MSJ-KMS Joint Meeting 2023

ABSTRACTPlenary Talks
Shouhei Honda (Tohoku Univ.)
Ricci curvature and Weyl’s law
In 1911, H. Weyl provided an asymptotic behavior of eigenfunctions λi (counted with themultiplicities) of the Dirichlet Laplacian −∆Ω on a bounded domain Ω in a Euclidean space
Rn of dimension at most 3, in terms of the size of the domain, namely
ωn n
N (λ)
=
L (Ω),
n/2
λ→∞ λ
(2π)n
lim

where N (λ) is the counting function deﬁned by N (λ) = ♯{i ∈ Z≥0 |λi ≤ λ} and ωn denotes
the volume of a unit ball in Rn . This result was later on extended to higher dimensional (notnecessarily ﬂat) smooth spaces, including closed Riemannian manifolds (M n , g). It is known
as Weyl’s law.
On the other hand, based on a joint work with Luigi Ambrosio (Scuola Normale Superiore)and David Tewodrose (Nantes University), it was conjectured that such Weyl’s law holds
even for nonsmooth spaces with Ricci curvature bounded below. The main purpose of thetalk provides counterexamples to this conjecture. In the constructions, we can see a newintersection between Gromov–Hausdorﬀ collapsing geometry and subRiemannian geometry.
This is a joint work with Xianzhe Dai (UC Santa Babara), Jiayin Pan (UC Santa Cruz) andGuofang Wei (UC Santa Babara).

Jinsung Park (KIAS)
Renormalized volume and Liouville action
The renormalized volume was mathematically deﬁned for conformally compact Einsteinmanifolds by Robin Graham. This invariant had been introduced and extensively studied
in the context of AdS-CFT correspondence by mathematical physics community. For a 3dimensional convex cocompact hyperbolic manifold M , the renormalized volume is an ﬁnite
real valued invariant of M if the renormalization process is deﬁned using the hyperbolic metric over conformal boundary ∂M . Hence, the renormalized volume can be considered as afunction deﬁned on the deformation space of convex cocompact hyperbolic structures. Frommathematical physics results, the renormalized volume is expected to be related to a quantity from conformal boundary ∂M . This quantity called Liouville action was mathematicallydeﬁned by Takhtajan and Zograf. In this talk, some basic results on the renormalized volume
and its relation with the Liouville action will be explained in the context of 2 and 3 dimensional hyperbolic manifolds. Some recent corresponding results for renormalized volume andthe universal Liouville action of Weil–Petersson curves will be also explained.

1

MSJ-KMS Joint Meeting 2023

Algebra Session
Yuri Yatagawa (Tokyo Inst. Tech.)Ramiﬁcation theory and characteristic cycles of rank one sheaves
The characteristic cycles for constructible sheaves on smooth varieties are algebraic cycles
introduced by Beilinson and Saito in 2016. They are deﬁned to be unique algebraic cycles on
the cotangent bundle satisfying the Milnor formula, which computes the total dimensions ofvanishing cycles, and compute the Euler characteristics of the sheaves by the index formula.
In this talk, we consider computations of characteristic cycles in terms of ramiﬁcation theoriesof the sheaves. When the varieties are curves, such are already known, and the index formula
deduces the classical Grothendieck–Ogg–Shafarevich formula. We restrict the sheaves tobe smooth of rank one, which means the sheaves correspond to characters of abelianizedfundamental groups, and we introduce recent developments for computations of characteristic
cycles with ramiﬁcation theories of the sheaves.

Byeong-Kweon Oh (Seoul National Univ.)
Recoverable quadratic forms
A (positive deﬁnite and integral) quadratic form f is called recoverable if any quadratic
form representing all proper subforms of f also represents f itself. If there is a quadratic
form F representing all subforms of f , except for f itself, then f is called irrecoverable and
F is called an isolation of f (from its subforms).
In this talk, we ﬁnd some (ir)recoverable quadratic forms and various isolations of irrecoverable quadratic forms.

