic function uf with respect to ϕ(∆), whose value is2

f on Dc , has a boundary limit for almost every ξ ∈ ∂D through a parabola that depends on ϕ and ϕ′ . Our
result includes the case ϕ(λ) = log(1 + λα/2 ). To prove our results, we use both the probabilistic and analytic
methods. In particular, the sharp Poisson kernel estimates are essential to our approach.

15:00–15:30
Tsung-Ju Lee (National Taiwan University)
Weil-Petersson metric on the moduli of Calabi-Yau varieties
Keywords: Weil-Petersson metric, Calabi-Yau, Mixed Hodge structures
Abstract: The moduli space of Calabi-Yau manifolds, with a ﬁxed polarization, exists due to a famous theoremof E. Viehweg. It’s also known that there is a canonical metric on it, which is called the Weil-Petersson metric.
It’s interesting to know whether the metric is complete or not. It can be shown that the ﬁnite distance property
somehow reﬂects the geometry on the central ﬁber.
In this talk, I will give the precise statement and review the proof of earlier results, due to Chin-Lung Wang,on the completion problem in the one parameter case. Then I will discuss some of my recent progresses on therelevant results which extend this theorem to some cases where the moduli spaces are of higher dimensions.

Jung-Hyun Bae (Sungkyunkwan University)
Eigenvalue results for nonlinear maximal monotone operators
Keywords: eigenvalue, maximal monotone operator, degree theory
Abstract: In this talk, we deal with some eigenvalue results for maximal monotone operators in reﬂexive Banach
spaces. A theory of maximal monotone operators has been developed by many researchers with applications
to operator equations and evolution equations. The study on nonlinear eigenvalue problems is mainly basedon degree theories for appropriate classes of nonlinear operators and a regularization method by means of the
duality operator.
Let X be a real reﬂexive Banach space with its dual X ∗ and let L be a dense subspace of X. Suppose that∗
T : D(T ) ⊂ X → 2X is a maximal monotone operator and C : D(C) ⊂ X → X ∗ is a densely deﬁned operator
with L ⊂ D(C). We consider an eigenvalue problem of the form
0 ∈ T x + λCx.
We present some eigenvalue results concerning operators of (S+ ) type, with the aid of a Kartsatos-Quarcoo
degree theory.

16:00–16:30
Yong Wei (Tsinghua University)
Sharp diameter estimates for compact manifold with boundary
Keywords: Diameter estimate, Manifold with boundary, Ricci curvature, Rigidity
Abstract: Let (N, g) be an n-dimensional complete Riemannian manifold. The classical Myers’ theorem saysthat if the Ricci curvature of (N, g) has a positive lower bound Ric ≥ n − 1, then the diameter of N is at mostπ. S.-Y. Cheng proved that if the diameter is equal to π, then N is isometric to the n-sphere Sn (1).
Recently, M. M. Li generalized the above theorems to manifold with boundary. He showed that if the Ricci
curvature of (N, g) is nonnegative and the mean curvature of the boundary ∂N satisﬁes H ≥ (n − 1)c0 > 0
for some constant c0 > 0, then the distance from any interior point x to the boundary ∂N has an uniform
upper bound 1/c0 . Moreover, if ∂N is compact, then N is also compact. If the distance from some point x
to the boundary achieves the maximum 1/c0 , then N is isometric to a ball of radius 1/c0 in an n-dimensional
Euclidean space Rn .
In the case that the manifold (N, g) has a negative lower bound Ric ≥ −(n − 1), and the mean curvatureof the boundary ∂N satisﬁes H ≥ (n − 1)c0 > (n − 1) for some constant c0 > 1, a standard argument can also
give the diameter boundsup d(x, ∂N ) ≤ coth−1 c0 .
(1)x∈N

If ∂N is compact, then (1) implies that N is also compact.
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In this talk, we show if the equality holds in (1), then N is isometric to a geodesic ball of radius coth−1 c0
in an n-dimensional hyperbolic space Hn (−1).

Yuri Yatagawa (The University of Tokyo)
Two invariants of ramiﬁcation
Keywords: ramiﬁcation, Swan conductor, Artin conductor, cleanliness
Abstract: In this talk, we study one diﬀerence between two invariants of ramiﬁcation, the reﬁned Swan
conductor and the reﬁned Artin conductor.
In number theory, these two conductors are deﬁned for the character of a representation of a Galois groupover a complete discrete valuation ﬁeld. We call this representation a Galois representation. A (complete)valuation ﬁeld is a generalization of an algebraic ﬁeld or an algebraic function ﬁeld over a ﬁnite ﬁeld. For an
algebraic ﬁeld (i.e. a ﬁnite extension of Q) and the integer ring (i.e. the integral closure of Z in the algebraic
ﬁeld), we can consider the problem “how prime ideals of Z decompose in the integer ring,” and we can also
consider for an algebraic function ﬁeld over a ﬁnite ﬁeld (i.e a ﬁnite extension of k(T ) where k is a ﬁnite ﬁeld)
and the integral closure of k[T ] in the algebraic function ﬁeld. Ramiﬁcation means this decomposition in number
theory.
In arithmetic geometry, we can treat discrete valuation ﬁelds with smooth varieties and divisors of them.
And in this case, a character of a Galois representation corresponds to a smooth sheaf over the complement of
a divisor. For this sheaf, we can generalize the two conductors above, and these are very important invariants
related to a formula of the Euler characteristic of the sheaf. In this talk, we consider a surface, and see one
diﬀerence between these two conductors with respect to the resolution of bad points.

16:40–17:10
Zhi-You Chen (National Central University)
On the Uniqueness and Structure of Solutions to the System Arising from Chern-Simons Models
Keywords: Uniqueness, Chern-Simons system
Abstract: Firstly, we review some well-known results about the Chern-Simons equation with single Higgs
particle:
N1∑
1∆u + 2 eu (1 − eu ) = 4πµi δpi in R2 .
εi=1Secondly, to the Chern-Simons equation with two Higgs particles:

N1
∑

1 v
u

µi δpi in R2 , ∆u + ε2 e (1 − e ) = 4π
i=1

N2
∑


νj δqj in R2 .
 ∆v + ε12 eu (1 − ev ) = 4π
j=1

we prove the uniqueness of topological multivortex solutions(i.e, (u(x), v(x)) → (0, 0) as |x| → ∞) if either the
Chern-Simons coupling parameter is suﬃciently small ε > 0 or suﬃciently large ε > 0. In addition, the sharp
range of ﬂux for non-topological radial solutions and structure of all entire solutions are also derived.

Hirokazu Shinjo (Tohoku University)
Coeﬃcients of L-functions of certain type of curves
Keywords: L-functions, Jacobian variety, GL2 -type, Formal group, Honda theory
Abstract: For algebraic curve C over Q, the L-function of C is important and interesting object in arithmetic
geometry. For example, let p be a good prime of C, counting the numbers of rational points on reduction modulo
p of C is equivalent to determine the coeﬃcients of polynomial Lp , the p-component of L. When C is elliptic
curve and p is good prime, Manin showed the congruency between the coeﬃcient of Lp and the Hasse-Wittmatrix. Honda showed this more globally by using the strong isomorphism of two formal groups, one is a formal
minimal model of C over Z, the other is attached to L. I will talk about the generalization of this congruency
to curves whose Jacobian variety is GL2 -type, and some example of calculating the coeﬃcients of Lp .

