Department of Mathematics - sci.osaka-u.ac.jp¼ˆ2019-2020）.pdf · geometry of nilpotent groups to the geometry of solvable groups. More precisely, I have studied the cohomology

Department of Mathematics

Department of MathematicsDepartment of Mathematics

Graduate School of Science2019-2020

SummaryMathematics, originally centered in the concepts of

number, magnitude, and form, has long been growingsince ancient Egyptian times to the 21st century.Through the use of abstraction and logical reasoning, itbecame an indispensable tool not only in naturalsciences, but also in engineering and social sciences.Recently, the remarkable development of computers isnow making an epoch in the history of mathematics.

Department of Mathematics is one of the sixdepartments of Graduate School of Science, OsakaUniversity. It consists of 6 research groups, all ofwhich are actively involved in the latest developmentsof mathematics. Our mathematics department hasranked among the top seven in the country.

The department offers a program with 32 newstudents enrolled annually leading to post-gradutedegrees of Masters of Science. The department alsooffers a Ph.D. program with possibly 16 new studentsenrolled annually.

Graduate courses are prepared so as to meet variousdemands of students. Besides introductory courses forfirst year students, a number of topics courses aregiven for advanced students. Students learn morespecialized topics from seminars under theguidance of thesis advisors.

Our department has our own library equipped withabout 500 academic journals and 50,000 books inmathematics, both of which graduate students can usefreely. Also by an online system, students as well asfaculty members can look up references throughInternet.

Research GroupsAlgebra, Geometry, Analysis, Global Geometry &Analysis, Experimental Mathematics, MathematicalScience.

Areas of ResearchNumber Theory, Ring Theory, Algebraic Geometry,Algebraic Analysis, Partial Differential Equations,Real Analysis, Differential Geometry, ComplexDifferential Geometry, Topology, Knot Theory,Discrete Subgroups, Transformation Groups, ComplexAnalysis, Complex Functions of Several Variables,Complex Manifolds, Discrete Mathematics, Probability

Theory, Dynamical Systems, Fractals, MathematicalEngineering, Information Geometry, MathematicalPhysics.

Faculty MembersProfessors (15)Shin-ichi DOI, Osamu FUJINO, Akio FUJIWARA, Ryushi GOTO, Nakao HAYASHI, Masashi ISHIDA, Seiichi KAMADA, Soichiro KATAYAMA, Takehiko MORITA, Hiroaki NAKAMURA, Shin-ichi OHTA, Hiroshi SUGITA, Atsushi TAKAHASHI, Takao WATANABE, Katsutoshi YAMANOI.

Associate Professors (14)Shinpei BABA, Ichiro ENOKI, Kento FUJITA, Hisashi KASUYA, Eiko KIN, Haruya MIZUTANI, Tomonori MORIYAMA, Tadashi OCHIAI, Shinnosuke OKAWA, Yuichi SHIOZAWA, Hideaki SUNAGAWA, Naohito TOMITA, Motoo UCHIDA, Seidai YASUDA.

Lecturer (1)Kazunori KIKUCHI.

Assistant Professors (9)Yasuhiro HARA, Takahisa INUI, Takao IOHARA, Ryo KANDA, Erika KUNO, Yoshihiko MATSUMOTO, Hiroyuki OGAWA,Koji OHNO, Kenkichi TSUNODA.

Cooperative Members in Osaka UniversityProfessors (7)Susumu ARIKI, Daisuke FURIHATA, Takayuki HIBI, Katsuhisa MIMACHI, Yoshie SUGIYAMA, Katsuhiro UNO, Masaaki WADA.

Associate Professors (5)Tsuyoshi CHAWANYA, Akihiro HIGASHITANI, Yuto MIYATAKE, Yoshiki OSHIMA, Kouichi YASUI.

Home Pagehttp://www.math.sci.osaka-u.ac.jp/eng/

Department of Mathematics Shinpei BABA

Low-dimensional Geometry, Topology

My research is centered on surfaces (i.e. 2-dimensional manifolds), which arefundamental object in geometry and topology.

In particular, I am interested in the relationsbetween geometric structures (locallyhomogeneous structures) on surfaces, andrepresentations of the fundamental groups ofsurfaces S (surface groups) into Lie groups G.

In the case that the Lie groups G = PSL(2,R) or PSL(2, C) and that representations havediscrete images, beautiful theories have beendeveloped extensively, in particular, inrelation to the classification and thedeformation theorem of 2- and 3 -dimensional manifolds.


Department of Mathematics Shin-ichi DOI

Partial Differential Equations

Partial differential equations have theirorigins in various fields such as mathematicalphysics, differential geometry, and technology.Among them I am particularly interested inthe partial differential equations that describewave propagation phenomena: hyperbolicequations and dispersive equations. A typicalexample of the former is the wave equation,and that of the latter is the Schroedingerequation. For many years I have studied basicproblems for these equations: existence anduniqueness of solutions, structure ofsingularities of solutions, asymptotic behaviorof solutions, and spectral properties. RecentlyI make efforts to understand how thesingularities of solutions for Schroedingerequations or, more generally, dispersive

equations propagate. The center of thisproblem is to determine when and how thesingularities of solutions for the dispersiveequations can be described by the asymptoticbehavior of solutions for the associatedcanonical equations.


Department of Mathematics Ichiro ENOKI

Complex Differential Geometry

A complex manifold is, locally, the world build out ofopen subsets of complex Euclid spaces andholomorphic functions on them. If two holomorphicfunctions are defined on a connected set and coincideon an open subset, then they coincide on the whole.Complex manifolds inherit this kind of property fromholomorphic functions. That is, they are stiff and hardin a sense. It seems to me that complex manifolds arenot metallically hard but have common warm feelingwith wood or bamboo, which have grain and gnarl.Analytic continuations, as you learned in the course onthe function theory of complex variables, is analogousto the process of growth of plants. Instead ofconsidering whole holomorphic functions, a class ofcomplex manifold can be build out of polynomials.This is the world of complex algebraic manifolds, themost fertile area in the world of complex manifolds.To complex algebraic manifolds, since they arealgebraically defined, algebraic methods are of courseuseful to study them. In certain cases, however,transcendental methods (the word "transcendental"

means only "not algebraic") are powerful. For example,one of the simplest proof for the fundamental theoremof algebra is given by the function theory of onecomplex variable. These two methods have beencompeting with each other since the very beginning ofthe history of the study of complex manifolds. Thiscompeting seems to me the prime mover of thedevelopment of the theory of complex manifolds.Comparing the world of complex manifold to the earth,the world of complex algebraic manifold is to compareto continents, and the boundary to continental shells.The reason I wanted to begin to study complexmanifolds was I heard the Kodaira embedding theorem,which characterizes complex algebraic manifolds inthe whole complex manifolds. The place I begin tostudy is, however, something like the North Pole or theMariana Trench. Now the center of my interest is inthe study of complex algebraic manifolds bytranscendental methods. (Thus I have reached land butI found this was a jungle.)


