# - The 4th COE Conference for Young Researchers -

### アブストラクト：プログラム

タイトル：Construction of a solution to the Gierer-Meinhardt system with saturation（サチュレーション効果のあるGierer-Meinhardt 系の解の構成）
アブストラクト：
We are concerned with stationary solutions to the Gierer-Meinhardt system with saturation. In this lecture, we study about the following. What is saturation effect? What is different from in no saturation case? How to construct of solutions.

タイトル：Higher q-Fock spaces and fusion products
アブストラクト：
Quiver varieties involve rich mathematical structures. For example, thier torus equivariant K-groups have actions of quantum loop algebras. On the other hand, q-Fock spaces play important roles in many areas of representation theory. In particular, bases so called "canonical bases" of q-Fock spaces sometimes mediate misteriously between completely different-looking objects. In this talk, we will intruduce isomorphisms between equivariant K-groups of quiver varieties of affine A type and q-Fock spaces. We will also discuss what would be expected beyond these isomorphisms.

タイトル：On the KAM theory
アブストラクト：
We review the KAM theory. Next, we give a new approach to the theory: We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency, without any nondegeneracy of the integrable part. We reconstruct the KAM theory, as a theory of the measure estimate for the initial values satisfying this sufficient condition. Our version is of the form closely analogous to the corresponding arguments for the integrable case.

タイトル：On the Navier-Stokes equations with spatially almost periodic data
アブストラクト：
We consider the Navier-Stokes equations when initial data may not decrease at spatial infinity so that Fourier sum of the form $\sum_{\lambda\in\Lambda}c_\lambda e^{i\lambda x}$ is allowed. It is proved that the local-in-time solution is almost periodic function and the frequency set is preserved in time. To solve this problem, uniformly $L^p$ local spaces, Besov spaces, modulation spaces and FM spaces are used.

タイトル：A characterization of Gel'fand-Shilov space $S^r_r$ and $(S^r_r)^{\prime}$ by Weyl transform
アブストラクト：
We give a characterization of Gel'fand-Shilov space $S^r_r$ and its dual space ($S^r_r$)^{\prime} by Weyl transform.

タイトル：On the turning point of the second kind of the Noumi-Yamada system $NY_{2m+1}$ （野海山田系方程式NY奇数系の第2種変わり点について）
アブストラクト：
The Noumi-Yamada system $NY_{l}$ is one of the Painlev$\acute{e}$ hierarchies. The structure of $NY_{l}$ depends on the parity of $l$. In this talk, we have the result concerning the number of the turning points of the second kind of $NY_{l}$ in the case of $l = 2m+1$.

タイトル：Classification of non-symplectic automorphisms of order 3 on $K3$ surfaces
アブストラクト：
We study non-symplectic automorphisms of order 3 on algebraic $K3$ surface over $\mathbb{C}$ which act trivially on the N\'{e}ron-Severi lattice. In particular we shall characterize their fixed locus in terms of the invariants of 3-elementary lattices.

タイトル：Asymptotic stability of planar stationary waves for damped wave equations in multi-dimensional half space
アブストラクト：
We consider the large-time behavior of solutions to the initial and boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space. We show that solutions to the problem converge to the corresponding planar stationary wave as time tends to infinity under the smallness condition on the initial perturbation. Moreover, the algebraic convergence rate is obtained by assuming that the initial perturbation decays algebraically in the normal direction.

タイトル：Physical states of quantum electrodynamics with an indefinite metric
アブストラクト：
We consider the quantized electromagnetic field interacting with an external source. We construct the quantized radiation field on an indefinite metric space. According to the Bleular-Gupta formalism we can choose the physical state space from the indefinite metric space in the case where the external source is infrared regular. If, however, the external source is infrared sigular, the physical state space is trivial.

タイトル：Poset transformations and Quasi symmetric functions
アブストラクト：
The quasi symmetric function is an algebraic invariant which encodes the flag f vector of a poset. Sometimes a transformation of a poset induces the corresponding transformation of the quasi symmetric function. There are explicit expressions for how the quasi symmetric function changes after applying operations, for example cartesian product, pyramid, prism. In this talk we will calculate more complicated cases.