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17:20–17:50
Kotaro Kawai (Tohoku University)
Deformations of associative submanifolds in nearly parallel G2 -manifolds
Keywords: Associative submanifolds, G2 geometry
Abstract: A Riemannian 7-manifold (Y, g) is called a nearly parallel G2 -manifold if the holonomy group of
its cone is contained in Spin(7). An associative submanifold M in Y is deﬁned to be a minimal 3-submanifold
which is related to the nearly parallel G2 -structure of Y .
A typical example of nearly parallel G2 -manifolds is a 7-sphere S 7 . In this talk, we assume that Y = S 7
for simplicity, and study submanifolds in S 7 . There are various classes of associative submanifolds arising fromother geometries. Special Legendrian 3-submanifolds in S 7 , arising from the Calabi-Yau geometry on R8 , areassociative, and Lagrangian submanifolds in a totally geodesic S 6 , arising from the G2 geometry on R7 , are
immersed into S 7 as associative submanifolds.
We study the inﬁnitesimal deformations of associative submanifolds, and compare deformation spaces inthese classes. Then we study the homogeneous associative submanifolds in S 7 explicitly.

Seongan Lim (Ewha Woman’s University)
On the Homomorphic Encryption NTRU
Keywords: Cryptography, public key encryption, homomorphic encryption, NTRU
Abstract: An homomorphic encryption scheme allows computing over ciphertexts and it is expected as avery useful cryptographic primitive for secure cloud services. Among many known homomorphic encryption
schemes, the homomorphic version of NTRU is the only one which manipulates ciphertexts for diﬀerent users
simultaneously. In this talk, we review the history of several published NTRU-variants and present an algebraic
analysis of the parameters for the homomorphic NTRU.

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Wednesday, 22 January
9:30–10:30
Jungkai Chen (National Taiwan University)
Geography of varieties of general type
Keywords:
Abstract: In classiﬁcation theory of algebraic varieties, we are interesting in the distribution of birational
invariants of varieties, such as genus, volume and Euler characteristics. A priori, these invariants could bearbitrarily large. However, they satisﬁes famous Miyaoka-Yau Inequality and Noether Inequality in dimension
2. In this talk, we will explore the three dimensional analogue of these inequalities.

10:40–11:40
Quanshui Wu (Fudan University)
Poisson (co)homology and duality
Keywords: Poisson algbera; Poisson (co)homology; Poicare duality
Abstract: Poisson algebras play an important role in Poisson geometry, non-commutative geometry and mathematical physics. Poisson homology and Poisson cohomology are important invariants of Poisson structures.
The set of Casimir elements of the Poisson structure is the 0-th cohomology; Poisson derivations modulo Hamiltonian derivations is the 1-st cohomology. In the talk, I will start from the basic deﬁnitions and examples, suchas Poisson algebras, poisson modules, Poisson homology and cohomology. Then I will talk about a (twisted)
duality theory between Poisson homology and Poisson cohomology, and prove a version of the twisted Poincare
duality between the Poisson homology and cohomology for any polynomial Poisson algebras with values in anarbitrary Poisson module, which generalizes the results in literature. In the case that the Poisson structure is
unimodular, the twisted Poincare duality reduces to the Poincare duality in the usual sense.

13:40–14:10
Koichi Shimada (The University of Tokyo)
Group Actions on von Neumann Algebras
Keywords: von Neumann algebra, group action
Abstract: In this talk, we will explain a classiﬁcation of group actions on von Neumann algebras. We consider
algebras consisting of linear operators on a Hilbert space. A von Neumann algebra is such an algebra which
is closed under a suitable topology. For example, the matrix algebra Mn (C), acting on the Hilbert space Cn
is a von Neumann algebra. Another example is essentially bounded functions L∞ (R), acting on the Hilbert
space L2 (R) by multiplication. Although these examples are not so interesting, by doing kinds of manipulations
on these examples, many examples of von Neumann algebras are obtained. We are interested in studying
properties of these von Neumann algebras. One of the methods to understand these von Neumann algebras is
to study their symmetry, that is, to study group actions on von Neumann algebras. The ﬁrst breakthrough inthis area has been done by A. Connes in 1970’s. He has classiﬁed the integer group actions on a von Neumann
algebra constructed by taking an inductive limit of matrix algebras. In this talk, we explain some progress ofclassiﬁcation of group actions on von Neumann algebras after Connes. In particular, the real number group
actions. Contrary to the integer group, the real number group is not discrete. We will explain the diﬃcult
point of the classiﬁcation of R-actions on von Neumann algebras and partial answers for this by many people,
including Kawahigashi, Kishimoto, Kawamuro, Masuda–Tomatsu and us.

Kameng Nip (Tsinghua University)
Combinations of Shop Scheduling and the Shortest Path Problem
Keywords: Approximation algorithm, combination of optimization problems, ﬂow shop, open shop, job shop,
scheduling, shortest path
Abstract: We consider several novel combinatorial optimization problems, which combine the classic shop
scheduling problems (namely, ﬂow shop scheduling, open shop scheduling or job shop scheduling) and the6shortest path problem. The objective of the obtained problems is to select a subset of jobs that forms a feasible
solution of the shortest path problem, and to execute the selected jobs on the shop (ﬂow shop, open shop or jobshop) machines to minimize the makespan. We show that these problems are NP-hard even if the number ofmachines is two, and cannot be approximated within a factor less than 2 if the number of machines is an input
unless P = NP. We design several approximation algorithms for these combination problems.

14:20–14:50
Ryunosuke Ozawa (Tohoku University)
Limit formulas for metric measure invariants and phase transition property(A joint work with Takashi Shioya (Tohoku University))
Keywords: Metric measure space, concentration, observable distance, separation distance, pyramid, phasetransition
Abstract: We consider Gromov’s compactiﬁcation of the set of metric measure spaces. An element of thecompactiﬁcation is called pyramid. In this talk, we generalize the observable diameter and the separation
distance for metric measure spaces to those for pyramids, and study the limit of these invariants for a convergent
sequence of pyramids. As an application, we obtain a criterion of the phase transition property for a sequenceof metric measure spaces or pyramids, and ﬁnd some examples of symmetric spaces of noncompact type withthe phase transition property. This talk is based on a joint work with Takashi Shioya (Tohoku University).

Sheng-Hua Chen (National Taiwan University)
Edge Roman Domination on Graphs(A joint work with Chun-Hung Liu and Gerard Jennhwa Chang)
Keywords: Graph theory, domination theory, discharging method
Abstract: Let G = (V, E) be a simple graph. An edge Roman dominating function on G is a function
f : E(G) → {0, 1, 2} satisfying that every edge e1 for which f (e1 ) = 0 is adjacent∑ to at least one edge e2 forwhich f (e2 ) = 2. The weight of a Roman dominating function is the value f (E) = e∈E(G) f (e). The minimum′
weight γR(G) of a edge Roman dominating function on a graph G is called the edge Roman domination numberof G. The concept of the edge Roman domination was ﬁrst introduced in [2].
In this talk, we investigate some upper bounds using discharging method and other skills.
[1] S. Akbari, S. Qajar, On the edge roman domination number of planar graphs, manuscript.
[2] P. Roushini Leely Pushpam, T. N. M. Nalini Mai, Edge Roman domination in graphs, J. Combin. Math.
Combin. Comput. 69 (2009), 175–182.

