SUMMARY

1. Peanry
Yasuyuki KAWAHIGASHI (Univ. of Tokyo)
Topological order, tensor categories and operator algebras
Two-dimensional topological order has attracted much attention in physics recently. It is related totopological quantum computing, for example, which is a target of a great deal of industrial research
today. This topic has many mathematical aspects related to diﬀerent ﬁelds in mathematics. In this talk,we present connections to operator algebras based on tensor categories with emphasis on a representation
theoretic viewpoint. We ﬁrst explain a general notion of a tensor category and then relate it to theory ofoperator algebras, particularly subfactors of the so-called hy- perﬁnite II1 factor in theory of Jones. With
these preliminaries, we explain recent progress on two-dimensional topological order, chiral conformal
ﬁeld theory and tensor networks. Abstract begins.
Inkang KIM(KIAS)
Signature, Toledo invariant, eta invariant, and the representation of surface groupe in real symplectic
In this talk, by using Atiyah-Patodi-Singer index theorem, we obtain a formula for the signature of aﬂat symplectic vector bundle over a surface with boundary, which is related to the Toledo invariant ofa surface group representation in the real symplectic group and the Rho invariant on the boundary. Asan application, we obtain a Milnor-Wood type inequality for the signature. In particular, we give a new
proof of the Milnor-Wood inequality for the Toledo invariant in the case of closed surfaces and obtainsome modiﬁed inequalities for the surface with boundary.
2. Algebra
Takehiko YASUDA(Osaka Univ.)
On the isomorphism problem for projective schemes
In this talk, I discuss the isomorphism problem for (quasi-)projective schemes over the ﬁeld of algebraic
numbers. This asks whether there is an algorithm to decide whether two projective schemes are isomorphic. The main result is decidability of isomorphism in the cases of (1) one-dimensional projective
schemes, (2) one-dimensional reduced quasi-projective schemes, (3) smooth irreducible projective varieties with canonical or anti-canonical divisor being big, and (4) K3 surfaces with ﬁnite automorphismgroups. As a closed related problem, I also discuss decidability of positivity properties of invertible
sheaves.
Sanghoon BAEK (KAIST)
Rationality and complexity for spaces of torsors under semisimple groups
Consider a generically free representation V of a semisimple group G over the complex numbers C.
Then, there exists a G-torsor over the quotient U/G (called the classifying space of G) for an open
subset U of V. Such a G-torsor is called versal as for every ﬁeld extension K/C, the K-points of theclassifying space parametrize all G-torsors over K. In this talk, we shall discuss two wide open problems:
the rationality problem of the classifying space and the problem of ﬁnding the minimum dimension of
the classifying space of G through all versal G-torsors. Also, we present some recent results on theunramiﬁed cohomology of the classifying space detecting the rationality and the exact values of theminimum dimension of the classifying space of the groups of type B.