タイトル：Application of non-commutative Groebner basis to a free resolution of a module over the Steenrod algebra
アブストラクト：
Groebner basis gives us an algorithm of computing some homotopy invariants. Adams spectral sequence is a powerful method to compute the homotopy set of maps, for example, homotopy groups. The E_2 terms of Adams spectral sequence is given by Ext, which can be computed by a free resolution of a module over the Steenrod algebra. I will explain how to compute that free resolution using non commutative Groebner basis.

タイトル：Initial boundary value problem for the Hasegawa-Wakatani equations
アブストラクト：
The Hasegawa-Wakatani equations are the nonlinear equations which describe the resistive drift wave turbulence in Tokamak. I proved the existence and uniqueness of a local strong solution for the initial boundary value problem for the Hasegawa-Wakatani equations. In this talk, I shall explain Tokamak and the Hasegawa-Wakatani equations and present my result.

タイトル：Optimal covariant measurement of momentum on a half line in quantum mechanics
アブストラクト：
We cannot perform the projective measurement of a momentum on a half line since it is not an observable. Nevertheless, we would like to obtain some physical information of the momentum on a half line. We define an optimality for measurement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo. Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self-adjoint and explicitly construct a model Hamiltonian for the measured and probe systems. By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure (POVM) of a momentum on a half line is reproduced. We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line.

タイトル：Bicolor-eliminable multiplicities on the braid arrangement
アブストラクト：
A mutiarrangement consists of a finite collection of linear hyperplanes with a multiplicity function on it. We can associate a reflexive module, called logarithmic vector fields, to a multiarrangement, and to consider its freeness is an important problem. We consider multiplicities induced from graphs with two-colored edges on the braid arrangement, and characterize their freeness by giving a generalization of the theory of eliminable, or chordal graphs which were originally introduced by Stanley.

タイトル：Study of quark confinement with Hodge decomposition approach
アブストラクト：
The quark confinement mechanism is one of the most important problems in particle physics, and is listed as the 7 difficult mathematical problems of the New Millennium by Cray Mathematics Institute. In this problem, it is suggested that the monopole singularity of the Yang-Mills field plays a crucial role. To extract the singularity, we introduce the Hodge decomposition of the Yang-Mills field, and numerically study the role of the singularity for confinement in the lattice gauge theory.

タイトル：Mathematics in information security (情報セキュリティ分野の数学)
アブストラクト：
As several kinds of information-technological applications are getting widespread, research areas concerning information security have been wider and more various as well, so are the mathematical theories applied there. This talk will survey how (even advanced) mathematics is used in recent researches on information security.

タイトル：The spectral analysis of the Hamiltonian with a singular perturbation in quantum fields theory
アブストラクト：
The GSB-Hamiltonian describes particle systems coupled to quantum fields. In this research, the GSB-Hamiltonian with a singular perturbation is considerd. It is proven that the Hamiltonian has a ground state and asymptotic fields exist, where ultraviolet and infrared cut-offs are imposed on.

タイトル：The approximation of instantons over an arbitrary closed oriented 4-manifold
アブストラクト：
A classical theorem of Runge asserts that a meromorphic function defined on a domain in the complex plain can be approximated, over compact subsets, by rational functions, i.e. by meromorphic functions on the entire Riemann sphere. In this talk we shall present two analogous results in which meromorphic functions are replaced by instantons or pseudoholomorphic curves. They generalize Donaldson's "Runge theorem for instantons over the four-sphere".

タイトル：$L^p$ energy method for scalar viscous conservation laws on the half line （単独粘性保存則に対するある初期・値境界値問題の解の漸近評価について）
アブストラクト：
We study the $L^p$ energy method of the solution to an initial boundary value problem on the half line for scalar viscous conservation law. In t his talk, it is proved that even for a wideｒ class of flux functions which are not necessarily convex, such the superposition of the stationary solution and the rarefaction wave have the same decay rate as the flux is convex, provided the rarefaction wave is weak.

タイトル：Chow rings of complex algebraic groups
アブストラクト：
The Chow ring is a certain invariant for a variety similar to the cohomology ring. In fact, in some cases it can be calculated from the cohomology ring, which is extensively studied by algebraic topologists. Combining the results on the cohomology of flag varieties with a tool from Schubert calculus, we determined the Chow ring of simple, simply-connected complex algebraic groups (joint with M.Nakagawa).

タイトル：Pin structures and spin structures on surfaces
アブストラクト：
For a closed orientable$($resp. non-orientable$)$ surface, we can always give finite numbers of spin$($resp. pin$^{\pm})$ structures.
And it is known that there are some algebraic or geometric counterpartsfor them. In this talk and article, we see their relationships and discuss some applications.