Department of Mathematics Osamu FUJINO

Algebraic Geometry

I am mainly interested in algebraic geometry.More precisely, I am working on thebirational geometry of higher-dimensionalalgebraic varieties. In the early 1980s,Shigefumi Mori initiated a new approach forhigher-dimensional birational geometry,which is now usually called the MinimalModel Program or Mori theory. Unfortunately,this beautiful approach has not beencompleted yet. One of my dreams is tocomplete the Minimal Model Program in fullgenerality. I am also interested in toricgeometry, Hodge theory, complex geometry,and so on.


Department of Mathematics Kento FUJITA

Algebraic Geometry

My research interest is algebraic geometry.Especially, I am interested in birationalbehavior of Fano varieties, a special class ofalgebraic varieties. I have researched theMukai conjecture, the existence of certaingood models over some reducible varieties,and an algorithm to classify log del Pezzosurfaces (joint work with Kazunori Yasutake),etc. Recently, I am interested in K-stability ofFano varieties; I (and independently Chi Li)gave a birational interpretation of K-stabilityof Fano varieties.


Department of Mathematics Akio FUJIWARA

Mathematical Engineering

"What is information?" Having this naiveyet profound question in mind, I have beenworking mainly on noncommutative statistics,information geometry, quantum informationtheory, and algorithmic randomness theory.

One can regard quantum theory as anoncommutative extension of the classicalprobability theory. Likewise, quantumstatistics is a noncommutative extension ofthe classical statistics. It aims at finding thebest strategy for identifying an unknownquantum object from a statistical point ofview, and is one of the most exciting researchfield in quantum information science.

Probability theory is usually regarded as abranch of analysis. Yet it is also possible toinvestigate the space of probability measures

from a differential geometrical point of view.Information geometry deals with a pair ofaffine connections that are mutually dual(conjugate) with respect to a Riemannianmetric on a statistical manifold. It is knownthat geometrical methods provide us withuseful guiding principle as well as insightfulintuition in classical statistics. I am interestedin extending information geometricalstructure to the quantum domain, admittingan operational interpretation.I am also delving into algorithmic and game-

theoretic randomness from an informationgeometrical point of view. Someday I wish toreformulate thermal/statistical physics interms of algorithmic information theory.


Department of Mathematics Ryushi GOTO

Geometry

My research interest is mostly in complexand differential geometry, which are closelyrelated with algebraic geometry andtheoretical physics. My own research startedwith special geometric structures such asCalabi-Yau, hyperKaehler, G2 and Spin(7)structures. These four structures exactlycorrespond to special holonomy groups whichgive rise to Ricci-flat Einstein metrics onmanifolds. It is intriguing that these modulispaces are smooth manifolds on which localTorelli type theorem holds. In order tounderstand these phenomena, I introduce anotion of geometric structures defined by asystem of closed differential forms andestablish a criterion of unobstructeddeformations of structures. When we apply

this approach to Calabi-Yau, hyperKaehler,G2 and Spin(7) structures, we obtain a unifiedconstruction of these moduli spaces. Atpresent I also explore other interestinggeometric structures and their moduli spaces.


Department of Mathematics Yasuhiro HARA

Topology

The field of my study is topology and,especially, I study the theory oftransformation groups. The Borsuk-Ulamtheorem is one of famous theorems abouttransformation groups. This theorem is oftentaken up as an application in elementarylectures about the homology theory. Thecontent of the theorem is as follows: for everycontinuous map from the n-dimensionalsphere to the n-dimensional Euclidian space,there exists a point such that the map takesthe same value at the point and at theantipodal point. A famous application of thistheorem is the following. ''On the earth, thereis a point such that the temperature andhumidity at the point are the same as those atthe antipodal point.'' We consider a free

action of a group of order two on the n-dimensional sphere to prove the Borsuk-Ulamtheorem. Then for any equivariant map (anycontinuous map which preserves the structureof the group action) from the sphere to itself,the degree of the map is odd. By using thisfact, we obtain the Borsuk-Ulam theorem. Inthe case of the Borsuk-Ulam theorem, weconsider spheres and free actions of a groupof order two. Actually, when we considerother manifolds and actions of other groups,there are some restrictions of homotopy typesof equivariant maps. I study such restrictionsof homotopy types of equivariant maps byusing the cohomology theory, and I studyrelationships between homotopy types ofequivariant maps and topological invariants.


Department of Mathematics Nakao HAYASHI


I am interested in asymptotic behavior in time ofsolutions to nonlinear dispersive equations (1 Dnonlinear Schreodinger, Benjamin-Ono,Korteweg-de Vries, modified Korteweg-de Vries,derivative nonlinear Schreodinger equations) andnonlinear dissipative equations Complex Landau-Ginzburg equations, Korteweg-de Vries equationon a half line, Damped wave equations with acritical nonlinearity). These equations haveimportant physical applications. Exact solutions ofthe cubic nonlinear Schreodinger equations andKorteweg-de Vries can be obtained by using theinverse scattering method. Our aim is to studyasymptotic properties of these nonlinear equationswith general setting through the functionalanalysis. We also study nonlinear Schreodingerequations in general space dimensions with acritical nonlinearity of order 1+2/n and the

Hartree equation, which is considered as a criticalcase and the inverse scattering method does notwork. On 1995, Pavel I. Naumkin and I started tostudy the large time behavior of small solutions ofthe initial value problem for the non-lineardispersive equations and we obtained asymptoticbehavior in time of solutions and existence ofmodified scattering states to nonlinearSchreodinger with critical and subcriticalnonlinearities. It is known that the usual scatteringstates in L2 do not exist in these equations.Recently, E.I.Kaikina and I are studying nonlineardissipative equations (including Korteweg-deVries) on a half line and some results concerningasymptotic behavior in time of solutions areobtained.