Osamu Iyama (The Univ. of Tokyo)
Tilting theory via g-fans in real Grothendieck groups
The notion of tilting complexes is basic to study the structure of derived categories ofrings. The class of silting complexes complements the class of tilting objects from a point ofview of mutation. For a ﬁnite dimensional algebra A over a ﬁeld, 2-term silting complexesof A give rise to a simplicial complex (called the g-simplicial complex ) and a nonsingular
fan in the real Grothendieck group of A (called the g-fan). For example, the g-fan of apreprojective algebra is the Coxeter fan, and the g-fan of a Jacobian algebra of a certain
quiver with potential is the g-fan of the corresponding cluster algebra. The g-fan of A is a
useful combinatorial invariant which has a lot of information about representation theory ofA, and therefore satisﬁes many nice properties. For example, the h-vector and the Dehn–
Sommerville equation of the g-simplicial complex has a tilting theoretic interpretation. Also
the g-fan of A is complete if and only if A has only ﬁnitely many 2-term silting complexes up
to isomorphism. One of the basic problems is to classify complete g-fans. We give an answer
for rank 2 case by showing that complete g-fans of rank 2 are precisely complete sign-coherent
fans of rank 2. As a by-product, for each positive integer N , we gives a ﬁnite dimensional
algebra A of rank 2 such that the Hasse quiver of the poset of 2-term silting complexes ofA has precisely N connected components. This talk is based on a series of joint works with
T. Aoki, A. Higashitani, R. Kase and Y. Mizuno (arXiv:2203.15213, 2301.01498).

Jinhyung Park (KAIST)
Syzygies of secant varieties of smooth projective curves
Exploring the interplay between the geometric properties of algebraic varieties and thealgebraic properties of the equations deﬁning algebraic varieties is an important subject
in algebraic geometry. The case of smooth projective curves of large degree is fairly wellunderstood. Two major results are Green’s (2g + 1 + p)-theorem and the proof of the gonality
conjecture by Ein–Lazarsfeld and Rathmann. On the other hand, there has been a great deal
2

MSJ-KMS Joint Meeting 2023

of research on secant varieties of projective varieties in the last three decades particularly
because some results on secant varieties have found some applications to algebraic statistics
and algebraic complexity theory. In this talk, I report recent progress on syzygies of secant
varieties of smooth projective curves obtained by joint work with Lawrence Ein and Wenbo
Niu and joint work with Junho Choe and Sijong Kwak. In the former, we extend Green’s
theorem to secant varieties of smooth projective curves by conﬁrming Sidman–Vermeire’s
conjecture, and in the latter, we prove a generalization of the gonality conjecture: the gonality
sequence of a smooth projective curve completely determines the shape of the minimal freeresolutions of secant varieties of the curve of large degree. Our results show that there is a“matryoshka structure” among secant varieties of a smooth projective curve.

Geometry and Topology Session
Mayuko Yamashita (Kyoto Univ.)
Algebraic topology and quantum ﬁeld theories
Recently, there has been a growing interest in the relations between algebraic topology
and theoretical physics. Algebraic topology is used to classify physical systems, and it canbe a very powerful tool to analyze physical problems in purely mathematical ways. Also,
physically motivated conjectures has lead to many interesting developments in homotopy
theory. In this talk, I explain these ideas and some of my related works, where we use
homotopy theory to study anomalies in string theory.
In my joint work with Yuji Tachikawa (IPMU, Physics) in 2021, we showed the absenceof anomaly in heterotic string theory using homotopy theory. Our strategy is to use theStolz–Teichner conjecture, which is one of the big conjectures connecting homotopy theory
and physics, to translate the physics problem into a purely mathematical problem and solve
it mathematically. Also, I explain the recent update on that story, where we study thesecondary morphism induced from the above vanishing. It turns out that this morphism isclosely related to Anderson self-duality of topological modular forms, which had only been
understood in a purely homotopy theoretical way. This leads us to new conjectures relating
vertex operator algebras and homotopy theory.

Jungsoo Kang (Seoul National Univ.)
A Floer–Gysin exact sequence for prequantization bundles
In this talk, I will discuss the Rabinowitz Floer homology (in short, RFH) of prequantization bundles. Prequantization bundles are circle bundles over closed integral symplectic
manifolds which have canonical contact structures. One natural setting in which RFH fora prequantization bundle is deﬁned is when it admits a Liouville ﬁlling. I will explain theconstruction of a Gysin-type long exact sequence relating RFH of a Liouville ﬁlling of a prequantization bundle and the quantum homology of the base manifold. Several applications of
this exact sequence will also be discussed. On the other hand, prequantization bundles carrynatural (non-exact) symplectic ﬁllings, namely the associated complex line bundles. In thiscase, the chain complex of RFH is more sophisticated than the one in the Liouville setting. I
will introduce ﬁltrations for the chain complex of RFH for complex line bundles and present
computational results. This is based on joint work with Peter Albers, Joonghyun Bae, andSungho Kim.

