[ 講演概要 ]
The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

[ 講演概要 ]
今日の情報科学および情報工学における発展トレンドの大きな流れは、「実用化
段階に入ったAI（人工知能）」、「演算速度のさらなる高速化」、「増大するサ
イバーセキュリティの重要性」の３つがあげられる。本連携講座では、このトレ
ンドに沿って、「AIと量子計算」を主テーマとし暗号理論の最前線にも触れなが
ら議論を深めていきたいと考えている。今回は、その第1回として、情報社会に
おいて情報収集拠点と共に情報発信拠点であるデータセンターの最新事情につい
て述べる。ここで起きている産業のデジタル化の中で、自動車産業が「CASE」
（Connected、Autonomous、Sharing & Services、Electrification）へと大転換
していることを取り上げデータセンターとの関係について述べることとする。

[ 講演概要 ]
A simple reaction-diffusion predator-prey model with Holling type IV functional response
and logistic growth in the prey is considered. The functional response can be interpreted as
a group defense mechanism, i.e., the predation rate decreases with resource density when the
prey density is high enough [1]. Such a mechanism has been described in diverse biological
interactions [2,3]. For instance, high densities of filamentous algae can decrease filtering
rates of filter feeders [4].
The model will be described and linked to plankton dynamics. Nonspatial considerations reveal that the zooplankton may go extinct or coexistence (stationary or oscillatory) between
zoo- and phytoplankton may emerge depending on the choice of parameters. However,
including space, the dynamics are more complex. In particular, spatiotemporal irregular
oscillations can rescue the predator from extinction. These oscillations can be characterized
as spatiotemporal chaos. The results provide a simple mechanism not only for the emergence
of inhomogeneous plankton distributions [5] but also for the occurrence of chaos in plankton communities [6]. Possible underlying mechanisms for this phenomenon will be discussed.
References
[1] Freedman, H. I., Wolkowicz, G. S. (1986). Predator-prey systems with group defence: the
paradox of enrichment revisited. Bulletin of Mathematical Biology, 48(5-6), 493–508.
[2] Tener, J. S.. Muskoxen in Canada: a biological and taxonomic review. Vol. 2. Dept. of Northern
Affairs and National Resources, Canadian Wildlife Service, 1965.
[3] Holmes, J. C. (1972). Modification of intermediate host behaviour by parasites. Behavioural
aspects of parasite transmission.
[4] Davidowicz, P., Gliwicz, Z. M., Gulati, R. D. (1988). Can Daphnia prevent a blue-green algal
bloom in hypertrophic lakes? A laboratory test. Limnologica. Jena, 19(1), 21–26.
[5] Abbott, M., 1993. Phytoplankton patchiness: ecological implicationsand observation methods.
In: Levin, S.A., Powell, T.M., Steele, J.H.(Eds.), Patch Dynamics. Lecture Notes in Biomathematics, vol. 96. Springer-Verlag, Berlin, pp. 37–49.
[6] Beninc`a, E. et al. (2008). Chaos in a long-term experiment with a plankton community. Nature,
451(7180), 822.