Department of Mathematics Takahisa INUI

Nonlinear Partial Differential Equations

My research subject is nonlinear partialdifferential equations. Especially, I aminterested in the global behavior of thesolutions to nonlinear dispersive equations ornonlinear wave equations.Dispersive equations describe the dispersion

phenomena of waves and are basic equationsin quantum physics. For example,Schroedinger equation and Klein--Gordonequation are typical dispersive equations,which appear in quantum physics orrelativistic quantum field theory. Waveequations express the properties of motion inwaves. Considering nonlinear interactionsbetween particles,we can treat various physical phenomena,

for example, optics, superconductivity, and

Bose-Einstein condensate, by the nonlineardispersive or wave equations. Nonlineardispersive or wave equations have twoproperties.One is dispersion and the other is

nonlinearity. They conflict each other. Thus,there are so many behaviors of the solutions. Istudy this subject to find all behavior.


Department of Mathematics Takao IOHARA


My research interest is concerned withnonlinear partial differential equationsappearing in fluid mechanics. The currentresearch topic is the equations of the motionof viscous incompressible fluid which hasfree moving surface. The motion of viscousincompressible fluid is governed by theNavier-Stokes equations, which are not easyto solve because of their nonlinearity. Thefree moving surface adds another nonlinearityto the problem and the study of it needs moreelaborate technique than the problems onfixed domain.


Department of Mathematics Masashi ISHIDA

Differential Geometry

My research interest is in geometry,particularly, interaction between topology anddifferential geometry. For instance, I amstudying the nonexistence problems ofEinstein metrics and Ricci flow solutions on4-manifolds by using the Seiberg-Wittenequations. I am also interested in thegeometry of the Yamabe invariant. Thecomputation of the Yamabe invariant for agiven manifold is a difficult problem ingeneral. By using the Seiberg-Wittenequations, I determined the exact value of theYamabe invariant for a large class of 4-manifolds which includes complex surfacesas special cases. Furthermore, I am alsointerested in both the Ricci flow in higherdimension and some generalized versions of

the Ricci flow like the Ricci Yang-Mills flow.The Ricci flow was first introduced by R.Hamilton in 1981 and used as the main tool inG. Perelman's solution of the Poincareconjecture in 2002. Perelman introducedmany new and remarkable ideas to prove theconjecture. The theory developed byHamilton and Perelman is now called theHamilton-Perelman theory. One of my recentinterests is to investigate geometric analyticalproperties of the generalized versions of theRicci flow from the Hamilton-Perelmantheoretical point of view.


Department of Mathematics Seiichi KAMADA

Topology

"charts".A quandle is an algebra with a binary

operation which satisfies three axiomscorresponding to Reidemeister moves in knottheory. A homology theory of quandles wasestablished and we can use it for constructionof invariants for surface-links. 2-dimensional braids, or surface braids, are ageneralization of classical braids. They are inone-to-one correspondence to orientedsurface-links in 4-space modulo certain basicsmoves. As a method of describing 2-dimensional braids, a chart description wasintroduced. This method has been extended ina general theory of charts. For example,charts can be used for studies of Lefschetsfibrations of 4-manifolds.

My research area is topology (geometrictopology-- knot theory, low dimensionaltopology). I have been working on surface-links in 4-space, including 2-knots and 2-links, 2-dimensional braids, braided surfaces,and 4-manifolds.Surface-links are closed surfaces smoothly

embedded in the 4-space, and they areconsidered equivalent if they are ambientlyisotopic. It is a quite difficult problem todecide whether two given surface-links areequivalent or not. Invariants are often usedto show that two surface-links are notequivalent. However, few invariants areknown for surface-links. I am interested instudying surface-links and their invariantsusing "quandles" and graphics so-called


Department of Mathematics Ryo KANDA

Ring Theory

My research area is ring theory. A ring is analgebraic structure which has addition, subtraction, andmultiplication. Typical examples are the set of integers,the set of polynomials, and the set of n-by-n matrices. Iam particularly interested in noncommutative rings,whose multiplication is noncommutative.

The notion of modules plays an important role in thestudy of a ring. It is a generalization of vector spacesappearing in linear algebra, but the only difference inthe definition is that the coefficients live in a ring, notnecessarily a field such as the field of real/complexnumbers. A vector space over an arbitrary coefficientfield is determined by the cardinality of its basis, up toisomorphism. On the other hand, in the case of a ring,even the nature of finitely generated modules highlydepends on the structure of the ring. For this reason,we can investigate the ring by looking the behavior ofits modules. Especially for noncommutative rings, thisapproach is often clearer than looking the ring itselfdirectly, and this leads us to deeper results.

An abelian category is a further generalization of the

collection of modules. For each ring, the collection ofits modules has the structure of an abelian category,and the ring can be almost recovered by the categoricalstructure. A similar thing holds for the abelian categoryconsisting of coherent sheaves on an algebraic variety.Hence the notion of abelian categories is a largeframework including (noncommutative) rings and(commutative) algebraic varieties. I have investigatedgeneral properties of certain classes of abeliancategories, and have revealed several new properties ofnoncommutative rings as consequences. An advantageof this general setting is that we can consider similarproblems for abelian categories which are not obtainedas module categories over rings. For example, thefunctor category, which is the category consisting offunctors from a given abelian category, is again anabelian category, and its structure reflects homologicalproperties of the original abelian category. I expectthat, by considering naive questions arising fromgeneral theory of abelian categories in a specificsetting, such as the functor category, we can extendexisting theories to new directions.


Department of Mathematics Hisashi KASUYA

Geometry

Until now, I have tried to extend thegeometry of nilpotent groups to the geometryof solvable groups. More precisely, I havestudied the cohomology theory ofhomogeneous spaces of solvable Lie groupsand complex geometry of non-Kählermanifolds. It seems that the gap betweennilpotent groups and solvable groups is small.But this gap contains a potential for geometry.By the growing out of left-invariance andnon-triviality of local system cohomology, Isucceeded in giving a great surprise.Recently, I am interested in the geometry

which relates to reductive or semi-simplegroups in contrast to nilpotent or solvablegroups In particular, I study non-abelianHodge theory, variations of hodge structures,

lattices in semi-simple Lie groups and locallyhomogeneous spaces.