15:00–15:30
Jiyoung Han (Seoul National University)
Distribution of integral lattice points in an ellipsoid with a diophantine center
Keywords: Equidistribution, lattice point counting and Heisenberg group
Abstract: Using representation theory and symplectic geometry combined with equidistribution theorem on
the Jacobi
Sp(n, R) ⋉ R2n }, we evaluate the mean square limit of exponential sums related with a rational{ group2
ellipsoid a1 x1 + · · · + an x2n = 1 , extending a work of Marklof. The result helps to study the asymptotic valuesof the normalized evaluations of the number of lattice points inside a rational ellipsoid and inside a rational
thin ellipsoid.

Myeongmin Kang (Seoul National University)
Inexact accelerated augmented Lagrangian method for linearly constrained ℓ1 -ℓ2 minimization
Keywords: Augmented Lagrangian method, Acceleration, Compressive sensing, inexact solution
Abstract: We consider the linearly constrained ℓ1 -ℓ2 minimization:min ∥x∥1 +
x

β
∥x∥22
2

subject to
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Ax = b.

In compressive sensing, a linearly constrained ℓ1 minimization problem, i.e., the problem (1) with β = 0,
where an underdetermined system of equations, has been used. Adding the ∥ · ∥22 in the basis pursuit problem
yields the tractable objective function, which is a strictly convex function. Thus, the linearly constrainedℓ1 -ℓ2 minimization problem has a unique solution and the dual problem is smooth. We apply augmented
Lagrangian method to the linearly constrained ℓ1 -ℓ2 minimization. But, the augmented Lagrangian method
has the subproblem and does not have the closed form solution. In this talk, we propose an inexact accelerated
augmented Lagrangian method for solving this minimization problem. In our algorithm, we give the stopping
conditions of subproblem. The proposed method is based on the extrapolation technique, which is used inaccelerated proximal gradient methods proposed by Nesterov. A convergence rate of O( k12 ) is proved for the
proposed method when it is applied to solve a more general linearly constrained nonsmooth convex minimization
problem. We numerically test our proposed method on a synthetic problem from compressive sensing. The
numerical results conﬁrm that our inexact accelerated augmented Lagrangian method is faster than state-ofarts algorithms.

16:00–16:30
Li Ming (Fudan University)
The interior estimate for convex solutions of k-Hessian equations and a rigidity theorem
Keywords: interior estimate, k-Hessian equations, Dirichlet problem
Abstract: The interior C 2 estimates for Monge-Ampére equations were studied ﬁrstly by A.V. Pogorelov.
Then K.S Chou and X-J. Wang extended Pogorelov s estimates to the case of k-Hessian equations. Explicitly,in their paper, for any function f not depending Du, for any small positive constant ϵ, the following estimateshold,(−u)1+ϵ ∆u ≤ C.
Here, constant C depends on the domain Ω, k, the function f , and supΩ |∇u|. We concentrate on the interior
C 2 estimates for the following Dirichlet problem for k-Hessian equations,{σk [D2 u] = f (x, u, Du),
(2)
u|∂Ω
=
ϕ.
Here, Ω ⊂ Rn is a bounded domain, u is a smooth function, Du and D2 u are its gradient and Hessian respectively.
The k-Hessian equation: σk {D2 u} = f in Ω, where 1 ≤ k ≤ n,
σk {D2 u} = σk {λ}λ = (λ1 , · · · , λn ) are the eigenvalues of the Hessian matrix (D2 u), and∑σk {λ} =λi1 · · · λik
i1 <i2 <···<ik

is the k-th elementary symmetric polynomial. The k-Hessian equation includes the Poisson equation −∆u = f ,(k= 1) and the Monge-Ampére equation det(D2 u) = f ,(k = n) as special examples. Firstly, I will talk aboutk = 2, i.e. for 2-Hessian equations, there is some constant β > 0, such that(−u)β ∆u ≤ C.
Here positive constants β and C depend on the domain Ω, the function f , supΩ |u| and supΩ |∇u|. Then, forthe general k, k-Hessian equations, if the solutions are k + 1 convex, namely, function u in k+1 convex cone,
we have,(−u)∆u ≤ C.
Here, positive constant C depends on the domain Ω, the function f , supΩ |u| and supΩ |∇u|. At last, as anapplication of interior estimates may to prove rigidity theorem for k-Hessian equations: the entire solutions in
k + 1 convex cone of the equations deﬁned in Rn with quadratic growth are quadratic polynomials, where k + 1
convex cone is Γk+1 = {u ∈ C ∞ (Ω)|σi > 0, 1 ≤ i ≤ k + 1}

Yasuhiro Ishituka (Kyoto University)
Some geometric aspects of 2-descent of hyperelliptic Jacobians

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Keywords: 2-descent, hyperelliptic curve, Jacobian variety
Abstract: Let us consider an abelian variety A over Q, e.g. an elliptic curve over Q, or the Jacobian variety
of a smooth hyperelliptic curve over Q. The rational points of A gives a ﬁnitely generated abelian group, called
the Mordell-Weil group of A. This group has been important and mysterious objects in arithmetic geometry.
We concentrate on the following question: how can we compute the rank of Mordell-Weil group explicitly? A
method called descent method gives a partial solution to this question. In this talk, we ﬁrstly describe the2-descent method for the Jacobian variety JC of a smooth hyperelliptic curve C over Q. In the process, we willobtain varieties possessing some arithmetic information of JC . We discuss the property of the varieties, or theirvariants.

