AY 2019 Undergraduate School Course Catalog

### Mathematics

 2019/10/17

開講学期／Semester 2019年度／Academic Year 　1学期・2学期 ／1st & 2nd Quarter 1st year 2.0 Shigeru Watanabe Takao Maeda, Kazuto Asai, Shigeru Watanabe, Cong-Thang Truong, Chunhua Su － －

更新日／Last updated on 2019/01/16 (ICTG class starts in Q3.And Prof. Asai and Prof. Cong-Thang Truong is in charge of the class.)In the basic course, there are two subjects in mathematics: one is linear algebra and the other is differential and integral calculus. In this class, the basic part of linear algebra will be taught. Students are required to study this class with differential and integral calculus, because these subjects are closely connected each other.Linear algebra is a field of mathematics that is based on both addition and scalar multiple, and vectors in high school mathematics are its basic parts. The main theme of linear algebra is eigenvalue problem that arose from the theory of simultaneous linear differential equations, while its historical origin is in solving simultaneous linear equations. The purpose of this class is to learn necessary notions and techniques to consider eigenvalue problem.Exercises will be also given side by side with the lectures. Students are expected to participate subjectively and positively.Linear algebra has many applications to computer science and engineering. For example, it is impossible to understand mechanisms of computer graphics without linear algebra. Further, some fields of mathematics arose from engineering and physics, and developed under the influences of them. These are the essential reasons why mathematics is required to learn. The purpose of this class is to learn necessary notions and techniques to consider eigenvalue problem, and the following contents will be dealt with.vectors in plane, vectors in space, matrices of (2,2)-type, matrices of (3,3)-type, general definition of matrix, elementary transformations, simultaneous linear equations, inverse matrices, determinants, linear spaces, dimension and basis, subspaces, linear mappingsAttainment targetsStudents will be able to understand rank and solve simultaneous linear equations.Students will be able to understand base and dimension of linear space and deal with linear subspaces. 1 Introduction to matrices (1)2 Introduction to matrices (2)3 Matrices and linear transformations (1)4 Matrices and linear transformations (2)5 Definition of matrix and operations - general theory6 Square matrices, regular matrices and linear mappings7 Elementary transformations and rank8 Simultaneous linear equations 9 Definition of determinant10 Properties of determinant11 Expansions of determinant12 Definition and properties of linear space13 Basis and dimension 14 Linear subspaces, linear mappings Masahiko Saito Introduction to linear algebra (in Japanese) University of Tokyo PressYoshihiro Mizuta Linear algebra (in Japanese) Saiensu-sha Asai's classFinal Exam. 100%. (More than 80% of homework assignments should be submitted.)Watanabe's classFinal 100%Maeda's classQuiz 10%, Midterm exam. 30%, Final exam.60% Asai's classDirectory for Asai's class: ~k-asai/classes/holm/ Handouts and Exercises for Asai's class: http://web-ext.u-aizu.ac.jp/~k-asai/classes/class-texts.html

開講学期／Semester 2019年度／Academic Year 　3学期・4学期 ／3rd & 4th Quarter 1st year 2.0 Shigeru Watanabe Takao Maeda, Shigeru Watanabe, Yodai Watanabe, Kazuya Matsumoto, Lothar M. Schmitt － －

更新日／Last updated on 2019/01/16 (ICTG class starts in Q1.And Prof. Schmitt is in charge of the class.)Linear algebra II is a continuation of linear algebra I and deals with eigenvalue problem. Students are required to study this class with differential and integral calculus because of the same reason that is described in the syllabus of linear algebra I. For example, matrices and determinants play important roles in differential and integral calculus of several variables. And eigenvalue problem gives a strong way to solve recurrence formulae of sequences. Students must know importance of understanding organic connection between linear algebra and differential and integral calculus. They will also learn bases of vector analysis that is necessary to learn electromagnetism. Besides, the fundamental policy does not change from the case of linear algebra I. Eigenvalue problem of matriceseigenvalues, eigenvectors, diagonalizationAttainment targetsStudents will be able to solve eigenvalues and eigenvectors.Students will be able to deal with diagonalization.Students will be able to deal with diagonalization of normal matrices by unitary matrices. 1 Inner products2 Metric linear spaces3 Orthogonalization4 Introduction to eigenvalue problem --- meaning of diagonalization5 Eigenvalues and eigenvectors (1)6 Eigenvalues and eigenvectors (2)7 Diagonalization (1)8 Diagonalization (2)9 Diagonalization of normal matrices by unitary matrices (1)10 Diagonalization of normal matrices by unitary matrices (2)11 Diagonalization of normal matrices by unitary matrices (3)12 Diagonalization of real symmetric matrices by orthogonal matrices13 Quadratic forms14 Quadratic curvesThe order of classes may be changed. Masahiko Saito Introduction to linear algebra (in Japanese) University of Tokyo PressYoshihiro Mizuta Linear algebra (in Japanese) Saiensu-sha Watanabe's classFinal 100%Maeda's classQuiz 10%, Midterm exam. 30%, Final exam.60%