Department of Mathematics Soichiro KATAYAMA


My research interest is in nonlinear partialdifferential equations. To be more specific, Iam working on the initial value problem fornonlinear wave equations (in a narrow sense),and also for partial differential equationsdescribing the nonlinear wave propagation ina wider sense, such as Klein-Gordonequations and Schroedinger equations.The initial value problem is to find a

solution to a given partial differentialequation with a given state at the initial time(a given initial value). However, in general, itis almost impossible to give explicitexpression of solutions to nonlinear equations.Therefore, in the mathematical theory, it isimportant to investigate the existence ofsolutions and also their behavior when they

exist.If we consider the initial value problem for

the equations mentioned above and if theinitial value is sufficiently small, theexistence of solutions up to arbitrary time (theexistence of global solutions) is mainlydetermined by the power of the nonlinearity.Especially, when the nonlinearity has thecritical power, the existence and non-existence of global solutions depend also onthe detailed structure of the nonlinear terms. Iam interested in this kind of critical case, andstudying sufficient conditions for theexistence of global solutions and theirasymptotic behavior.


Department of Mathematics Kazunori KIKUCHI

Differential Geometry

I have been studying topology of smooth four-dimensional manifolds, in particular interested inhomology genera, representations ofdiffeomorphism groups to intersection forms, andbranched coverings. Let me give a simpleexplanation of what interests me the most, orhomology genera. The homology genus of a smoothfour-dimensional manifold M is a map associating toeach two-dimensional integral homology class [x] ofM the minimal genus g of smooth surfaces in M thatrepresent [x]. For simplicity, reducing thedimensions of M and [x] to the halves of themrespectively, consider as a two-dimensionalmanifold the surface of a doughnut, or torus T, and aone-dimensional integral homology class [y] of T.Draw a meridian and a longitude on T as on theterrestrial sphere, and let [m] and [l] denote thehomology classes of T represented by the meridianand the longitude respectively. It turns out that [y] =a[m] + b[l] for some integers a and b, and that [y] is

represented by a circle immersed on T with onlydouble points. Naturally interesting then is thefollowing question: what is the minimal number n ofthe double points of such immersions representing[y]? Easy experiments would tell you that, forexample, n = 0 when (a,b) = (1,0) or (0,1) and n = 1when (a,b) = (2,0) or (0,2). In fact, it is proved withtopological methods that n = d−1, where d is thegreatest common divisor of a and b. It is the minimalnumber n for T and [y] that corresponds to theminimal genus g for M and [x]. The study on theminimal genus g does not seem to proceed with onlytopological methods; it sometimes requires methodsfrom differential geometry, in particular methodswith gauge theory from physics; though moredifficult, it is more interesting to me. I have beentackling the problem on the minimal genus g withsuch a topological way of thinking as to see thingsas if they were visible even though invisible.


Department of Mathematics Eiko KIN

Topology, Dynamical Systems

I am interested in the mapping class groups onsurfaces. The most common elements in the mappingclass groups are so called pseudo-Anosovs. I try tounderstand which pseudo-Anosovs are the most simplein the mapping class groups. I describe my goal moreclearly. Pseudo-Anosovs possess many complicated(and beautiful) properties from the view points of thedynamical systems and the hyperbolic geometry. Thereare some quantities which reflect those complexities ofpseudo-Anosovs. Entropies and volumes (i.e., volumesof mapping tori) are examples. We fix the topologicaltype of the surface and we consider the set of entropies(the set of volumes) coming from the pseudo-Anosovelements on the surface. Then one can see that thereexists a minimum of the set. That is, we can talk aboutthe pseudo-Anosovs with the minimal entropies(pseudo-Anosovs with the minimal volumes). I wouldlike to know which pseudo-Anosov achieves theminimal entropy with the minimal volume. Recently,Gabai, Meyerhoff and Milley determined hyperbolic

closed 3-manifolds and hyperbolic 3-manifolds withone cusp with very small volume. Intriguingly, theresult implies that those hyperbolic 3-manifolds areobtained from the single hyperbolic 3-manifold byDehn filling. Some experts call the single 3-manifoldthe ``magic manifold". Said differently, the magicmanifold is a parent manifold of the hyperbolicmanifolds with very small volume. It seems likely thatwe have the same story in the world of pseudo-Anosovs with the very small entropies. This conjectureis based on the recent works of myself and otherspecialists. We note that there are infinitely manytopological types of surfaces (for example, the familyof closed orientable surface with genus g). For themapping class group of each surface we know thatthere exists a pseudo-Anosov element with theminimal entropy. Thus, of course, there exist infinitelymany pseudo-Anosov elements with the minimalentropies. It might be true that all minimizers areobtained from the magic 3-manifold. When I work onthis project, I sometimes think of our universe.


Department of Mathematics Erika KUNO

Topology

My research theme is mainly mapping classgroups of surfaces (including non-orientablesurfaces) and 3-dimensional handlebodies. Inparticular, I am interested in exploring themfrom the viewpoints of geometric grouptheory. Geometric group theory is a new fieldamong a lot of areas of mathematics and it isprogressing significantly. One of the mostimportant problems in geometric grouptheory is classifying finitely generated groupsby “quasi-isometries”. Two finitely generatedgroups are quasi-isometric if roughlyspeaking, their word metrics are the same upto linear functions. An interesting part of thegeometric group theory is that the propertiesof the infinite groups are revealed one by oneby measuring with a coarse scale of quasi-

isometries, but not isometries. Currently,groups which are quasi-isometric to mappingclass groups have hardly been found. Thenwhat I am wondering is the question "Whichgroups are quasi-isometric to the mappingclass group?". Based on this big theme, Iwould like to elucidate properties of mappingclass groups, and deepen their understanding.


Department of Mathematics Yoshihiko MATSUMOTO

Differential Geometry, Several Complex Variables

Working on differential geometry, partly with some flavorof function theory of several complex variables. I’ve beenmainly studying geometry of “asymptotically complexhyperbolic spaces,” with emphasis on a partial differentialequation called Einstein’s equation on them. While being ageneralization of geometry of bounded strictly pseudoconvexdomains in function theory of several complex variables, it’sbeyond the scope of the field of complex geometry. Based onthis experience, I’m now aiming toward some more generaltheory that applies to other “spaces that converge to oneswith much symmetry,” which are called “asymptoticallysymmetric spaces.”