16:40–17:10
Chong Tian (Fudan University)
Unstability of pseudoharmonic maps between pseudo-Hermitian manifolds
Keywords: Pseudo-Hermitian manifold, Weingarten map, unstable pseudoharmonic map
Abstract: As known to all, a harmonic map is a critical point of the energy integral. A harmonic map iscalled stable if it has nonnegative second variation, that is, the index of the map is 0. The stability problem
is an important problem in the theory of harmonic maps. R.T. Smith estimated the index of the identity mapof a Riemannian manifold, in particular, he showed that the identity map on S m is m + 1. Y.L. Xin provedthat for m ≥ 3, any nonconstant harmonic map f : S m → N n is unstable. A result of Leung states that any
nonconstant map from a compact Riemannian manifold to the sphere is unstable too. R. Howard and S. W.
Wei extended the Leung’s result to the case that the target manifold is a compact immersed submanifold of
Euclidean space.
In recent yeas, some pseudoharmonic maps were introduced in the ﬁeld of pseudo-Hermitian geometry. Petit
introduced a new horizontal energy functional. He derived the ﬁrst variation formula and called a critical pointof the energy a pseudoharmonic map. Note that there is an extra condition on the pull-back of the torsion on thetarget manifold. Dong modiﬁed Petit’s variational problem slightly by restricting the variational vector ﬁeld tobe horizontal. The critical point of the restricted variational problem about is referred to as a pseudoharmonic
too. Among other results, Dong derived the second variation formula of pseudoharmonic maps into Sasakian
manifold and proved that any nonconstant horizontal pseudoharmonic map from a closed pseudo-Hermitian
manifold into the odd dimensional sphere is unstable.
In today’s talk, I want to introduce some work with my advisor Professor Dong and Y. B. Ren. We extend theabove results to the case that the target manifold is an isometric embedded CR manifold or a pseudo-Hermitian
immersed submanifold of Heisenberg group and give a low bound of the index of identity map I : S 2n+1 → S 2n+1
with n ≥ 1. Firstly, we derive the second variation formula of pseudoharmonic maps into any pseudo-Hermitian
manifolds. Then we give a condition on Weingarten map which implies that there is no nonconstant horizontal
pseudoharmonic map from a closed pseudo-Hermitian manifold into an isometrically embedded CR manifold.
Next we consider the identity map I : S 2n+1 → S 2n+1 . From the above result we known that it is unstable.
Following the result of R.T.Smith we discuss the degree of the unstability and derive that Index(I) ≥ 2n + 2. In
the end, we give a condition on CR Weingarten map which also implies there is no nonconstant pseudoharmonic
map from a closed pseudo-Hermitian manifold into a pseudo-Hermitian immersed submanifold of Heisenberg
group.

Kazuki Sato (Tohoku University)
The solubility of diagonal cubic equations
Keywords: Rational point, cubic surface, Selmer group
Abstract: Let V be a smooth cubic surface a1 X13 + a2 X23 + a3 X33 + a4 X43 = 0 over Q, where the ai are nonzero
rational integers. Assume that V is everywhere locally soluble, i.e. V has a Qp -rational point for all prime
numbers p.
The Hasse principle does not hold for cubic surfaces in general, so we are concerned with the solubility over
Q of the surface V . Conjecturally, the existence of rational points is controlled by the Brauer group (somegeometric object) of V . Subject to the assumption that the Tate-Shafarevich group of every elliptic curve is
ﬁnite, Swinnerton-Dyer showed that under a certain local condition on the coeﬃcients ai , the surface V has apoint over Q.
In this talk, we give another condition on the coeﬃcients for the solubility of V , assuming the ﬁniteness of
the Tate-Shafarevich group. In order to ﬁnd a rational point, ﬁrst of all we reduce V to two cubic curves. And
then, we calculate the relevant Selmer groups of the Jacobians of two curves.
9

17:20–17:50
Jungchan Lee (Sungkyunkwan University)
Some variational problem on almost Hermitian structures(Joint work with JeongHyeong Park (Sungkyunkwan University)and Kouei Sekigawa (Niigata University))
Keywords: Critical almost Hermitian structure, Einstein-Hilbert functional, ﬁrst Chern number
Abstract: Let M be an even dimensional compact smooth manifold admitting an almost∫ complex structure.
Let (λ, µ) ∈ R2 \ (0, 0). We discuss the critical points of the functional Fλ,µ (J, g) = M (λτ + µτ ∗ )dvg on
the space of all almost Hermitian structures AH(M ) on M and its subspace AHc (M ) with a certain positive
constant c, where τ and τ ∗ are the scalar curvature and the ∗-scalar curvature of (J, g), respectively. We shallgive several characterizations of Kähler structure for some special classes of almost Hermitian manifolds, interms of the critical points of the functionals Fλ,µ (J, g) on AH(M ). Further, we provide the almost Hermitian
analogy of the Hilbert’s result on AHc (M ). We establish an integral formula for the ﬁrst Chern number of acompact almost Hermitian surface. As applications of above variational problem, we derive curvature identities
from the integral formula based on the fundamental fact that the ﬁrst Chern number is a topological invariant.

Chiu-Ju Lin (National Tsing Hua University)
Competition of phytoplankton species for light with wavelength
Keywords: Competition for light, light spectrum, competitive exclusion, coexistence
Abstract: We study the competition of phytoplankton species for light with wavelength in a well mixed watercolumn. Stomp et al. 2007 presented a reasonable model about phytoplankton species competition for light andeach species prefers diﬀerent interval of light spectrum, hence the speciﬁc absorption spectrum of each species
is diﬀerent. Their experimental results show that colorful phytoplankton species coexist.
We ﬁrst consider the case that all species have the same speciﬁc absorption spectrum and diﬀerent growth
function and death rate. In this case, we conclude that the superior species competitively excludes others.
Secondly, we consider two identical species competition model with diﬀerent speciﬁc absorption spectrum. For
simplicity, we assume that the growth function is linear. The conclusion is that either one species competitively
excludes the other, or two species coexist. Bistability of two species is impossible.

18:00–18:30
Chan Yong Kim (Sungkyunkwan University)
Bertrand curves in R3 and S 3 (1)Keywords: Spherical curve, Bertrand curve, spherical Bertrand curve
Abstract: A Frenet curve C is called a Bertrand curve if there exists another Frenet curve C̄, distinct fromC, and a bijection f between C and C̄ such that the same principal normal lines of C and C̄ at corresponding
points coincide. Here, C̄ is called a Bertrand mate of C.
It is well-known that a Frenet curve C in R3 is a Bertrand curve if and only if there exists a linear relationaκ + bτ = 1, where κ is a curvature function of C and τ is a torsion function of C when a and b are non-zeroconstant real numbers.
In this talk, We show many examples of curves on the unit 2-sphere S 2 (1) in R3 and the unit 3-sphere3
S (1) in R4 . We study whether its curves are Bertrand curves or spherical Bertrand curves and provide someexamples illustrating the resultant curves.

Liping Li (Fudan University)
Regular subspaces of Dirichlet forms
Keywords: Markov processes, regular Dirichlet form, subspaces, independent coupling
Abstract: Let E be a locally compact separable Hausdorﬀ space, m be a σ-ﬁnite Radon measure with supp m =E on E. Then L2 (E, m) is a Hilbert space. A non-negative deﬁnite symmetric bilinear form E densely deﬁned on
L2 (E, m) is called a Dirichlet form if it is closed and Markovian, F is the domain of E. We always signed (F, E)
for a Dirichlet form, resp. (Tt )t>0 , (Gα )α>0 for its semigroup and resolvent. E1 (u, v) ≜ E(u, v) + (u, v)m , ∀u, v ∈10