[ 講演概要 ]
The possible control of competitive invasion by infection of the invader and multiplicative
noise is studied. The basic model is the Lotka-Volterra competition system with emergent
carrying capacities. Several stationary solutions of the non-infected and infected system are
identied as well as parameter ranges of bistability. The latter are used for the numerical
study of diusive invasion phenomena. The Fickian diusivities, the infection but in particular the white and colored multiplicative noise are the control parameters. It is shown
that not only competition, possible infection and mobilities are important drivers of the
invasive dynamics but also the noise and especially its color and the functional response of
populations to the emergence of noise.
The variability of the environment can additionally be modelled by applying Fokker-Planck
instead of Fickian diusion. An interesting feature of Fokker-Planck diusion is that for spatially varying diusion coecients the stationary solution is not a homogeneous distribution.
Instead, the densities accumulate in regions of low diusivity and tend to lower levels for
areas of high diusivity. Thus, the stationary distribution of the Fokker-Planck diusion can
be interpreted as a re
ection of dierent levels of habitat quality [1-5]. The latter recalls the
seminal papers on environmental density, cf. [6-7]. Appropriate examples will be presented.
References
[1] Bengfort, M., Malchow, H., Hilker, F.M. (2016). The Fokker-Planck law of diffusion and
pattern formation in heterogeneous media. Journal of Mathematical Biology 73(3), 683-704.
[2] Siekmann, I., Malchow, H. (2016). Fighting enemies and noise: Competition of residents
and invaders in a stochastically fluctuating environment. Mathematical Modelling of Natural
Phenomena 11(5), 120-140.
[3] Siekmann, I., Bengfort, M., Malchow, H. (2017). Coexistence of competitors mediated by
nonlinear noise. European Physical Journal Special Topics 226(9), 2157-2170.
[4] Kohnke, M.C., Malchow, H. (2017). Impact of parameter variability and environmental noise
on the Klausmeier model of vegetation pattern formation. Mathematics 5, 69 (19 pages).
[5] Bengfort, M., Siekmann, I., Malchow, H. (2018). Invasive competition with Fokker-Planck
diusion and noise. Ecological Complexity 34, 134-13.
[6] Morisita, M. (1971). Measuring of habitat value by the \environmental density" method. In:
Spatial patterns and statistical distributions (Patil, C.D., Pielou, E.C., Waters, W.E., eds.),
Statistical Ecology, vol. 1, pp. 379-401. Pennsylvania State University Press, University Park.
[7] N. Shigesada, N., Kawasaki, K., Teramoto, E. (1979). Spatial segregation of interacting species.
Journal of Theoretical Biology 79, 83-99.

[ 講演概要 ]
In this talk, I will survey some recent progress towards understanding the slopes of modular forms, with or without level structures. This has direct application to the conjecture of Breuil-Buzzard-Emerton on the slopes of Kisin's crystabelline deformation spaces. In particular, we obtain certain refined version of the spectral halo conjecture, where we may identify explicitly the slopes at the boundary when given a reducible non-split generic residual local Galois representation. This is a joint work in progress with Ruochuan Liu, Nha Truong, and Bin Zhao.

（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は北京からの中継です．）

[ 講演概要 ]
Let G be a finite cyclic subgroup of GL(n,C). Then Cn/G is a cyclic quotient singularity. In the case n = 2, Cn/G possess the unique minimal resolution, and it is obtained by Hirzubruch-Jung continued fraction. In this talk, we show a sufficient condition of existence of crepant desingularization for Gorenstein abelian quotient singularities in all dimensions by using Ashikaga’s continuous fractions. Moreover, as a corollary, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant desingularization.

[ 講演概要 ]
(Joint work with Yusuke Kuno.) Given an oriented surface S and a simple closed curve C in S, the "Dehn twist" along C is the homeomorphism of S defined by "twisting" S around C by a full twist. If the curve C is not simple, this transformation of S does not make sense anymore, but one can consider two possible generalizations: one possibility is to use the homotopy intersection form of S to "simulate" the action of a Dehn twist on the (Malcev completion of) the fundamental group of S; another possibility is to view C as a curve on the top boundary of the cylinder S×[0,1], to push it arbitrarily into the interior so as to obtain, by surgery along the resulting knot, a new 3-manifold. In this talk, we will relate two those possible generalizations of a Dehn twist and we will give explicit formulas using a "symplectic expansion" of the fundamental group of S.

[ 講演概要 ]
大規模な量子コンピュータが出現すれば，これまで広く使われてきた公開鍵暗号が破られる危険性が指摘されている．それに対する対策として，量子コンピュータでも効率的に解けない数学問題の困難性に基づいて，新しい暗号を提案する動きが活発化している．それらは，耐量子計算機暗号と呼ばれるが，格子，符号，多変数多項式，同種写像などといったそれぞれ異なる数学問題の計算困難性をよりどころにした方式が知られている．本講演では，その動向の概略と共に，私が主に取り組んでいる同種写像暗号について説明する．特にSIDH鍵共有と種数1,2のCGLハッシュ関数を紹介する．

[ 講演概要 ]
The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

[ 講演概要 ]
Two intimately connected topics, namely, regularization of gradient catastrophes of all orders for the Burgers-Hopf equation via the Jordan chain and the singularities of the plane into plane mappings
associated with two-component hydrodynamic type systems of parabolic type are discussed.
It is shown that the regularization of all gradient catastrophes (generic and higher orders) for the Burgers-Hopf equation is achieved by the step by step embedding of the Burgers-Hopf equation into multi-component parabolic systems of quasilinear PDEs with the most degenerate Jordan blocks. Infinite parabolic Jordan chain provides us with the complete regularization. This chain contains Burgers and KdV equations as particular reductions.
It is demonstrated that the singularities of the plane into planes mappings associated with the two-component system of quasilinear PDEs of parabolic type are quite different from those in hyperbolic and elliptic cases. Impediments arising in the application of the original Whitney's approach to such case are discussed. It is shown that flex is the lowest singularity while higher singularities are given by ( k+1,k+2) curves which are of cusp type for k=2n+1, n=1,2,...,. Regularization of these singularities is discussed.