Asymptotically hyperbolic spaces, which are the most basicexamples of asymptotically symmetric spaces, are lackingsymmetries such as the homogeneity or the isotropicity of thegenuine hyperbolic spaces in the strict sense. However, as apoint in the space moves more away from a fixed one, itsneighborhood looks more like an open set of the hyperbolicspace. Recall the “natural” conformal structure on the sphereat infinity of the hyperbolic space—the boundary at infinityof an asymptotically hyperbolic space is equipped with aconformal structure in the same way. In the case ofasymptotically complex hyperbolic spaces, whose model has

little less isotropicity than that of the hyperbolic space, theassociated geometric structure on the boundary at infinity isthe CR (Cauchy–Riemann) structure.

The fundamental question of geometry of asymptoticallysymmetric spaces is the following: what property of thespace reflects the geometric structure at infinity and thetopology of the space, and in what way? This includes thequestion whether or not there is a space with some certainproperty under a given condition on the structure at infinityand the topology. The existence problem of Einstein metricsis a typical example.

Although asymptotically hyperbolic spaces have beenstudied for several decades, many fundamental questionsremain unsolved. And, if we think of understandings fromthe general viewpoint of geometry of asymptoticallysymmetric spaces, our field still seems to be kind of awilderness. I’d cultivate it by going back and forth betweenanalyses on special cases and abstract considerations.


Department of Mathematics Haruya MIZUTANI


The Schrödinger equation is the fundamentalequation of physics for describing quantummechanical behavior. I am working on themathematical theory of the Schrödingerequation and my research interest includesscattering theory, semiclassical analysis,spectral theory, geometric microlocal analysisand so on. My current research has focusedon various estimates such as decay orStrichartz inequalities, which describedispersive or smoothing properties ofsolutions and are fundamental for studyinglinear and nonlinear dispersive equations. Inparticular, I am interested in understandingquantitatively the influence of the geometryof associated classical mechanics on the

behavior of quantum mechanics, via suchinequalities.


Department of Mathematics Takehiko MORITA

Ergodic Theory, Probability Theory, Dynamical System

I specialize in ergodic theory. To be moreprecise, I am studying statistical behavior ofdynamical systems via thermodynamicformalism and its applications.Ergodic theory is a branch of mathematicsthat studies dynamical systems withmeasurable structure and related problems. Itsorigins can be found in the work ofBoltzmann in the 1880s which is concernedwith the so called Ergodic Hypothesis.Roughly speaking the hypothesis wasintroduced in order to guarantee that thesystem considered is ergodic i.e. the spaceaverages and the long time averages of thephysical observables coincide. Unfortunately,it turns out that dynamical systems are notalways ergodic in general. Because of such a

background, the ergodic problem (= theproblem to determine a given dynamicalsystem is ergodic or not) has been one of theimportant subjects since the theory came intoexistence. In nowadays ergodic theory hasgrown to be a huge branch and hasapplications not only to statistical mechanics,probability, and dynamical systems but alsoto number theory, differential geometry,functional analysis, and so on.


Department of Mathematics Tomonori MORIYAMA

Number Theory

I am interested in automorphic forms ofseveral variables. A classical automorphic(modular) form of one variable is aholomorphic function on the upper half planehaving certain symmetry. Such functionsappear in various branches of mathematics,say notably number theory, and have beeninvestigated by many mathematicians.There is a family of manifolds calledRiemannian symmetric spaces, which is ahigher-dimensional generalization of theupper half plane. The set of isometries of aRiemannian symmetric space forms a Liegroup G. Roughly speaking, an automorphicform of several variables is a function on aRiemannian symmetric space satisfying therelative invariance under an "arithmetic"

subgroup of G and certain differentialequations arising from the Lie group G.Studies on automorphic forms of severalvariables started from C. L. Siegel's works in1930s and have been developed throughinteraction with mathematics of the day.Currently I am working on two themes: (i)the zeta functions attached to automorphicforms and (ii) explicit constructions ofautomorphic forms, by employingrepresentation theory of reductive groupsover local fields. One of the joy in studyingthis area is to discover a surprisingly simplestructure among seemingly complicatedobjects.


Department of Mathematics Hiroaki NAKAMURA

Number Theory

Theory of equations has a long history ofthousands of years in mathematics, and,passing publication of the famous Cardano-Ferrari formulas in Italian Renaissance,Galois theory in the 19th century establisheda necessary and sufficient condition for analgebraic equation to have a root solution interms of its Galois group. My researchinterest is a modern version of Galois theory,especially its arithmetic aspects. In the lastcentury, the notion of Galois group wasgeneralized to "arithmetic fundamentalgroup" by Grothendieck, and Belyi'sdiscovery (of an intimate relationshipbetween Galois groups of algebraic numbersand fundamental groups of topological loopson hyperbolic curves) undertook a new area

of "anabelian geometry". Here are importantproblems of controlling a series of covers ofalgebraic curves and their moduli spaces, andIhara's theory found deep arithmeticphenomena therein. Related also toDiophantus questions on rational points,fields of definitions and the inverse Galoisproblem, nowadays, there frequently occurimportant developments as well as newunsolved problems. I investigate these topics,and hope to find new perspectives for deeperunderstanding of the circle of ideas.


Department of Mathematics Tadashi OCHIAI

Arithmetic Geometry

I study Number theory and ArithmeticGeometry. In the research of Number theory,we study not only properties of integers andrational numbers, but every kinds of problemsrelated to integers. For example, we are verymuch interested in rational points of algebraicvarieties defined over the field of rationalnumbers. Number theory has a long historyand we had a great progress, which is typicalin the proof of Fermat's conjecture by Wilesand the proof of Mordell's conjecture byFaltings in 20th century. At the beginning ofmy career, my subject of research was thestudy of the l-adic etale cohomology ofvarieties over local fields. More recently, I aminterested in the study of special values ofzeta functions via the philosophy of Iwasawa

theory. Precisely speaking, my project is tostudy Iwasawa theory from the view point ofGalois deformations. I think that Numbertheory is full of surprise as we see a lot ofunexpected relations between different kindsof objects.