F. We then denote C(E) the space of all real continuous functions on E and its subspace Cc (E) (resp.
Cb (E), Cc1 (E)) with compact support (resp. bounded continuous functions, one order continuous diﬀerentiable
functions). A Dirichlet form (F, E) is called regular if F ∩ Cc (E) is dense in F with E1 -norm and dense in
Cc (E) with uniform norm. The regular subspace (E ′ , F ′ ) of a Dirichlet form (E, F) is the regular Dirichletforms that inherit the original form but possess smaller function spaces, i.e. F ′ ⊂ F and E ′ (u, v) = E(u, v) for
any u, v ∈ F ′ . The two problems we are concerned are: (1) the existence of regular subspaces of some given
regular Dirichlet form; (2) the characterization of the regular subspaces of the given Dirichlet form if exists.
In this talk, one of our main results is that regular subspaces must inherit the jumping and killing measuresof the original one. As it is well known that every regular Dirichlet form
type decomposition∫ has a Beurling-Deny
1
(c)
2
which
is
composedbyastrongly
local
part
E
,
the
non-local
part(u(x)−u(y))
J(dxdy)and the killing part2∫
u(x)2 k(dx) where J and k are called the jumping and killing measures of the Dirichlet form. We will show that
if F ′ is a regular subspace of Dirichlet form (E, F), then their jumping and killing measures in Beurling-Denydecomposition must be the same.
We also proved that an arbitrary ﬁnite dimensional Lévy type Dirichlet form with present strong local
part always have proper regular subspaces. This is an extended result of one dimensional Brownian motion
or diﬀusions by Fang, Fukushima and Ying in 2005. In particular, the regular subspaces of multidimensional
Brownian motion can be generated by the coupling of one-dimensional case. In use of the killing, time change,and spatial transformations, we can also extend similar results to planar Brownian motions on any domains.
We also proved that the Dirichlet forms of arbitrary symmetric ﬁnite-dimensional uniform ellipticity diﬀusion
always possess proper regular subspaces.

11Thursday, 23 January
9:00–9:30
Yong Hu (Fudan University)
Inequality of Noether type for smooth minimal 3-folds of general type
Keywords: Noether inequality, 3-folds
Abstract: In algebraic geometry, it is very interesting to ﬁnd the relationships between the birational invariantsof algebraic varieties. Given a nonsingular projective variety X, denote by KX the canonical divisor. Let C
be a smooth projective curve of genus g, then we have the equality deg(KC ) = 2(g − 1). Let S be a smoothminimal projective surface of general type, then we have the classical Noether inequality KS2 ≥ 2χ(OS ) − 6.
Since Noether inequality plays a very important role in the surface theory, especially in studying the geographyof algebraic surfaces, we want to ﬁnd the inequality of Noether type in high dimension. Let X be a smoothminimal projective 3-fold of general type. We give a sharp inequality of Noether type between the canonical3and χ(ωX ):volume KX
43KX≥ χ(ωX ) − 2.
3

Kota Uriya (Tohoku University)
Asymptotic behavior of a solution to a system of quadratic nonlinear Schrödinger equations in two dimensions
Keywords: Nonlinear Schrödinger equation, Scattering theory
Abstract: We are concerned with the asymptotic behavior in time of a solution to a system of quadratic
nonlinear Schrödinger equations in two dimensions. We say that the solution of nonlinear Schrödinger equation
is asymptotically free when the solution behaves a solution of the free Schrödinger equation as time tends toinﬁnity. In general, quadratic nonlinearities are critical in two dimensions, that is, it is not clear whether there
exists an asymptotically free solution or not. It is known that the parameters which stand for the mass of theparticles inﬂuence to the behavior of the solutions. Under the special relation for that parameter, we show theexistence of the solution to the system which is neither asymptotically free nor modiﬁed free solution.

9:40–10:10
Jian Xiao (Fudan University)
Weak transcendental holomorphic Morse inequalities on compact Kähler manifolds
Keywords: Holomorphic Morse inequalities, Kähler current, big cohomology class
Abstract: Let X be an n-dimensional compact complex manifold and let α, β be two cohomology classes in thenef cone of X. For n-dimensional projective varieties, a k-cycle is called nef if its intersection number with any
eﬀective (n − k)-cycle is non-negative, and it is called big if it lies in the interior of cone generated by eﬀective
k-cycles. We also have similar nefness or bigness deﬁned by metric over general compact complex manifolds.
If X is a smooth complex algebraic variety and α, β are Chern classes of line bundles, a basic theorem of
Siu tells us if αn − nαn−1 β > 0 then α − β is big. Similar to Siu’s theorem, X.Y.Yuan has also proved the same
big criterion for arithmetic varieties.
In my talk, I will introduce Demailly’s remarkable holomorphic Morse inequalities. We shall see Siu’s result
can be derived from holomorphic Morse inequalities. But rationality is important in the previous results. For
general k, I will give a big criterion for the class αk − β k on compact Kähler manifolds without projective or
rational conditions. Our results partially improve the recent results of Boucksom-Demailly-Paun-Peternell.

Xu Qing (Fudan University)
Backward Stochastic Schrödinger Equations and related topics
Keywords: Backward stochastic diﬀerential equation, schrödinger equation
Abstract: In this speech, a class of backward stochastic diﬀerential equations of schrödinger type is studied.
First, we will introduce the ﬁeld of Backward Stochastic Diﬀerential Equations (BSDE, for short) and give thedeﬁnition of the adapted solution of the Backward Stochastic Schrödinger Equations. In the linear case, both12

of the additive and multiplicative noises are considered. We derive the global existence and uniqueness of theadapted solution via the Galerkin method. We also consider some nonlinear cases. In these cases, the local
existence and uniqueness of the solution is derived via the Strichartz estimate of the schrödinger operator. We
will also present some other results, e.g., how the solution behaves when the time approaches the terminal.

10:20–10:50
Yen-Wen Fan (National Taiwan University)
An Optimal Gap Theorem in a Complete Strictly Pseudoconvex CR (2n + 1)-Manifold
Keywords: Gap Theorem, Linear Trace Harnack, CR manifold
Abstract: In the Kaehler geometry, it is conjectured that a complete noncompact Kaehler manifold M of
positive holomorphic bisectional curvature of complex dimension m is biholomorphic to Cm . The ﬁrst result
concerning this conjecture was obtained by Mok-Siu-Yau. They proved that M is isometrically biholomorphic
to Cm with the standard ﬂat metric under assumptions of maximum volume growth condition V (r) ≥ δrm andthe scalar curvature R decays as R = o(r−2−ϵ ). We call this theorem as a gap theorem. (The “gap” indicates“how big the curvature can be” between ﬂat and not ﬂat manifold and try to ﬁnd an constraint for curvature sothat the manifold is ﬂat.) For a long time, it has been conjectured the maximum volume growth condition was
not necessary. In 2012, L. Ni obtained an optimal gap theorem on M with nonnegative bisectional curvature
without the maximum volume growth condition, provided the average of scalar curvature decay as o(r−2 ). The
same question can be asked in the CR manifold category. Now let M be a (2n + 1)-manifold with a contact
structure and a complex structure J on the contact plane. One can view it as a boundary of a convex domain
in Cm . Under the same curvature assumption, we are able to generalize the optimal gap theorem to the CR
manifold when torsion is vanishing. The classiﬁcation of CR geometry is rare and most result is compact. This
result is the ﬁrst result about the complete manifold with CR structure. In this talk I will give a review of
CR geometry and history. And then I will sketch the idea of the proof. This is an application of linear trace
Li-Yau-Hamilton inequality of Lichnerowicz-Laplacian heat equation.