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SUMMARY

Akio TAMAGAWA (Kyoto Univ.)“Ghosts” in families of abelian varieties with a common isogeny factor (joint work with Anna Cadoret)
We discuss the following question raised by Rössler and Szamuely. Given a family of abelian varieties
over a variety over a ﬁeld, assume that all the closed ﬁbers have a nontrivial common isogeny factor.
Does this imply that the generic ﬁber has a nontrivial isotrivial isogeny factor? This question can bereformulated in terms of ℓ-adic representations and the diﬃcult case is when the base ﬁeld is ﬁnite.
In this setting, we introduce the notion of ghosts in terms of ℓ-adic representations. Ghosts control
completely the problem, namely the original question is positive if and only if the ghost is trivial. We
prove the triviality of the ghost in various cases and, in general, it is predicted by Zarhin’s microweights
conjecture.
Jihun PARK (POSTECH)
Sasaki-Einstein 5-manifolds from an algebro-geometric point of view
We present the state of the art concerning classiﬁcation of closed simply connected Sasaki-Einstein
5-manifolds from an algebro-geometric point of view.
3. Geometry and Topology
Koichi NAGANO (Univ. of Tsukuba)
Topological regularity theorems for CAT(0) spaces
In this talk, I would like to focus on topological regularity problems of when CAT(0) spaces are homeomorphic to Euclidean spaces. I will give an overview of a study on asymptotic topological regularity ofCAT(0) spaces. I will present conditions for CAT(0) spaces of small volume growth to be homeomorphic
to Euclidean spaces. I will also report the answer to the question of Gromov for CAT(0) spaces whether
there exist CAT(0) topological manifolds other than Euclidean spaces in the remaining 4- dimensionalcase. The part on the answer to the question of Gromov for CAT(0) spaces is a joint work with Alexander
Lytchak (University of Cologne) and Stephan Stadler (Max Planck Institute for Mathematics).
Yunhyung CHO (Sungkyunkwan Univ.)
MONOTONE LAGRANGIANS AND MUTATIONS IN FLAG VARIETIES
Using a Gorenstein Fano toric degeneration, one can construct a Lagrangian torus ﬁbration on a smooth
Fano variety X. Such a ﬁbration has a unique central ﬁber which becomes a monotone Lagrangian torus
and its Maurer- Cartan space contains a complex torus equipped with a Laurent polynomial W : (C∗ )n →C called a Fukaya-Oh-Ohta-Ono’s disc potential. When a degeneration is good, the potential function
W coincides with the Landau-Ginzburg superpotential of X which captures the multiplicative structureof the quantum cohomology ring of X. In this talk, I will explain this picture for ﬂag varieties. Also I
will explain how those potential functions are related by cluster mutations.
Hiroshi OHTA (Nagoya Univ.)
Analytic aspect of Fukaya category
For a symplectic manifold (X, ω) we have a ﬁltered A∞ category called Fukaya category, whose objects
are Lagrangian submanifolds. I will discuss some analytic aspect of certain meromorphic connection
arising from deformations of Fukaya categories.
Cheol-Hyun CHO (SNU)
Berglund-Hübsch conjecture for invertible curve singularities
Given one of the polynomials W = xp + yq,xp + xyq,xp y + xyq and a diago- nal symmetry group
G, Berglund-Hübsch mirror symmetry conjectures that the symplectic geometry of W is related to thecomplex geometry of transpose polynomial W T with the transpose symmetry group GT . We ﬁrst
deﬁne a new Fukaya category of the Landau-Ginzburg orbifold (W,G), from the the wrapped Fukaya
category of its Milnor ﬁber together with monodromy information. We formulate and prove a full version

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of BH homological mirror symmetry con- jecture in these cases. Furthermore, for ADE singularities,
Auslander-Reiten quivers for indecomposable matrix factorizations were known from 80’s, and we ﬁnd
the corresponding Lagrangians as well as surgery exact triangles.

4. analysis
Neal BEZ (Saitama Univ.)
The nonlinear Brascamp-Lieb inequality
The Brascamp-Lieb inequality provides a natural generalisation of the multilinear Holder inequality, theLoomis-Whitney inequality and the Young convolution inequality. For a number of reasons, including
applications to harmonic analysis and PDE, it is of interest to establish a nonlinear variant of the
Brascamp-Lieb inequality. I will explain some recent work with Jonathan Bennett, Stefan Buschenhenke,
Michael Cowling and Taryn Flock, where we establish such a nonlinear Brascamp-Lieb inequality in fullgenerality.
Soonsik KWON (KAIST)
Blow-up dynamics of the self-dual Chern-Simons-Schrödinger equations
This abstract and talk is based on joint works with Kihyun Kim and Sung-Jin Oh [11, 12, 13]. We
study a quantitative description of the blow-up dynamics of solutions near the vortex solutions Q to
the self-dual Chern-Simons-Schrödinger equation under equivariant symmetry. In this abstract, we ﬁrst
introduce the self-dual Chern-Simons-Schrödinger equations and then announce our results on their blowup dynamics. All the statements of results are informal here. For more precise statements and details,
we refer to original papers.
Takayoshi OGAWA (Tohoku Univ.)
End-point maximal regularity and free boundary problem of the Navier-Stokes equations
We consider maximal regularity for the initial boundary value problem of the heat and the Stokes equations. Maximal regularity generally ensures that regularity of solutions for parabolic partial diﬀerential
equations can be obtained up to regularity for the external forces. Theory of maximal regularity iswell developed in the framework of UMD (unconditional martingale diﬀerences) Banach spaces and theequivalent characterization for the well-posedness is established for not only the external force but for
the initial and boundary traces. Since UMD Banach spaces are necessarily reﬂexive, maximal regularity
in non-reﬂexive Banach spaces is not clear in general. Here we consider maximal time-L1 -regularity
over a non-reﬂexive Banach space with inhomogeneous boundary condition in the half Euclidean space
and apply it to the well-posedness issue for free boundary problems of the incompressible Navier-Stokes
equations. This talk is based on a joint work with Prof. Senjo Shimizu (Kyoto University).
Kyungkeun KANG (Yonsei Univ.)
Local regularity of Stokes system and Navier-Stokes equations near boundary
We review some results about local regularity for the Stokes system and the Navier-Stokes equations
near boundary. In a half-space, it was shown in [8] that a solution generated by a compactly supported,Hölder continuous boundary ﬂux from a far may have unbounded normal derivatives near the boundary.
These phenomena are mainly due to nonlocal eﬀect caused by singular ﬂux far away from a boundary
region under consideration. Such singular solutions turn out to be of ﬁnite global energy and they can bealso constructed to the Navier-Stokes equations as well (see [9]). It is also possible to construct solutions
that are bounded but the gradients of velocity ﬁelds are singular in any Lp , p > 1 in a local neighborhoodnear boundary (see [2]). Furthermore, using the construction of singular solutions whose gradients are
not in L2loc , it was proved in [3] that the Caccioppoli’s inequality in general fails for both Stokes system
and Navier-Stokes equations. The local maximal regularity including boundedness of gradient of velocity
was analyzed in [2] as well.
Works are partly collaborated with T.-K. Chang or with B-Lai, C.-C. Lai and T.-P. Tsai.