Department of Mathematics Hiroyuki OGAWA

Algebraic Number Theory

I have an interest in periodic objects.Expanding rational numbers into decimalnumbers is delightful. The decimal numberexpansion becomes the repeat of a sequenceof some integers. I have an appetite forcontinued fraction expansions, never get tiredto calculate it, and want to find continuedfractions with sufficiently long period. It is onthe way to Gauss' class number oneconjecture. Recently, I am studying iterationof rational functions. For a rational functiong(x) with rational coefficients, a complexnumber z with g(g(...g(z)...))=z is called aperiodic point on g(x) and is an algebraicnumber. I expect that number theoreticalproperties which such an algebraic number zhas is described by the rational function g(x).

This does not seem to work out anytime, butone can find many rational functions g(x) thatdescribe the Galois group, the class number,the class group, and so on of a periodic pointof g(x). I think that this should be surelyuseful, and calculate like these every day.


Department of Mathematics Koji OHNO

Algebraic Geometry

When I was a student, I thought I knew numbertheory, geometry, but algebraic geometry wasunfamiliar for me. One may say algebra andgeometry are different fields, but you know thetheory of quadratic curves and are aware ofefficiency of algebraic methods for solvinggeometric problems. The field called algebraicgeometry lies on such a line. When I was studyingthe theory of quadratic curves, I wondered, ''Why dothey only treat special equations like quadratics?There are many other equations. But how can theybe treated?". When I discovered the answer might lieon this field, I decided to enter this field. The easiestnon-trivial equation has the form such as "the secondpower of y = an equation of x of degree three",which defines the so called "an elliptic curve". Thetheory of elliptic curve was one of the greatestachievements of nineteenth century and keepsdeveloping today. Recently, the famous Fermat's

conjecture has been solved using this theory. Thetheory of quadratic and elliptic curves involve onlytwo variables x, y. It is natural to think of theequations with many variables. In fact the algebraicgeometers are expanding the theory, curves tosurfaces and higher dimensional cases these days.Two dimensional version of elliptic curves are calledK3 surfaces, which can be treated only using thetheory of linear algebra(!) thanks to the Torelli'stheorem. These days, the 3-dimensional versions,which is called Calabi-Yau threefolds are fascinatingfor algebraic geometers like me. Somehowtheoretical physicists are also interested in this field.To study Calabi-Yaus by specializing these to oneswith a fiber structure (on which field, I'm nowworking) might be one method, but I have beenthinking that a new theory is needed. These days,many intriguing new theories have appeared and onemay find more!


Department of Mathematics Shin-ichi OHTA

Differential Geometry, Geometric Analysis

My research subject is geometry, especiallydifferential geometry and geometric analysisrelated to analysis and probability theory. Akeyword of my research is “curvature” whichrepresents how the space is curved. As seen inthe difference between the sums of interiorangles of triangles in a plane and a sphere, thebehavior of the curvature influences variousproperties of the space (the shapes oftriangles, the volume growth of concentricballs, how heat propagates, the behavior ofentropy, etc.). This powerful and versatileconception has been applied to Riemannianmanifolds, metric spaces, Finsler manifolds,Banach spaces, as well as discrete objectssuch as graphs.


Department of Mathematics Shinnosuke OKAWA

Algebraic Geometry

Geometric objects which are described as thecollection of solutions of algebraic equation(s), suchas ellipses, parabolas, and hyperbolas, are calledalgebraic varieties. These are the subject of study inthe field of algebraic geometry, and I have beenworking on various problems in this area.In the early days, I was studying topics aboutGeometric Invariant theory (GIT) and birationalgeometry. GIT is a theory about quotients ofalgebraic varieties by algebraic group actions, and inbirational geometry certain "transformation(modification)" of algebraic varieties is studied. Thedefinition of a GIT quotient depends on a choice of aparameter, which is called a stability condition, andone obtains different quotients by changing thestability conditions. Typically these quotients are allbirational to each other, and in good situations itturns out that the birational geometry of the quotientvariety is complete described in this way. I haveproved several properties of this class of varieties.

Recently I am mainly investigating algebraicvarieties from categorical points of view. One of myinterests is the "irreducible decomposition" of thederived category of coherent sheaves. This in fact ismotivated by birational geometry, and importanttechniques of birational geometry, e.g. the canonicalbundle formula, are used. I am also studying non-commutative deformations of algebraic varieties andtheir moduli spaces. Derived categories again playsa central role, but other interesting topics such asGIT, geometry of elliptic curves, and birationaltransformation of non-commutative algebraicvarieties also show up. Through the study, I foundan interesting and unexpected relationship with anold invariant theory which goes back to the end ofthe 19th century. I am also trying to understand towhat extent the derived category of coherent sheaveskeeps the geometric information of the originalvariety.


Department of Mathematics Yuichi SHIOZAWA

Probability Theory

My research area is probability theory. Inparticular, I am working on the sample pathanalysis for symmetric Markov processesgenerated by Dirichlet forms. Dirichlet formis defined as a closed Markovian bilinearform on the space of square integrablefunctions. The theory of Dirichlet forms playsimportant roles in order to construct andanalyze symmetric Markov processes.I am interested in the relation between theanalytic information on Dirichlet forms andthe sample path properties of symmetricMarkov processes. I am also interested in theglobal properties of branching Markovprocesses, which are a mathematical modelfor the population growth of particles bybranching.


Department of Mathematics Hiroshi SUGITA

Probability Theory

I specialize in Probability theory. In particular, Iam interested in infinite dimensional stochasticanalysis, Monte-Carlo method, and probabilisticnumber theory. Here I write about the Monte-Carlo method. One of the advanced features of themodern probability theory is that it can deal with"infinite number of random variables". It was E.Borel who first formulated "infinite number ofcoin tosses" on the Lebesgue probability space,i.e., a probability space consisting of [0,1)-intervaland the Lebesgue measure. It is a remarkable factthat all of useful objects in probability theory canbe constructed upon these "infinite number of cointosses". This fact is essential in the Monte-Carlomethod. Indeed, in the Monte-Carlo method, wefirst construct our target random variable S as afunction of coin tosses. Then we compute asample of S by plugging a sample sequence of

coin tosses --- , which is computed by a pseudo-random generator, --- into the function. Now, aserious problem arises: How do we realize apseudo-random generator? Can we find a perfectpseudo-random generator? People have believed itto be impossible for a long time. But in 1980s, anew notion of "computationally secure pseudo-random generator" let people believe that animperfect pseudo-random generator has somepossibility to be useful for practical purposes.Several years ago, I constructed and implementeda perfect pseudo-random generator for Monte-Carlo integration, i.e., one of Monte-Carlomethods which computes the mean values ofrandom variables by utilizing the law of largenumbers.