Makoto Fujiwara (Tohoku University)
Uniform and constructive provability in reverse mathematics
Keywords: Reverse mathematics, Uniformity of proofs, Constructive mathematics, Metatheorem
Abstract: Reverse mathematics is a research program in foundation of mathematics, which was initiated by
H. Friedman in the 1970’s and extensively developed by S. Simpson and others. In reverse mathematics, weseek the necessary set existence axiom for proving each theorem in ordinary mathematics. In this way, mostof ordinary theorems are classiﬁed into three kinds. However, it is known that the classiﬁcation will slightly
change if we stand on the position that a proof of theorem “For any X satisfying some conditions, there exists
Y satisfying some conditions” is required to give the uniform construction of Y from X. We say that a proof
is uniform if such a construction is given.
Recently, some interesting metatheorems, which are useful to show the constructive unprovability of theorems
by using the uniform consideration in Friedman-Simpson reverse mathematics, have been established. Thiskind of metatheorems can be a bridge between the research of reverse mathematics and that of constructive
mathematics. In this talk, I will present an overview of such metatheorems and their application to actual
mathematical theorems in some examples.

11:00–11:30
Masatoshi Kitagawa (The University of Tokyo)
Stable branching laws for spherical varieties
Keywords: Lie group, spherical variety, multiplicity-free representation
Abstract: Let G be a connected reductive algebraic group over C (e.g. GL(n, C), Sp(n, C), . . . ), and X bea quasi-aﬃne G-variety over C. We consider⊕ a (C[X], G)-module V , and decompose V into the direct sum ofirreducible representations of G as V = λ m(λ)πλ . Here, C[X] is the ring of regular functions on X. Our
main concern is to compute the multiplicities m(λ). X is said to be spherical if C[X] is multiplicity-free Gmodule (i.e. m(λ) ≤ 1 for V = C[X]). F. Sato gave a method to describe m(λ) for ‘suﬃciently large’ λ, when13

X is a aﬃne spherical homogeneous space and V is the space of global sections of a G-equivariant vector bundle
on X.
In this talk, we generalize Sato’s theorem for any quasi-aﬃne spherical variety X and (C[X], G)-module V .
As an application, we will treat branching laws of inﬁnite dimensional unitary representations of Lie groups,
called holomorphic discrete series representations.

Jinyeong Park (Seoul National University)
Practical synchronization of Kuramoto system with an intrinsic dynamics
Keywords: Kuramoto model, intrinsic dynamics, external force, practical synchronization
Abstract: We study the practical synchronization of the Kuramoto dynamics of processors distributed over
networks. The processor dynamics on the nodes of the network are governed by the interplay between theirown intrinsic dynamics and Kuramoto coupling dynamics. We present two suﬃcient conditions for practicalsynchronization under homogeneous and heterogeneous forcing. For practical synchronization estimates, weemploy the conﬁguration diameter as a Lyapunov functional, and derive a Gronwall-type diﬀerential inequality
for this value.

11:40–12:10
Yuichiro Tanaka (The University of Tokyo)
Visible actions on generalized ﬂag varieties and a generalization of the Cartan decomposition
Keywords: multiplicity-free representation; compact Lie group; ﬂag variety; visible action; Cartan decomposition
Abstract: I would like to talk about normalization of matrices motivated by the theory of visible actions oncomplex manifolds, which was introduce by T. Kobayashi.
Let H be a Lie group and X a complex manifold with a holomorphic H-action. We say the action H ↷ X
is strongly visible if• there is a real submanifold S of X such that X ′ := H · S is an open subset of X, and• there is an anti holomorphic diﬀeomorphism σ of X ′ such that σ(H · x) ⊂ H · x for x ∈ X ′ and σ|S = idS .
If X is a homogeneous space G/L of a matrix group G, the visibility of the H-action on G/L implies that any
element of G can be normalized to certain form corresponding to the totally real submanifold S by using theright L-action and the left H-action on G.
In this talk, we consider normalization of elements of compact Lie groups (the unitary group, the orthogonalgroup, the symplectic group, etc) under the left and right actions of Levi subgroups (block diagonal matrix
subgroups).

Eun-Kyung Kim (Ewha Woman’s University)
Relations between Multilinear Maps and Fully Homomorphic Encryption Scheme
Keywords: Ideal lattice, Multilinear map, Fully homomorphic encryption
Abstract: Bilinear maps are one of the most important and useful tools in cryptography including one-round
3-way secret key exchange protocol which was introduced by Joux in 2000. Multipartite key exchange protocol is
one of the main interest of many researchers, so from the tripartite key exchange protocol construction of Joux,
they expect that κ-multilinear maps may allow to have one-round (κ + 1)-way secret key exchange protocol.
However, extending bilinear maps to multilinear had been an important open problem.
Another important and useful tools in cryptography is a fully homomorphic encryption. Like multilinear
maps, it also has many applications. The ﬁrst fully homomorphic encryption scheme was introduced by Gentry
in 2009, and it also used ideal lattice. After that, in 2010 Dijk et. al constructed simpler fully homomorphic
encryption scheme over integers. The ﬁrst plausible candidate of multilinear maps was constructed by Garg et.
al in 2013 and it is from ideal lattice and then Coron et. al presented a diﬀerent construction that works overthe integers in 2013.
In this talk, we review the multilinear map and fully homomorphic encryption scheme over integers brieﬂy
and present some relations between these schemes.

14

Friday, 24 January
9:30–10:30
Shanjian Tang (Fudan University)
Linear-Quadratic Optimal Control: from deterministic to stochastic
Keywords: linear-quadratic regulator, Kalman ﬁlter, Riccati equation, Hamilton system, explicit solution,
backward stochastic Riccati equation
Abstract: In this talk, I would begin with the historical contribution of R. E. Kalman on linear quadratic
regulator for a deterministic ordinary diﬀerential system, where in particular the solution of Riccati equation is
explicitly represented via the transition matrix of the related deterministic Hamilton system. Then, we consider
the stochastic diﬀerential system, recall Kalman ﬁlter, and review the related works by W. M. Wonham, J. M.
Bismut, and others on the stochastic linear quadratic (SLQ) control. We shall show that how the solution of
the Riccati equation should satisfy a system of matrix-valued NONLINEAR ODEs when the coeﬃcients of thestochastic linear quadratic control problem are all deterministic.
Finally, we consider a robust SLQ control problem and expose the diﬃcult features of the associated backward
stochastic Riccati equation.

10:40–11:40
Seok-jin Kang (Seoul National University)
Cyclotomic categoriﬁcation theorem and 2-representation theory
Keywords: categoriﬁcation, Khovanov-Lauda-Rouquier algebra, 2-representation theory
Abstract: In this talk, we will ﬁrst explain the main idea of categorical representation theory. One of the mostprominent applications is LLT-Ariki theory where aﬃne Hecke algebras and their cyclotomic quotient categorify
quantum aﬃne algebras of type A and their highest weight modules.
In 2008, Khovanov-Lauda and Rouquier independently introduced a family of graded algebras which are now
called the Khovanov-Lauda-Rouquier algebras or quiver Hecke algebras. The KLR algebras can be regarded asa vast generalization of aﬃne Hecke algebras. It was shown by Khovanov-Lauda and Rouquier that they give acategoriﬁcation of quantum Kac-Moody algebras. Recently, Kang and Kashiwara showed that the cyclotomic
quotients of KLR algebras provide a categoriﬁcation of integrable highest weight modules over quantum KacMoody algebras.
Finally, we will discuss possible applications of these categoriﬁcation theorems and future developments inthis area.