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5. Probability Theory and Applied Mathematics
Kenta ISHIMOTO (Kyoto Univ.)
On Flows and Form of Microswimmers
Microswimmers are microscopic self-propelled particles in ﬂuids. In this article, we give an introductory
review on the hydrodynamics of microswimmers as an example of a moving boundary problem in ﬂuid
dynamics research. In particular, we focus on the dynamics of a microscopic object in a ﬂuid ﬂow andprovide a new formula for a chiral active particle such as swimming bacteria.
Ildoo KIM (Korea Univ.)
An Lp -theory for second-order stochastic partial diﬀerential equations
In this talk, we introduce a short history of Lp-theories for second-order stochastic partial diﬀerential
equations (SPDEs). Especially, we present ex- istence, uniqueness, and sharp regularity results of solution
to the stochastic partial diﬀerential equation (SPDE) of the form of
du = (aij (ω, t, x)uxi xj + f )dt + (σ ik (ω, t, x)uxi + g k )dwtk , u(0, x) = u0 ,where {wtk : k = 1, 2, · · · } is a sequence of independent Brownian motions. It is notable that if thecoeﬃcients are independent of the space variable x, then they are allowed to be merely measurable in(ω, t) and can be unbounded and fully degenerate, that is, coeﬃcients aij , σ ik merely satisfy)(∞
1 ∑ ikjkijij
σ (ω, t)σ (ω, t) ≥ 0.
(α (ω, t))d×d := a (ω, t) −2
k=1

Yuzuru INAHAMA (Kyushu Univ.)
Heat trace asymptotics for equiregular sub-Riemannian manifolds
We discuss a “div-grad type” sub-Laplacian with respect to a smooth measure and its associated heat
semigroup on a compact equiregular sub-Riemannian manifold. We prove a short time asymptotic
expansion of the heat trace up to any order. Our main result holds true for any smooth measure on
the manifold, but it has a spectral geometric meaning when Popp’s measure is considered. Our proof isprobabilistic. In particular, we use S. Watanabe’s distributional Malliavin calculus.
Myungjoo KANG (SNU)
Some applications using Deep Learning Techniques
Since 1990, many mathematicians studied the image processing based on partial diﬀerential equationsand variational methods. Using TV(total variation) and some optimization techniques, there were a lotof improvement in image processing areas. Until deep learning coming out, those were the state of art
methods in these areas. But once using deep learning techniques, it turns out that in almost every areas
in image processing ﬁelds, deep learning is the best method. I will compare the results between thevariational image processing techniques and deep learning. Also, I will give some industrial applications
using deep learning.