タイトル：Siegel disks with bounded type rotation number
アブストラクト：
A Siegel disk is a rotation domain in the Riemann sphere. We consider the topology and geometry of the boundary of Siegel disks. A Siegel disk of some polynomial with bounded type rotation number has the quasicircle boundary containing its critical point. In order to construct such a Siegel disk not of a polynomial but of a rational function, we consider some Blaschke product and employ the quasiconformal surgery.

タイトル：Non-uniform asymptotic stability for linear time-varying second-order differential equations
アブストラクト：
We obtain sufficient conditions that the equilibrium of second-order differential equation is "not"uniformly asymptotically stable. It is known that the equilibriums of the linear equations $$x''+\frac{1}{1+t}x'+x=0, \qquad x''+\frac{1}{1+t}x'+\frac{1}{1+t}x=0$$ are asymptotically stable respectively. We discuss that they are uniformly asymptotically stable or not.

タイトル：On toric face rings
アブストラクト：
A toric face ring is a generalization of Stanley-Reisner ring and affine semigroup rings, which was defined by Stanley, and recently is more generated by Bruns, Koch and R\"omer. In this talk, we will introduce this notion, and present some results obtained in joint work with Kohji Yanagawa.

タイトル：Homotopy type of the box complexes of graphs without 4-cycles
アブストラクト：
For a graph G, an abstract free simplicial Z_2-complex B(G), called the box complex of G, is defined by J. Matousek and G.M.Ziegler. The Z_2-index of B(G) give us a lower bound for the chromatic number of G. We are interested in the relation between the topology of B(G) and the combinatorics of G. In this speech, the box complexes of graphs without 4-cycles are characterized by a 1-dimensional Z_2-subcomplex of B(G).

タイトル：Generalized arcsine law in an infinite measure system
アブストラクト：
It is known that intermittent phenomena, such as the Rayleigh-Benard convection and Belousov-Zhabotinsky reaction, are described by infinite measure dynamical systems. We show that the time average of some observation functions converges to the generalized arcsine distribution in an infinite measure dynamical system.

タイトル：The modeling in the equations of fluid mechanics at the dawn or Navier{-}Stokes （黎明期における流体力学ないしはNavier{-}Stokes方程式のモデリング）
アブストラクト：
The original Navier-Stokes equations (NS) or Navier equations were introduced in the prime of the second period of molecules in 1821 and also Stokes’ in 1845 respectively. The heated arguments for modeling in the formulations of NS, based on or not on the then current topics on actions of molecules among the g´eom`etres about the elastic solid/fluid, were published in the then jounals. Helmholtz and Thomson discovered the theories of vorticies in 1850s, in not using NS. Moreover, “until Reynold’s and Boussinesq’s studies of turbulent flow in 1880s, NS remained completely irrelevant to hydraulics” (Darrigol). We would like to report on these modeling after fluid mechanics at the dawn, from the three points of view : two parameters in the equations of the elastic solid/fluid, the boundary conditions and vortices.

タイトル：A noncommutative algebro geometric characterization of representation type of a quiver （クイバーの表現型の非可換代数幾何学的特徴付け）
アブストラクト：
Let Q be a quiver and let \Pi be the preprojective algebra of Q. We prove that the category of representations of Q and the category \qgr\Pi are derived equivalent where \qgr\Pi is the abelian category which is considered as the category of coherent sheaves on noncommutative projective scheme \proj\Pi associated to \Pi.As a corollary we obtain "a noncommutative geometric characterization of representation type of a quiver"

タイトル：On the boundary of the moduli space of log Hodge structure
アブストラクト：
In recently work of Kazuya Kato and Sampei Usui, they constructed partial compactifications of the arithmetic quotients of Griffiths period domains by marrying the original work of Griffiths school on degenerations of Hodge structure with the theory of log geometry. They introduced a notion of log Hodge structure and showed that these partial compactifions are fine moduli of polarized log Hodge structure. I talk about some result on the boundary of this moduli spaces.

タイトル：A mathematical principle of evolutionary game theory （進化ゲーム理論の数理）
アブストラクト：
The basic concepts in the modern economics of today was constructed by the great mathematician (von Neumann, Nash, Smale, etc). In my talk, we deal with the EVOLUTIONARY GAME THEORY, pursue the new developments. It is said that it can explain the anomaly in the economics and it will be important field in the future. It is formulated rigorously and rearranged by my researches. We propose the direction of the new research (Equilibrium Concept, etc).