13:40–14:10
Yuuya Takayama (Research Institute for Mathematical Sciences)
Introduction to bow varieties
Keywords: hyper-Kähler manifold, bow variety, quiver variety
Abstract: (M, g, I, J, K) is called a hyper-Kähler manifold, if M is a 4n-dimensional manifold, I, J, K(= IJ)
are complex structures and g is a Kähler metric for I, J, K. Since this deﬁnition is strict, so few explicit examples
are well-known. On the other hand, we can construct some other hyper-Kähler manifolds from explicit examples
by hyper-Kähler quotients. And quiver varieties and bow varieties, appearing in Keywords, are classes of hyperKähler quotients. Then there exists a question what manifolds are constructed by hyper-Kähler quotients. The
answer is well-known for quiver varieties, but it was not known for bow varieties. In this talk, I will explain theway to construct bow varieties, the reason to introduce them, and a partial answer to the above question forthem.

Ye Li (Tsinghua University)
A class of nonconforming quadrilateral ﬁnite elements for incompressible ﬂow
Keywords: Incompressible ﬂow, ﬁnite element method, nonconforming quadrilateral elements, optimal error
estimates
15

Abstract: I have been studying the numerical methods for PDEs. The ﬁnite element method is a class ofnumerical methods for solving PDEs. It is a combination of variational methods and subdivision interpolation
technique and it is widely used in engineering. We say an element is nonconforming if its ﬁnite element spacedoes not belong to the original space for the continuous variational problem. The Navier-Stokes equations areused to describe incompressible ﬂow. In this talk I will focus on the low-order nonconforming rectangular andquadrilateral ﬁnite elements approximation of incompressible ﬂow. Especially I consider the problem: whichkinds of basis functions are to be chosen in order to have an optimal low-order error estimate on quadrilateralmeshes?
Beyond the previous research works, we propose a general strategy to construct the basis functions. Under
several speciﬁc constraints, the optimal error estimates are obtained, i.e., the ﬁrst order accuracy of the velocitiesin H1 -norm and the pressure in L2 -norm, as well as the second order accuracy of the velocities in L2 -norm.
Besides, we clarify the diﬀerences between rectangular and quadrilateral ﬁnite element approximation. In
addition, we give several examples to verify the validity of our error estimates.

14:20–14:50
Yusuke Nakamura (The University of Tokyo)
Minimal model program and singularity theory
Keywords: Minimal model program, termination of ﬂips, singularity theory
Abstract: One of the most important problem in algebraic geometry is the birational classiﬁcation of algebraic
varieties. The minimal model program is a “program” whose goal is to ﬁnd a variety which is as simple aspossible in a birational equivalent class. In this area, the conjecture of terminations of ﬂips is one of the mostimportant conjecture, which certiﬁes that the program really ends in the ﬁnite steps.
The conjecture of terminations of ﬂips is a global geometrical problem in essentials. The minimal log
discrepancy (mld for short) was introduced by Shokurov, in order to reduce the conjecture of terminations ofﬂips to a local problem about singularities. Recently, this has been a fundamental invariant in the minimal model
program. There are two related conjectures about mld’s, the ACC (ascending chain condition) conjecture andthe LSC (lower semi-continuity) conjecture. Shokurov showed that these two conjectures imply the conjectureof terminations of ﬂips.
In this talk, we will explain the above conjectures and the relation to the termination of ﬂips. After that,
we will explain the known results about these conjectures.

p(·)

L

Jihoon Ok (Seoul National University)-regularity theory for the gradient of weak solutions for elliptic and parabolic equations in divergence form

Keywords: Lp(·) -estimate, elliptic equation, parabolic equation
Abstract: Recently there have been systematic and extensive research activities on generalized Lebesgue andSobolev spaces with variable exponents, Lp(·) and W 1,p(·) . These spaces have been developed from the classic
Lebesgue and Sobolev spaces. They appear in the literature dealing with physical phenomena where the energies
have the nonstandard growths in the area of various applications, for examples, the eletrorheological ﬂuid, thethermistor problem, image processing models and so on. In this talk, I will show an optimal Lp(·) -regularity
theory for the gradient of weak solutions for elliptic and parabolic equations in divergence form with measurable
coeﬃcients in a nonsmooth domains. With a variable function p(·) satisfying log-Hölder continuity, we prove thatthe gradient of weak solution is as integrable as the nonhomogeneous term in Lp(·) spaces under the assumptions
that the coeﬃcients are merely measurable in one variable while have a small bounded mean oscillation(BMO)
in the other variables, and that the domain is assumed to be δ-Reifenberg ﬂat.

15:00–15:30
Yuhei Suzuki (The University of Tokyo)
Amenable minimal Cantor systems of free groups arising from diagonal actions
Keywords: C∗ -algebras, amenable actions, free groups, Cantor systems
Abstract: The notion of the amenability for topological dynamical systems is a natural generalization of theone for discrete groups. Amenability of dynamical system is characterized by the nuclearity (a notion of theamenability in C∗ -algebra theory) of an associated C∗ -algebra, called the reduced crossed product. Amenability
16also provides other nice properties for the reduced crossed product. From this, we are interested in amenable
dynamical systems in particular. In this talk, we consider amenable minimal Cantor systems of free groups.
The motivation comes from the following two natural asks. The ﬁrst one is how well the crossed products
remember the original dynamical systems. The second one is giving new (tractable and useful) presentations
for Kirchberg algebras. Here a Kirchberg algebra is a C∗ -algebra satisfying certain nice properties. (Moreprecisely, a C∗ -algebra A is said to be a Kirchberg algebra if it is simple, separable, nuclear, purely inﬁnite, andsatisfying the UCT.) It is known that Kirchberg algebras are completely classiﬁed by K-theoretic invariants.
(The classiﬁcation theorem of Kirchberg and Phillips.) Hence they are very important and interesting. To
ﬁnd the answers of both questions, it is important to give many concrete examples whose crossed products arecompletely determinable. However, until now, only a few examples were known. (E.g., the boundary action,
the dynamical system constructed by Elliott-Sierakowski.) In my recent work, I construct continuum many
examples of amenable minimal Cantor systems of free groups whose crossed products are Kirchberg algebras
and their K-groups are completely determined (hence their isomorphism classes themselves are also completely
determined thanks to the classiﬁcation theorem of Kirchberg and Phillips). Moreover, with a little eﬀort, we also
construct such systems for virtually free groups. Then, as a consequence, we obtain certain (continuum many)
Kirchberg algebras are decomposed as a crossed product of an amenable minimal (topologically free) Cantor
system of arbitrary virtually free groups. Furthermore, in this computation, the technique of computations
for K-groups for certain amenable minimal Cantor systems are also established. I also want to talk about anapplication to the classiﬁcation for certain amenable minimal Cantor systems of free groups.