Presentation is based on two publications:

1. B. Konopelchenko and G. Ortenzi, Parabolic regularization of the gradient catastrophes for the Burgers-Hopf equation and Jordan chain, J. Phys. A: Math. Theor., 51 (2018) 275201.

2. B.G. Konopelchenko and G. Ortenzi, On the plane into plane mappings of hydrodynamic type. Parabolic case. Rev. Math. Phys.,32 (2020) 2020006. Online access. arXiv:1904.00901.

[ 講演概要 ]
In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.
In this talk, we present a family of Rankin-Selberg integrals (the twisted doubling method, in joint work with Friedberg, Ginzburg, and Kaplan) for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. This can be viewed as a generalization of the doubling integrals of Piatetski-Shapiro and Rallis. Time permitting, we will discuss the twisted doubling integrals for Brylinski-Deligne covers of classical groups.

[ 講演概要 ]
双曲性は通常の力学系（1パラメータ群作用の研究）において最も基本的な重要性を持つ概念です．それは，十分な豊かさ（拡大性や正エントロピー）を持ちながらも，同時に制御可能（安定性や適切な意味での良い測度の一意性）な力学系の例を与えます．ではこれを群作用に拡張できるでしょうか？

ナイーブには困難です．例えば $Z^2$ の作用を考えましょう（つまり可換な 2 パラメータ作用）・簡単にわかるのは，有限次元のコンパクト多様体に $Z^2$ が可微分に作用するとき，その $Z^2$ 作用としてのエントロピーはゼロになります．つまり，通常の有限次元の状況には，豊かな $Z^2$ 作用は存在しません．言い換えると，十分に豊かな群作用を得るためには無限次元の世界に行かざるを得ません．しかし，無限次元の世界でどのような構造を見出せばよいのでしょうか？

この講演では，このような方向性にアプローチする際に，平均次元と呼ばれる量が大きな役割を果たす可能性を説明します．特に，次のような原理についてお話します：

$Z^k$（可換な $k$ パラメータ群）が空間 $X$ に何らかの「双曲性」を持って作用するとき，$Z^k$ のランク $k-1$ の部分群 $G$ の部分作用に対する平均次元が制御できる．

この講演はTom Meyerovitch，篠田万穂との共同研究に基づきます．

[ 講演概要 ]
We study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem. We especially study the examples of non-pluriharmonic loci in smooth toric varieties. I would like to explain the relation of non-pluriharmonic loci and polytopes.

[ 講演概要 ]
Over the last decade it became clear that the role of discrete Painlevé equations in applications has been steadily growing. Thus, the question of recognizing a certain non-autonomous recurrence as a discrete Painlevé equation and understanding its position in Sakai’s classification scheme, recognizing whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such example, becomes one of the central ones. Fortunately, Sakai’s geometric theory provides an almost algorithmic procedure of answering this question.
In this work we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for orthogonal polynomials can be expressed in terms of solutions of some discrete Painlevé equation. In this work we study orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form.
This is joint work with Galina Filipuk (University of Warsaw, Poland) and Alexander Stokes (University College, London, UK)

[ 講演概要 ]
The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

[ 講演概要 ]
Let K be a knot and G be a group. The representation space of K for the group G means the space of homomorphisms from the knot group to G and is defined by using the group ring C[G], where C[G] is the ring of functions on G and has a commutative Hopf algebra structure. This construction can be generalized to any commutative Hopf algebras.

In this talk, we extend this construction to any braided Hopf algebras with braided commutativity. A typical example is BSL(2), which is the braided SL(2) introduced by S. Majid. Applying the above construction to BSL(2), we get the space of BSL(2) representations, which provides a quantization of SL(2) representations of a knot. This is joint work with Roloand van der Veen.