Department of Mathematics Hideaki SUNAGAWA


My research field is Partial DifferentialEquations of hyperbolic and dispersive type.They arise in mathematical physics asequations describing wave propagation, sothere are a wealth of applications and plentyof problems to be studied. Of my specialinterest is the nonlinear interactions ofhyperbolic waves. Since the analysis ofnonlinear PDE is still a developing subject,there are few general conclusions about that.To put it another way, it means that there arepossibilities for coming across wonderfulphenomena which no one has ever seenbefore.


Department of Mathematics Atsushi TAKAHASHI

Complex Geometry, Algebra, Mathematical Physics

My current interests are mathematical aspects ofthe superstring theory, in particular, algebraicgeometry related to the mirror symmetry.More precisely, I am studying homologicalalgebras and moduli problems for categories of"D-branes" that extend derived categories ofcoherent sheaves on algebraic varieties.Indeed, I am trying to construct Kyoji Saito'sprimitive forms and their associated Frobeniusstructures from triangulated categories defined viamatrix factorizations attached to weightedhomogeneous polynomials.For example, I proved that the triangulatedcategory for a polynomial of type ADE isequivalent to the derived category of finitelygenerated modules over the path algebra of theDynkin quiver of the same type.Now, I extend this result to the case when thepolynomial corresponds to one of Arnold's 14

exceptional singularities and then showed the"mirror symmetry" between weightedhomogeneous singularities and finite dimensionalalgebras, where a natural interpretation of the"Arnold's strange duality" is given.


Department of Mathematics Naohito TOMITA

Real Analysis

My research field is Fourier analysis, and Iam particularly interested in the theory offunction spaces. Fourier series wereintroduced by J. Fourier(1768-1830) for thepurpose of solving the heat equation. Fourierconsidered as follows:"Trigonometric seriescan represent arbitrary periodic functions".However, in general, this is not true. Then,we have the following problem: "When canwe write a periodic function as an infinite (orfinite) sum of sine and cosine functions?".Lebesgue space which is one of functionspaces plays an important role in this classicalproblem. Here Lebesgue space consists offunctions whose p-th powers are integrable.In this way, function spaces are useful forvarious mathematical problems. As another

example, modulation spaces were recentlyapplied to pseudodifferential operators whichare important tool for partial differentialequations, and my purpose is to clarify theirrelation.


Department of Mathematics Kenkichi TSUNODA

Probability Theory

My research field is probability theory. Inparticular I am interested in problems relatedto so-called "Hydrodynamic limit", which is acertain type of space-time scaling limits.Hydrodynamic limit means a method whichdetermines a macroscopic quantity of amicroscopic system such as particles systems.To tackle a difficult problem related toHydrodynamic limit, it is necessary to invokeresults on functional analysis or partialdifferential equations, and to use specificarguments for particle systems and wideknowledge of probability theory.Hydrodynamic limit is formulated as Law oflarge numbers for a macroscopic quantitysuch as the number of particle systems or thecurrent for a microscopic system. I am

working on related Central limit theorem andLarge deviation principle.In recent years, my another interest isRandom topology, which has arisen from thedevelopment of Topological data analysis inapplied mathematics. I am working on thisnew research area with my probabilistictechnique although this theme is not relatedto above Hydrodynamic limit deeply.


Department of Mathematics Motoo UCHIDA

Algebraic Analysis, Microlocal Analysis

My research field is algebraic analysis andmicro-local analysis of partial differentialequations. The view point of micro-localanalysis (with cohomology) is a newimportant point of view in analysisintroduced by Mikio Sato in the early 1970s.Thinking from a micro-local point of viewhelps us to well understand a number ofmathematical phenomena (at least for PDE)and to find a simple hidden principle behindthem. Even for some classical facts (scatteredas well known results) we can sometimes finda new unified way of understanding from amicro-local or algebro-analytic viewpoint.


Department of Mathematics Takao WATANABE

Algebraic Number Theory

My current interest is the Geometry of Numbers. TheGeometry of Numbers was founded by HermannMinkowski in the beginning of the 20th century.Minkowski proved a famous theorem known as"Minkowski's convex body theorem", which assertsthat "there exists a non-zero integer point in V if V isan o-symmetric convex body in the n-dimensionalEuclidean space whose volume is greater than 2^n".When V is an ellipsoid, this theorem is refined asfollows. Let A be a non-singular 3 by 3 real matrix andK(c) the ellipsoid consisting of points x such that theinner product (Ax, Ax) is less than or equal to c > 0.For i = 1,2,3, we define the constant c_i as theminimum of c > 0 such that K(c) contains i linearlyindependent integer points. Then c_1, c_2, c_3satisfies the inequality c_1c_2c_3 <= 2|det A|^2. Thisis called "Minkowski's second theorem". A similarinequality holds for any n-dimensional ellipsoid.Namely, if A is a non-singular n by n real matrix andK(c) is the n-dimensional ellipsoid defined by (Ax,Ax) <= c, we can define c_i for i = 1,2, ..., n as the

minimum of c > 0 such that K(c) contains i linearlyindependent integer points. Then the inequalityc_1c_2...c_n <= h(n)|det A|^2 holds for any A. Theoptimal upper bound h(n) does not depend on A, and iscalled Hermite's constant. We know h(2) = 4/3, h(3) =2, h(4)= 4, ..., h(8) = 256, but h(n) for a general n is notknown. A recent major topic of this research area is thedetermination of h(24). In 2003, Henry Cohn andAbhinav Kumar proved that h(24) = 4^24.(Incidentally, h(3) was essentially determined byGauss in 1831, and h(8) was determined by Blichfeldtin 1953. If you would determine h(9), then your namewould be recorded in treatises on the Geometry ofNumbers.) Now I study (an analogue of) the Geometryof Numbers on algebraic homogeneous spaces. One ofmy results is a generalization of Minkowski's secondtheorem to a Severi-Berauer variety. In addition, I aminterested in the reduction theory of arithmeticsubgroups, automorphic forms, the algebraic theory ofquadratic forms and Diophantine approximation.