開講学期／Semester 2019年度／Academic Year 　1学期・2学期 ／1st & 2nd Quarter 1st year 2.0 Hiroshi Kihara Hiroshi Kihara, Yoshiko Ogawa, Nobuyoshi Asai, Peng Li － －

更新日／Last updated on 2019/03/10 (ICTG class starts in Q4.And Prof. Li, P. is in charge of the class.)Calculus I is devoted to the former half of (differential and integral) calculus. Calculus and linear algebra are essential in the study of mathematical sciences. Since even the vector calculus is needed in the course of physics, the student should get familiar with calculus and linear algebra and understand the relation between them as soon as possible.Differential and integral calculus started from an understanding of basic objects such as areas of figures and tangent lines to curves, and was based on the Newton's mechanical investigations. Of course, the basic notions in calculus are constructed using that of a limit.Calculus I deals with calculus of one variable. The basic calculatioal techniques are reviwed and new notions and results are introduced; the rigorous treatment (epsilon definition) of a limit is also (partly) introduced.Exercises are also offered. Establishing the foundation of the basic calculational techniques studied in high school, we introduce advanced notions and results such as inverse trigonometric functions, expansion of a function, and recurrence relation of integrals.Calculus I is foundation for an understanding of Calculus II, Probability and Statistics, Fourier analysis, Complex function theory, Physics, and all fields of computer sciences. 1. Set of real numbers2. Limit of a sequence3. Limit of a function and continuous functions4. Derivative, exponential function, and logarithmic function5. Trigonometric functions, inverse trigonometric functions, and higher derivatives6. Euler's formula7. Derivative, the mean-value theorem, and increase or decrease of a function8. Taylor's theorem and expansion of a function9. Indefinite integral and rerecurrence relation 10. Integral of rational functions11. First and second order linear differential equations12. Definition and properties of a definite integral13. Calculation of definite integrals, extention of the definition of a definite integral, and measurement of figures14. Review of the course Minoru Kurita, Shinkou bisekibungaku, Gakujutsutosho, 1442 yenGen Yoneda, Rikoukeinotameno bibunsekibun nyuumon, Science sha, 1890 yen Test : Report = 8 : 2 None Instructed in the lectures

開講学期／Semester 2019年度／Academic Year 　3学期・4学期 ／3rd & 4th Quarter 1st year 2.0 Hiroshi Kihara Hiroshi Kihara, Takeaki Sampe, Takahiro Tsuchiya, Yodai Watanabe, Kenta Ofuji, Xiang Li － －

更新日／Last updated on 2019/03/08 (ICTG class starts in Q2. And Prof. Li, X. is in charge of the class.)Calculus II deals with calculus of several variables.Differential and integral calculus of several variables reduces to calculus of one variable. If you understand it, you can easily master the main part of Calculus II.See also the syllabuses of Calculus I and Linear algebra II. The main objective of this course is to master differential and integral calculus of several variables.The notion of derivative of a function of one variable is extended to that of partial derivative of a function of several variables. It is applied to solve problems of local minimun/maximum.The notion of definite integral of a function of one variable is extended to that of multiple integral of a function of several variables. Especially, the technique of transformation of variables, which corresponds to that of substitution integral, is important.You also study the basics of series. Especially, the notion and results of series of functions are foundation of Fourier analysis and Complex function theory. 1. Parametrized curves2. Partial differential coefficients3. Differentiation of composite functions4. Total differential and expansion of a function5. Local minimum/maximum of a function6. Implicit functions, curves, and surfaces7. Multiple integrals and their calculations8. Technique of transformation of variables9. Areas and volumes10. Differential 1-forms and integrals11. Convergence and absolute convergence of a series12. Sum and product of series and limit of a sequence of functions13. Power series14. Review of the course Minoru Kurita, Shinkou bisekibungaku, Gakujutsutosho, 1442 yenGen Yoneda, Rikoukeinotameno bibunsekibun nyuumon, Science sha, 1890 yen Test : Report = 8 : 2 Calculus I, Linear Algebra I Instructed in the lectures