Jinsu Kim (Seoul National University)
Eﬃcient Fully Homomorphic Encryption
Keywords: ElGamal, Goldwasser-Micali, Naccache-Stern, Hybrid Scheme, Multiplicative Homomorphic Encryption, Additive Homomorphic Encryption, Fully Homomorphic Encryption, Decryption Circuit, Exponentiation, Bootstrapping
Abstract: The concept of computation on encrypted data without decryption was ﬁrstly introduced by Rivest,
Adleman and Detourzos in 1978. After thirty years, Gentry proposed a fully homomorphic encryption (FHE)
based on ideal lattices. This scheme is far from being practical due to its large computation costs and large
ciphertexts. Since then, lots of eﬀorts has been done to devise more eﬃcient schemes and their successors
become much more eﬃcient. However, most FHE schemes still have huge ciphertext size, at least millions ofbits for a single ciphertext. This is a big bottleneck when it is deployed in practice.
In this talk, we explore how to obtain eﬃcient hybrid homomorphic encryption schemes by combining apublic key encryption with small ciphertext size and a somewhat homomorphic encryption (SHE) that could
evaluate the decryption circuit of the PKE. For eﬃcient construction, we consider SHE for hybrid scheme rather
than FHE.

16:00–16:30
Bo Li (Tsinghua University)
Inﬁnitely many solutions for the prescribed curvature problem of polyharmonic operator
Keywords: Polyharmonic operator, critical exponents, inﬁnitely many solutions
Abstract: We study the existence of inﬁnitely many solutions for the prescribed curvature problem Dm u =∗K|u|m −2 u in SN . The number m∗ = N 2N
−2m is called the critical exponent, where N ≥ 2m + 1, m ≥ 1. Dm ism∏the polyharmonic operator of 2m order given by Dm =(−∆g + 41 (N − 2k)(N + 2k − 2)).
k=1

Given a compact Riemannian manifold (SN , g) and a smooth function K deﬁned on SN , the prescribed4
curvature problem of polyharmonic operator is to ﬁnd a conformally invariant metric gu = u N −2m g such thatthe scalar curvature is just K under this new metric gu .
Under what condition of K shall we obtain inﬁnitely many solutions to the prescribed curvature problem ofpolyharmonic operator? In this talk, we will give the positive answer to this question. First, we assume K has
a positive maximum between the poles, and show the existence of inﬁnitely many non-radial positive solutions
by energy expansion and min-max methods. Then, we assume K is a positive constant instead, the problem
allows inﬁnitely many non-radial sign-changing solutions by glueing large amounts of bubble solutions.

Tse-Chung Yang (National Taiwan University)
Monomial, Gorenstein, and Bass orders
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Keywords: Maximal orders, monomial orders, Gorenstein orders, Bass orders
Abstract: In the integral theory for central simple algebras over non-Archimedean local ﬁelds, we have thefollowing important classes of orders: maximal orders, hereditary orders, Bass orders and Gorenstein orders.
Many theories and relations are investigated by many authors for understanding these classes of orders, as wellas their module structures. It is well-known that they form the following proper inclusions:(maximal orders) ⊂ (hereditary orders) ⊂ (Bass orders) ⊂ (Gorenstein orders)
In this talk, we study a class of orders called monomial orders in a central simple algebra over a nonArchimedean local ﬁeld. Monomial orders are easily represented and they may be also viewed as a direct
generalization of Eichler orders in quaternion algebras. A criterion for monomial orders to be Gorenstein or to
be Bass is given. It is shown that a monomial order is Bass if and only if it is either a hereditary or an Eichler
order of period two.

16:40–17:10
Sz-Sheng Wang (National Taiwan University)
Calabi-Yau threefolds with small contractions
Keywords: Calabi-Yau threefold, primitive contraction
Abstract: Let X be a projective Calabi-Yau 3-fold. A primitive contraction of X is a birational contraction onto
a normal projective variety which cannot be further factored into birational morphisms of normal varieties. In
order to study the behaviour of the Kähler cone under deformations, P.M.H. Wilson gives a coarse classiﬁcationof such contractions. A primitive contraction is called of type I if it is a small contraction, i.e. it contracts aﬁnite number of curves to points. In this talk, we will prove that a primitive type I contraction is irreducible,
i.e. every ﬁber is either a point or a irreducible curve, and study Calabi-Yau threefolds with small contractions.
We will also discuss applications to extremal transition if time allowed.

Nobuaki Naganuma (Tohoku University)
Asymptotic error distributions of the Crank-Nicholson scheme for SDEs driven by fractional Brownian motion
Keywords: Fractional Brownian motion, Stochastic diﬀerential equation, Crank-Nicholson scheme, Exact rateof convergence
Abstract: We investigate the error between the solution to a stochastic diﬀerential equation(SDE) driven by afractional Brownian motion and the approximation by the Crank-Nicholson scheme associated to the equation.
In preceding results, researchers deal with numerical schemes solving SDEs, for example, the Crank-Nicholson
scheme and the Euler scheme. They are interested in the convergence rates of the errors and the limit distribution, which is the limit of the errors with some weights.
In this talk, we consider the error of the Crank-Nicholson scheme as stochastic processes and determine
the convergence rate and the limit distribution in the Skorohod topology. The limit distribution is expressedin terms of the solution to the equation and the Itô integral with respect to a standard Brownian motion
independent of the driving process of the equation. The key ingredients in our proof are asymptotic behavior
of weighted Hermite variations as stochastic processes.

17:20–17:50
Qizhi Wang (Fudan University)
Conformal Positive Mass Theorem for Asymptotically Flat Manifolds with Inner Boundary
Keywords: Mass, conformal, spin
Abstract: Inspired by Witten’s insightful spinor proof of positive mass theorem in general relativity, in thistalk, we will use the spinor method to derive higher dimensional type conformal positive mass theorems on
asymptotically ﬂat spin manifolds with inner boundary, which states that under a condition about the plus
(minus) relation between the scalar curvatures of the original and the conformal manifold, in addition withsome boundary condition, we will get the associated positivity of their ADM masses. The rigidity part ofthe plus part without boundary is used in the proof of black hole uniqueness theorems. These theorems arealso related with quasi-local mass and the spectrum of Dirac operator. We will talk about the spinor method18

ﬁrst, and use it to prove the main theorem, then we will brieﬂy talk about their relation with quasi-local mass
(Brown-York mass) and Dirac spectrum in asymptotically ﬂat manifold with inner boundary.

Cheng-Fang Su (National Central University)
The existence of solutions of 2-dimensional incompressible Navier-Stokes equations on a moving domain in an
optimal Sobolev space
Keywords: Incompressible Navier-Stokes equations, ALE formulation, moving domain
Abstract: We establish the existence of a solution to the Navier-Stokes equations on a moving domain withsurface tension in an optimal Sobolev space for the case of two space dimension. No compatibility conditions
are required to guarantee the existence of a solution.
In this talk, we will introduce the ALE formulation which can transform our moving domain into a ﬁxed
domain and simplify our problem. However, the ALE formulation could make nonlinear terms of Navier-Stokes
equations stronger. So we will give an simple example to explain the diﬃculties of our study.

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