Department of Mathematics Katsutoshi YAMANOI

Complex Analysis, Complex Geometry

My research interest is Complex geometryand Complex analysis, both from the viewpoint of Nevanlinna theory. In the geometricside, I am interested in the conjectural secondmain theorem in the higher dimensionalNevanlinna theory for entire holomorphiccurves into projective manifolds. Also I aminterested in the behavior of Kobayashipseudo-distance of projective manifolds ofgeneral type. These problems are related to analgebraic geometric problem of bounding thecanonical degree of algebraic curves inprojective manifolds of general type by thegeometric genus of the curves. In the analyticside, I am interested in classical problems ofvalue distribution theory for meromorphicfunctions in the complex plane.


Department of Mathematics Seidai YASUDA

Number Theory

Systems of polynomials with integral coefficients arestudied in number theory. It is often very difficult tofind the integral solutions of such a system. Instead, wesimultaneously deal with the solutions in variouscommutative rings. The solutions in various ringsforms a scheme, which provides geometric methodsfor studying the system of polynomials.Hasse-Weil L-functions of an arithmetic scheme aredefined using geometric cohomology. I am interestedin the special values of these L-functions. The specialvalues are believed to be related to motiviccohomologies, which are defined by using algebraiccycles or algebraic K-theory and are usually they hardto know explicitly. It is a very deep prediction toexpect that such abstract objects should be related tomore concrete L-functions.It is expected that motives are related to automorphicrepresentations. The expectation is important since wehave various methods for studying automorphic L-functions. Some relations between motives and

automorphic representations are realized by usingShimura varieties. In a joint work with Satoshi Kondo,I have proved a equality relating motivic cohomologiesand special values of Hasse-Weil L-functions for somefunction field analogues of Shimura varieties.Hasse-Weil L-functions are defined via some Galoisrepresentations. We need to study such Galoisrepresentations. For some technical reasons it isimportant to study Galois representations of p-adicfields with p-adic coefficients, and p-adic Hodgetheory provides some tools for studying suchrepresentations. For recent years there have been muchdevelopment in p-adic Hodge theory, and a lot ofbeautiful theories have been constructed. However thetheory is not fully established and many aspects of thetheory remains mysterious. I am now trying to makethe integral p-adic Hodge theory more convenient forpractical study.





●ALGEBRA

Number TheorySeminar

Number theory seminar at Osaka Universityis a seminar for faculty members andgraduate students of Osaka University orresearchers studying nearby Osaka University.The seminar is usually held on Fridays, onceevery two weeks. The subject of the seminarcovers wide topics concerning Number theory,especially, algebraic number theory andanalytic number theory, modular forms,arithmetic geometry, representation theoryand algebraic combinatorics. In this seminar,we have reports of new results on these topicsand we exchange ideas and technics of ourresearch.

SEMINARS and COLLOQUIA




Algebraic GeometrySeminar

The seminar is held two or three times amonth and each time one speaker gives a talkof 90 minutes. After a talk, we have time forquestions and discussion. The purpose of theseminar is to learn important results by activeresearchers in Algebraic Geometry andrelated fields, providing new perspectives onthe areas through lectures and discussions.We also have survey lectures by experts forgraduate students and young researchers. Wehave guest speakers not only from domesticuniversities but also from foreign countries,reflecting various aspects of the research area.



●GEOMETRY

Geometry Seminar

This seminar on Mondays is intended fortalks that will be of interest to a wide range ofgeometers. Topics discussed include Riemann-ian, complex, and symplectic geometry;PDEs on manifolds; mathematical physics.



Topology Seminar

In our research group of topology, we holdthree kinds of specialised seminars regularly:the low-dimensional topology seminarfocusing on the knot theory, three-manifolds,and hyperbolic geometry; the seminar ontransformation groups; and the seminar on 4-manifolds and complex surfaces.We also sometimes hold a topology seminarencompassing all fields of topology, where allof us meet together.




●ANALYSIS

Seminar ofDifferential Equations

Our seminar is held on every Friday from 15:30 to17:00. One of the features of the seminar is to covera wide variety of topics on Qualitative Analysis ofDifferential Equations. In fact, we are interested inordinary differential equations, partial differentialequations, linear differential equations, nonlineardifferential equations and so on. Lecturers areinvited from not only domestic universities but alsoforeign countries and present us their original resultsor survey of recent development of their fields.Furthermore, this seminar provides opportunities togive a talk for our colleagues and Ph.D. studentsmajoring in differential equations. Moreover, weshould mention that we are pleased to haveparticipants from other universities located closed toours. In this way, we communicate with each otherand try to contribute to the progress of the theory ofdifferential equations.




Seminar on Probability

Probability theory group, the graduate school ofscience and the graduate school of engineeringscience, organizes "Seminar on Probability" onTuesday evening. The topics on this seminar arethe following:(1) Probability theoryStochastic analysis and infinite dimensionalanalysis, problems arising from other areas ofmathematics such as real analysis, differentialequations and differential geometry.(2) Research fields related to Probability theory,Ergodic theory, dynamical system, stochasticcontrol and mathematical finance.We welcome visits and talks by manyresearchers from other universities, domestic andabroad.



Dynamics and Fractals Seminar

Researchers and students working onvarious fields related to dynamical systemsand fractals attend this seminar. We meetonce a month for approximately 90 minutes.Each talk on his/her research is followed bydiscussions among all participants.



●COLLOQUIA

Mathematics Colloquium

Colloquia take place on Monday afternoonat 16 : 30 in Room E404. They are directedtoward a general mathematical audience. Inparticular, one of the functions of theseColloquia is to inform non-specialists andgraduate students about recent trends, ideasand results in some area of mathematics, orclosely related fields.


Documents

Department of Mathematics - sci.osaka-u.ac.jp¼ˆ2019-2020）.pdf · geometry of nilpotent groups to the geometry of solvable groups. More precisely, I have studied the cohomology