2019年度年会(於:東京工業大学)

SUMMARY: The boundary layer analysis for high Reynolds number flows is a fundamental theme in fluid mechanics. The concept of the boundary layer was first proposed by L. Prandtl in 1904. The basic idea is to decompose the fluid region into two parts: the one is called the inner region or the boundary layer, where the effect of the viscosity remains significant, while the other is called the outer flow, in which the flow is approximated by the inviscid flow. The aim of this talk is to review recent progress on the mathematical analysis of this problem, with particular focus on the stability/instability of the shear boundary layers.

SUMMARY: In the history of the theory of singularities of smooth mapping, the notion of \(\mathcal {A}\)-equivalence (i.e. right-left equivalence or isomorphism) among smooth map germs in the sense of Mather is the most natural equivalence from the view point of differential topology. In order to solve the structural stability problems of Thom, Mather (around 1970) also introduced the notion of \(\mathcal {K}\)-equivalence, which played a key role in his theory. We remark that map germs can be considered as local sections of trivial (vector) bundles and \(\mathcal {K}\)-equivalence is naturally interpreted as an equivalence relation among section germs of a vector bundle.

In this talk, we consider the case when the target space or the corresponding vector bundle have geometric structures. Firstly, we consider equivalence relations among smooth map germs with respect to \(G\)-structures on the target space germ. These equivalence relations are natural generalization of \(\mathcal {A}\)-equivalence depending on geometric structures on the target space germ. Unfortunately, these equivalence relations are not necessarily geometric subgroups in the sense of Damon (1984). This means that the Thom–Mather theory of singularities cannot work properly. However, we have several interesting applications of these equivalence relations, including differential geometry of singular surfaces. We also consider equivalence relations among smooth section germs of vector bundle germs with respect to structure groups. This equivalence relation is a slight generalization of \(G\)-equivalence introduced by Tougeron (1972) as a generalization of \(\mathcal K\)-equivalence. However, Tougeron had never mentioned examples of \(G\)-equivalence except the original \(\mathcal K\)-equivalence and \(\mathcal R\)-equivalence. We give several interesting applications of this equivalence, including quantum physics (chemistry), determinantal singularities, etc. This equivalence is a geometric subgroup in the sense of Damon, so that the main techniques of the Thom–Mather theory can work properly.

SUMMARY: Mathematical onocology is a fusion of mathematical science and medical science. I show several mathematical methods using data science, mathematical modeling, and numerical methods, then medical outcomes in both fundamental and clinical, that is, maglignant signaling, drug resistence, bone metabolism, and angiogenesis.

SUMMARY: Classical logic is a standard framework for formalizing mathematical reasoning. From semantical point of view, it divides mathematical statements into true statements and false ones, in principle. On the other hand, if one will apply classical logic to the analysis of logical reasonings in general, e.g. reasonings in everyday life, one may sometimes feel it uncomfortable and inappropriate. This comes from the fact that the truth in these reasonings will depend often on time, situations, resources and accessible information. Nonclassical logic is a syntactic and semantical study of various logical reasonings, which attracts interest from philosophers to computer scientists nowadays. In my talk, beginning with examples of logical reasonings of different kind I will show how algebraic methods are effectively applied and lead us to a unified understanding of these reasonings.

SUMMARY: Now that mathematical science is all the rage, and growing rapidly to be an everyday tool to support human prediction, forecast, and decision making in many business fields. Mathematically, we usually solve classical, well-known, and even trivial problems using computers. But interestingly enough, when we pursuit speed, precision or stability, that are usually required in the business context, we cannot commit anything without exploiting special technical tips, algorithms or theory. These are the great achievements our forerunners have left behind in the mathematical community. In this talk, we introduce the achievements that help us greatly on the implementation of mathematical algorithms.

SUMMARY: Weakening the additivity for classical measures, which is a natural assumption, to monotonicity, we can describe various processes that have uncertainties controlled by complicated interactions. Lebesgue type integral theory cannot be applied to a monotone measure because of its non-additivity, so several types of integrals including our inclusion-exclusion integral have been proposed and used in many applied studies because they have some preferable and useful properties. Moreover, they allow interpretation by the measure. In this talk, I introduce definitions of nonlinear integrals and describe various relationships between these integrals and other several concepts in other areas corresponding to non-linear integral. In the latter half, I show a method of data analysis applied a non-linear integral using general statistical tools, and results of our attempt of applying to machine learning for big data analysis.

SUMMARY: Knot theory and 3-dimentional topology theory have been successfully applied to polymer science. In this talk we will discuss application to polymer material design, DNA topology, and topological polymers.

We will discuss the structure of polymer materials using decompositions of the 3-dimensional torus. Poly-continuous pattern of block copolymers can be studied and characterized by using networks, branched surfaces and decomposition of the 3-dimensional torus. Enzymatic action of DNA and structure of multi-cyclic polymers can be analyzed using knot theory and spatial graph theory. We will discuss some of the recent developments.

SUMMARY: The arithmetic (resp. geometric) Bogomolov conjecture is a problem in Diophantine geometry over a number field (resp. a function field). Let \(X\) be a closed subvariety of an abelian variety over a number field (resp. a function field). Then this conjecture asserts that if \(X\) has a dense set of points with small canonical height, then it is a torsion (resp. special) subvariety. While the arithmetic Bogomolov conjecture was established by Ullmo and Zhang as a theorem in 1998, the geometric Bogomolov conjecture was widely open at that time.

The proof of the arithmetic version of the conjecture uses a measure theoretic approach on complex analytic spaces associated to an archimedean place of a number field. The key was an equidistribution theorem of small points, which asserts that a dense set of points with small height are equidistributed in the complex space with respect to the so-called canonical measure.

It was natural to wish an analogue of the proof of the arithmetic version to work in the geometric setting. However, since a function field never has an archimedean place, there is no way to use complex analytic spaces in the geometric setting. To make an analogue, therefore, we need counterparts of complex analytic spaces and the canonical measures on the spaces over an archimedean place.

Here, nonarchimedean analytic geometry, which is developed over a nonarchimedean valued field, can provide us with powerful tools. Since a function field has nonarchimedean places, we can enjoy it (even) in the geometric setting. In fact, the usage of analytic spaces in the sense of Berkovich and measures introduced by Chambert–Loir on them has made remarkable contributions to the conjecture.

In this talk, we will begin by recalling what the Bogomolov conjecture is, and then we will review the approach used in the proof of arithmetic version. Finally, we will explain the nonarchimedean counterparts that can be used in the proof of a partial but important result on the geometric Bogomolov conjecture.

SUMMARY: Conformal field theory admits an infinite family of commuting Hamiltonians known as integrals of motion (IM). After \(q\)-deformation, quantum toroidal algebras emerge as the symmetry underpinning their integrability. We give a survey about this subject, including the construction of deformed IM and the description of their spectrum, as well as open questions.