Kimihiko Motegi (Nihon Univ.)
Dehn ﬁlling and knot group: the Property P conjecture and beyond
For a given knot K in the 3–sphere, removing its tubular neighborhood to obtain theexterior E(K), which has a torus boundary. Gluing back a solid torus to E(K) along theirboundaries so that a meridian of the solid torus represents a slope r (i.e., an unoriented
isotopy classes of simple loops on ∂E(K)), we obtain a closed 3–manifold. This operation
3

MSJ-KMS Joint Meeting 2023

introduced by Max Dehn is called r–Dehn surgery on K, and the gluing part is called r–Dehnﬁlling of E(K). Slopes are parametrized by rational numbers. The Property P conjecture,settled by Kronheimer and Mrowka, asserts that every non-trivial knot satisﬁes Property P:
for any slope r ∈ Q, there exists a non-trivial element g ∈ π1 (E(K)) which remains nontrivial after r–Dehn ﬁlling. (This is a paraphrase of the original Property P.) In this talkwe discuss “extension”, “enhancement” and “reﬁnement” of Property P from group theoretic
perspective, and demonstrate somewhat surprising ﬂexibilities of Dehn ﬁlling trivializations.
This is joint work with Tetsuya Ito (Kyoto University) and Masakazu Teragaito (Hiroshima
University).

JungHwan Park (KAIST)
Seifert surfaces in the 4-ball
We answer a question of Livingston from 1982 by producing Seifert surfaces of the same
genus for a knot in the 3-sphere that do not become isotopic when their interiors are pushed
into the 4-ball. We give examples where the surfaces are not topologically isotopic in the
4-ball, as well as examples that are topologically but not smoothly isotopic. These lattersurfaces are distinguished by their associated cobordism maps on Khovanov homology, andour calculations demonstrate the stability and computability of these maps under certain
satellite operations. This is joint work with Kyle Hayden, Seungwon Kim, Maggie Miller,
and Isaac Sundberg.

Analysis Session
Ryo Takada (The Univ. of Tokyo)
Large time behavior of global solutions to the Navier–Stokes equations with the Coriolis force
In this talk, we consider the large time behavior of global solutions for the initial value
problem of the incompressible Navier–Stokes equations with the Coriolis force in R3 . We
shall show the Lp temporal decay estimates with the dispersion eﬀect of the Coriolis force
for global solutions when the initial velocity u0 belongs to L1 (R3 ). Furthermore, under
the additional assumption that |x|u0 ∈ L1 (R3 ), we establish the large time asymptotics ofglobal solutions behaving like the ﬁrst-order spatial derivatives of the integral kernel of thecorresponding linear solution. This talk is based on the joint work with Takanari Egashira(Kyushu University).

Seick Kim (Yonsei Univ.)
On Schauder type estimates for elliptic and parabolic PDEs in double divergence form
We consider the elliptic operator L∗ of the form
∗

L u=

d
∑

Dij (a u) −
ij

i,j=1

d
∑

Di (bi u) + cu,

i=1

The operator L∗ is called double divergence form operator and is the formal adjoint of theelliptic operator of non-divergence form L given by
Lv =

d
∑

ij

a Dij v +

i,j=1

d
∑

bi Di u + cu.

i=1

An important example of a double divergence form equation is the stationary Kolmogorov
equation for invariant measures of a diﬀusion process.
We are concerned with regularity of weak solutions of L∗ u = 0 and show that Schauder
type estimates are available when the coeﬃcients aij are of Dini mean oscillation and bi and4

MSJ-KMS Joint Meeting 2023

c belong to certain Lebesgue classes. We will also discuss some applications and parabolic
counterparts.

Yasuhito Miyamoto (The Univ. of Tokyo)
Structure of radial solutions for supercritical elliptic equations
We are interested in a bifurcation structure of positive solutions for supercritical elliptic
Dirichlet problems in a ball, i.e., ∆u + λf (u) = 0 in B. Since a growth rate of a nonlinear
term is larger than the critical Sobolev growth, the corresponding energy functional is not
well-deﬁned in H 1 . It is diﬃcult to use variational approaches, and a solution structure in a
general domain remains not so clear as subcritical cases. In this talk we restrict a domain to aball. Using ODE approaches, we can derive detailed information about a solution structure.
Singular solutions play a crucial role in this study, and hence those are studied in detail.

Moon-Jin Kang (KAIST)
Stability of small BV solutions to compressible Euler equations in the class of inviscid limits
from Navier–Stokes
The convex integration shows that the compressible Euler system in multi-D is ill-posedin the class of entropy solutions, especially, non-uniqueness of entropy solutions containing
a shock wave. Recently, for the 1D isentropic case, we showed the uniqueness and stabilityof entropy solutions of small BV in the class of vanishing physical viscosity limits, that is,inviscid limits from the associated Navier–Stokes system. These results use the so-called ‘acontraction with shifts’ method for handling the uniform stability of a viscous shock. In thistalk, I will explain on the method of a-contraction with shifts for a single viscous shock, andextension of the method to more general situations for two shocks and small BV solutions.
This is joint work with Geng Chen and Alexis Vasseur.