[ 講演概要 ]
G. Schmieder-M. Shiba observed conformal embeddings of a fixed open Riemann surface of positive finite genus into closed Riemann surfaces of the same genus, and they showed the range of each diagonal element of the period matrices. Now we shall consider a smooth deformation of open Riemann surfaces with a complex parameter. In this talk, we show the rigidity of directional moduli induced by elements of the period matrices on pseudoconvex domains fibered by open Riemann surfaces of the same topological type.

[ 講演概要 ]
The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

[ 講演概要 ]
Keller-Segel方程式に対して，有限体積法の保存型の非線形的なスキームを提案した．まず離散解の存在性を示し，半群理論を用いて誤差評価を行った．特に，1次収束を示すために必要な離散解の事前評価を示した．さらに，自明な定常解に収束する場合に適用する離散Laypunov汎関数を定義し，Laypunov不等式を証明した．最後に爆発解について，離散Laypunov汎関数やスキームの提案について少し話したい．

[ 講演概要 ]
In this talk, we present a new staggered hybridization technique for discontinuous Galerkin methods to discretize linear elastodynamic equations and nonlinear Stokes equations. The idea of hybridization is used extensively in many discontinuous Galerkin methods, but the idea of staggered hybridization is new. Our new approach offers several advantages, namely energy conservation, high-order optimal convergence, preservation of symmetry for the stress tensor, block diagonal mass matrices as well as low dispersion error. The key idea is to use two staggered hybrid variables to enforce the continuity of the velocity and the continuity of the normal component of the stress tensor on a staggered mesh. We prove the stability and the convergence of the proposed scheme in both the semi-discrete and the fully-discrete settings. Numerical results confirm the optimal rate of convergence and show that the method has a superconvergent property for dispersion.

[ 講演概要 ]
There exist many numerical methods for solving the fluid dynamics equations, the main difference between them lies in the partitions of geometric domain and the discrete forms of governing equations. Due to the discontinuous piecewise polynomial subspaces, DG and HDG methods can be easily implemented on highly unstructured meshes, e.g. general polygonal mesh, and volume integrals could be calculated on physical elements, without reference elements and mappings between physical and reference elements. In this talk, DG and HDG methods employed to a class of variational inequality problems arising in hydrodynamics are studied. Some theoretical results will be shown, as well as the implementations of these methods are also put into practice.

[ 講演概要 ]
We propose a new hybridizable discontinuous Galerkin (HDG) method for steady-state diffusion problems. In our method, both the trace and flux of the exact solution are hybridized. The Lehrenfeld-Schöberl stabilization is implicitly included in the method, so that the orders of convergence in all variables are optimal without postprocessing and computation of any projection. Numerical results are present to show the validation of our method.

[ 講演概要 ]
We consider the Stokes–Darcy interface problem supplemented with the Beavers– Joseph–Saffman condition on the interface separating two domains. This condition allows for discontinuity in the tangential velocities and in the pressures along the interface. To effectively express it, we propose to use discontinuous linear finite elements to approximate all of the velocities/pressures in the Stokes/Darcy regions. The continuity of velocity in the normal direction is weakly enforced by adopting either the penalty method or Nitsche’s method. We present stability and error estimates for the proposed scheme, taking into account the situation where a curved interface is approximated by a polygonal curve or polyhedral surface.

[ 講演概要 ]
The immune system against tumor is a complex dynamical process showing a dual role. On the one hand, the immune system can activate some immune cells to kill tumor cells, such as cytotoxic T lymphocytes (CTLs) and natural killer cells (NKs), but on the other hand, more evidence shows that some immune cells can help tumor escape, such as regulatory T cells (Tregs). In this talk, we propose a tumor immune interaction model based on Tregs mediated tumor immune escape mechanism. When HTCs stimulation rate by the presence of identified tumor antigens below the critical value, the interior equilibrium P* is always stable in the region of existence. When HTCs stimulation rate higher than the critical value, the Inhibition rate of ECs by Tregs can destabilize P* and cause Hopf bifurcations and produce limit cycle. This model shows that Tregs might play a crucial role in triggering the immune escape of tumor cells. Furthermore, we introduce the adoptive cellular immunotherapy (ACI) and monoclonal immunotherapy as the treatment to boost the immune system to fight against tumors. The numerical results show that ACI can control more tumor cells, while monoclonal immunotherapy can delay the inhibitory effect of Tregs on effector cells (ECs). The results also show that the combination immunotherapy can control tumor cells and reduce the inhibitory effect of Tregs better than single immunotherapy.