タイトル：Large time decay of solutions to isentropic gas dynamics
アブストラクト：
We study the large time behavior of solutions to isentropic gas dynamics. Our goal in this talk is to derive the decay of the $L1$ norm of pressure. To do this, we investigate approximate solutions constructed by a difference scheme.

タイトル：Irrational rotation algebras and real quadratic fields
アブストラクト：
Irrational rotation algebras are objects in $C^*$-algebra theory. They are also called noncommutative tori. Y. Manin proposed the use of noncommutative tori as geometric framework for the study of abelian class field theory of real quadratic fields. In this talk, we introduce $C^*$-algebra theory and explain the reason why Y. Manin proposed.

タイトル：Generating functions for multiple zeta values with fixed weight, depth and generalized height
アブストラクト：
We consider multiple zeta values defined by the Euler sums with non-strict inequalities in the summation. The notion of i-height is introduced for every natural number i. We define generating functions for sum of multiple zeta values with fixed weight, depth and i-height and prove that the function can be expressed in terms of generalized hypergeometric functions.

タイトル：Classification of polarized manifolds containing Castelnuovo manifolds as their hyperplane sections
アブストラクト：
Given a variety $A$, it is a naive problem in algebraic geometry to determine the structures of projective varieties $X$ containing $A$ as ample divisors. The problem has been extensively studied by many mathematicians when $A$ has some special structure. In this conference, I would like to treat the case where $A$ is a Castelnuovo manifold, and to announce a classification result of $X$. Here a Castelnuovo manifold is a higher-dimensional analog of a curve of maximal genus. Typical examples of those manifolds are Del Pezzo and Mukai manifolds, which are Fano manifolds whose anti-canonical divisors are high multiples of ample divisors.

タイトル：On sub-Riemannian manifolds（サブリーマン多様体の無限小自己同型について）
アブストラクト：
My subject is a sub-Riemannian geometry. A sub-Riemannian geometry is a manifold endowed with a distribution and a fibre inner product on the distribution. A distribution here means a linear subbundle of the manifold. This Riemannian fibre metric is called a Carnot Caratheodry metric, which is related an Optimal Control Theory. Now we suppose that a distribution is bracket generating (i.e., the collection of all vector fields generated by Lie brackets spans the whole tangent bunndle). In this talk we study a local classification of sub-Riemannian manifolds.

タイトル：Analysis of the alpha particle orbits in Heliotron type fusion device（ヘリオトロン型核融合装置におけるα粒子の軌道解析）
アブストラクト：
The aim of this work is to investigate the 3.5 MeV alpha particle orbits in the high beta plasma. It has been found that the so-called "chaotic orbit" alpha particles are lost before they heat fuels, i.e., their lifetimes are much shorter than the alpha-electron relaxation time. The average lifetime of all alpha particles is characterized mainly by the short lifetimes of the chaotic orbit alpha particles.

タイトル：The generalized Feynman-Kac formula with a Lebesgue-Stieltjes measure and random variables
アブストラクト：
This Feyman-Kac formula was obtained for an arbitrary Borel measure. For this talk, I will concentrate on the more physically relevant case of a measure with a finitely supported discrete measure and for simplicity, on the case of a continuous measure + a Dirac mass at a single time \tau. I will show that the function defined by the function u(t) associated with the corresponding generalized Feynman-Kac functional satisfies a suitable differential equation and integral equation. I will then deduce that u(t) has a discontinuity at \tau. Then I will randomize those discontinuities \tau and the Dirac mass.

タイトル：Boundary element analysis of tokamak plasma current profile using quasi radial basis functions with variable scaling factors
アブストラクト：
The Grad-Shafranov equation, which is the axisymmetric form of the 3-DPoisson equation, has been transformed into a boundary integral equation byintroducing the "quasi radial basis function" expansion of itsinhomogeneous source term. It has been found that excellent accuracy of thesolution can be realized when adopting space-dependent scaling factors forthe above basis functions

タイトル：Hook formulas for a generalized Young diagram
アブストラクト：
The number of standard Young tableaux of a given shape is given by the classical hook formula. In this talk, we talk on a far-reaching generalization (colored hook formula, and q-Hook formula) for a generalized Young diagram in the sense of D.Peterson.　The colored hook formula is new even for a classical Young diagram.