Probability Theory and Applied Mathematics Session
Takahito Kashiwabara (The Univ. of Tokyo)
Error estimates of the ﬁnite element method in smooth domains
The ﬁnite element method (FEM) is one of the most commonly-used numerical methods
to approximate solutions of partial diﬀerential equations. One of its advantages, often mentioned in textbooks etc., is the applicability to a general domain Ω with a curved boundary.
In fact, it is true that the FEM does not require cartesian grids. However, since it is impossible to exactly partition such Ω into triangulations with ﬂat faces, the approximate problem
is actually constructed on an approximated domain Ωh . Therefore, from the mathematical
viewpoint, we need to address “domain perturbation errors” caused by the discrepancy between Ω and Ωh , a general theory of which seems less developed. In this talk, we present a
strategy to deal with domain perturbation errors arising in FEM, which is applied to some
ﬂuid problems, to maximum-norm error estimates, and to a generalized Robin boundary
condition involving the Laplace–Beltrami operator.

Un Cig Ji (Chungbuk National Univ.)
Analytic representations of CAP operators canonically associated with random variables
Let X be a random variable with ﬁnite moments of all orders. By the three-term recurrence relation of the orthogonal polynomials associated with the distribution of X, therandom variable X as a multiplication operator is represented as a sum of three linear operators called creation, annihilation and preservation operators (CAP-operators). In this talk,
we ﬁrst discuss analytic representations of the CAP operators in terms of diﬀerential operators with polynomial coeﬃcients. This approach extends the usual Boson quantum mechanics
5

MSJ-KMS Joint Meeting 2023

corresponding to the Gaussian measure to the quantum mechanics associated with randomvariables. Secondly, we introduce the notion of ﬁnite type random variable and characterize
type-2 and type-3 real-valued random variables. We also discuss a necessary condition for arandom variable to be of ﬁnite. This allows to prove that the Beta and the uniform distributions are of inﬁnite type. This talk is based on a joint work with L. Accardi, A. Ebang Ellaand Y. G. Lu.

Masanori Hino (Kyoto Univ.)
Martingale dimensions of diﬀusion processes associated with Dirichlet forms
The concept of martingale dimension is deﬁned for diﬀusion processes or more-general
stochastic processes, and is interpreted as the multiplicity of ﬁltration. Such a kind of concept
was introduced at least in the 1960s. In a typical situation where the process is a solution to astochastic diﬀerential equation on the Euclidean space, the martingale dimension corresponds
to the number of independent noises that the process possesses. For diﬀusion processes
on fractal-like sets, however, determining the martingale dimension is a diﬃcult problem.
The ﬁrst result in this direction was obtained in a 1989 paper, where it was shown thatthe martingale dimension of the Brownian motion of any dimensional Sierpinski gasket isone. This may seem counter-intuitive. The relationship between the martingale dimension
and other kind of dimensions is not yet fully understood. In this talk we give a survey ofdevelopments that followed and discuss the cases of diﬀusions on general state spaces.

Byungjoon Lee (Catholic Univ. of Korea)
Super-convergence of the Poisson solver with adaptive grid conﬁgurations
The Hodge decomposition, that is an important feature of incompressible ﬂuid ﬂows,
is orthogonal and the projection taking its incompressible component is therefore stable.
The decomposition is implemented by solving the Poisson equation. In order to simulate
incompressible ﬂuid ﬂows in a stable manner, it is desired to utilize a Poisson solver thatattains the orthogonality of the Hodge decomposition in a discrete level.
When a Poisson solver induces the orthogonality, its associated linear system is necessarily
symmetric. With this regard, the symmetric Poisson solvers by Losasso et al. [1, 2] are moreadvantageous not only to eﬃciently solving the linear system but also to stably simulating
ﬂuid ﬂows than nonsymmetric ones. Their numerical solutions were empirically observed to
be ﬁrst and second order accurate, respectively. One may expect that each of their numerical
gradients has convergence order that is one less than that of its numerical solution.
However, we in this work show that super-convergence holds true with both Poissonsolvers. Rigorous analysis is presented to prove that the diﬀerence is one half, not one
between the convergence orders of numerical solution and gradient in both solvers. The
analysis is then validated with numerical results. We furthermore show that both Poissonsolvers, being symmetric, indeed satisfy the orthogonal property in the discrete level andyield stable implementations of the Hodge decomposition in octree grids.
[1] F.Losasso, F.Gibou, and R.Fedkiw, Simulating water and smoke with an octree data
structure, ACM SIGGRAPH, 2004.
[2] FRSF.Losasso, R.Fedkiw, and S.Osher, Spatially adaptive techniques for level set methods and incompressible ﬂow , Computer and ﬂuid, 2006.

6