タイトル：Generalized Schur operators on the vector space spaned by rooted planar binary trees
アブストラクト：
We introduce new families of operators on the vector space spaned by rooted planar binary trees. We show that they are generalized Schur operators.

タイトル：Geometry of tangential distribution (接分布の幾何学)
アブストラクト：
Distributions on manifolds are studied by many researcher. In particular, Goursat and Cartan studied some distributions deeply using differential system and Lie algebra. In this talk, we explain results on geometric structure associated with distributions.

タイトル：The generic smoothness of the Gauss map and the reflexivity for a projective variety
アブストラクト：
The Gauss map is a rational map from a projective variety to the Grassmannian which assigns to a smooth point its projective tangent space. In this presentation, we study the Gauss map from the viewpoint of positive characteristic geometry. We discuss the relationship between the generic smoothness of the Gauss map and the reflexivity with respect to projective duality.

タイトル：Moment maps of isotropy representations of Hermitian symmetric spaces and the isoparametric hypersurfaces in spheres （Hermite対称空間 の等方表現の運動量写像と球面内の等径超曲面）
アブストラクト：
The isotropy representations of real Grassmannian manifolds of rank two are Hamiltonian actions. In this talk, we explain that (weighted) square norms of their moment maps are isoparametric functions. We expect that isoparametric hypersurfaces with four distinct principal curvatures in spheres are related to moment maps of group actions.

タイトル：On the shape of the stable patterns for activator-inhibitor systems in a disk
アブストラクト：
It is well-known that all the inhomogeneous steady states to a scalar reaction-diffusion equation on a convex domain in R^n are unstable. Hence if a steady state is stable, then it should be constant. When the domain is an interval, if a steady state of a shadow system of activator-inhibitor type is stable, then it should be monotone or constant. However, the shape of stable steady state of shadow systems seems to be unknown in the case of high-dimensional domains. We study the shape when the domain is a disk.

タイトル：Some relations in universal enveloping algebras of three dimensional Lie algebras
アブストラクト：
We let L be a three dimensional Lie algebra which generated by two elements x, y. And the universal enveloping algebra of L is denoted by U(L). When L is the spaclal linear Lie algebra $sl_2$ or the Heisenberg Lie algebra $H$ it has been proved that a concrete basis of U(L) is given by some $x^iy^jx^k$. We explain several relations to obtain a similar basis when L is neither $sl_2$ nor $H$. Then we also show certain characterizations of $sl_2$ and $H$.

タイトル：The plethystic program
アブストラクト：
The Plethystic Program is an efficient methhod of counting single-trace and multi-trace gauge invariant operators (GIO) of gauge theory proposed by Hanany et al. Systematically,there are three ways to count GIO, if Calabi-Yau (CY) manifold is given.The key to this program is the map called "The Plethystic Exponential " and its inverse map "The Plethystic Logarithm".This program also includes fascinating mathematical structures ;Syzygies ,Young Tableaux and Hilbert scheames.

タイトル：Study of neoclassical transport of LHD plasmas using the DCOM/NNW neoclassical transport database
アブストラクト：
We have developed a neoclassical transport database for LHD plasmas, DCOM/NNW. The mono-energetic diffusion coefficients are evaluated based on the Monte Carlo method by DCOM code and the mono-energetic diffusion coefficient database is constructed using a neural network technique. In this study, the database construction related to the plasma beta is investigated. The improved DCOM/NNW database is applied finite beta LHD plasma, and the neoclassical transport coefficients and the ambipolar radial electric field are evaluated.

タイトル：An introduction of the theory of reproducing kernel Hilbert spaces
アブストラクト：
This talk is oriented to the theory of reproducing kernel Hilbert spaces. First, I shall present the definition of Hilbert spaces and then I will develop a brief theory of the reproducing kernel Hilbert spaces. Some elementary examples are presented in this talk. If time permits, I will allude to my own results.

タイトル：A distinguished bounded realization and a symmetry characterization of homogeneous bounded domains
アブストラクト：
Stefan Bergman defined the Bergman mappings for bounded domains, which give the Riemann mappings in the one-dimensional case. We show that if a bounded domain is homogeneous, its Bergman mapping works very well. As an application, we prove that a homogeneous bounded domain is symmetric if and only if the image of the Bergman mapping is convex.