開講学期／Semester 2019年度／Academic Year 　1学期 ／First Quarter 2nd year 2.0 Takao Maeda Takao Maeda, Takahiro Tsuchiya, Xiang Li, Kazuyoshi Mori － －

更新日／Last updated on 2019/03/05 (ICTG class starts in Q3.And Prof. Li, X. is in charge of the class.)The origin of Fourier analysis is a study of the heat equation (differential equation) by Fourier in early 19th century.  The basic idea was “express arbitrary functions by using trigonometric functions well!”  This way of thinking led to many useful results in many application fields, but it was uncertain whether the conclusions were mathematically justified, since the concept of convergence had not been discovered.  In the present day, a proper theory has been developed for conventional sciences and technologies.  Fourier analysis is an indispensable basic theory for not only processing differential equations of Fourier’s era, but also modern applications, e.g., signal processing (including for image information and sound information, etc.)  The technique for handling the given function by using trigonometric functions and exponential functions is given though we will not examine of mathematical proofs in this lecture.  Through the answers to the exercises, students will become familiar with calculations and skilled at using basic theorems of Fourier analysis. Part1. Fourier series expansionStudents will understand the orthonormal system of the space of the functions defined over finite intervals and Bessel's inequality.    Students will understand the Fourier series of trigonometric functions and will calculate them for the functions expressed by polynomials, exponential functions or trigonometric functions.Part2. Properties of Fourier seriesStudents will understand conditions of convergence of Fourier series, the relationship between a given function and the Fourier series derived from it.  They will understand Parseval’s theorem and apply it. They will also understand Weierstrass' theorem.  Part3. Fourier integralStudents will understand that the Fourier integral is obtained through a heuristic method that extends finite intervals to infinite ones and calculates Fourier transforms and Fourier integrals (Fourier inverse transforms) of given functions.  Target functions are not only elementary functions described above, but also the functions of the type \$e^{-x^2})\$.  They will also understand a convolution of the functions, the relationship between the convolutions and the Fourier transforms of the functions and calculate themPart4. Laplace transformStudents will understand the Laplace transform as a transform based on Fourier transform and apply it to solving a certain kind of ordinary differential equations.  The method by which the solution is expressed by a convolution of the Laplace transform is included.  Part5. Discrete Fourier transformThrough the heuristic consideration of the stepping functions by applying the theory of Fourier series, students will understand the theory of discrete Fourier transform (DFT) similar to the theory introduced in Part 2.  They will also understand the method of the fast Fourier transform (FFT) as an efficient algorithm of DFT and appreciate its speed. 1: Part1. Fourier series expansion (Orthogonal system of the function space) 2: Part1. Fourier series expansion (Fourier series of trigonometric functions) 3: Part1. Fourier series expansion (Exercise) 4: Part2. Properties of Fourier series (Convergence condition of Fourier series) 5: Part2. Properties of Fourier series (Parseval's theorem, Weierstrass' theorem) 6: Part2. Properties of Fourier series (Exercise) 7: Part3. Fourier integral (Introduction from Fourier series, Fourier transform) 8: Part3. Fourier integral (Parseval's theorem, convolution) 9: Part3. Fourier integral (Exercise) 10: Part4. Laplace transform (Introduction from Fourier transform) 11: Part4. Laplace transform (Ordinary differential equations of constant coefficients) 12: Part4. Laplace transform (Exercise) 13: Part5. Discrete Fourier transform (Introduction from Fourier series) 14: Part5. Discrete Fourier transform (FFT(Fast Fourier Transform)) Handouts by each instructor and the following books:Tsuchiya class, Mori class, Maeda classGen-ichiro Sunouchi; Fourier analysis and its applications (SAIENSU-sha) Li classNakhle H. Asmar; Partial Differential Equations with Fourier Series and Boundary Value Problems: Third Edition (Dover Books on Mathematics) Students will be assessed based on regular quizzes and report with an emphasis on the final examination. Formal prerequisite: M-3 Calculus I or M-4 Calculus II, M-1 Linear algebra or M-2 linear algebra IIImportant related courses: M-6 Complex analysis, A-3 Image processing, A-8 Digital signal processing Transnational college of Lex (ed.), “Adventure of Fourier” Hippo family clubKen-ichi Kanaya, “Applied mathematics”, Kyoritsu Syuppan

開講学期／Semester 2019年度／Academic Year 　4学期 ／Fourth Quarter 2nd year 2.0 Kazuto Asai Masahide Sugiyama, Kazuto Asai, LI Xiang － －

更新日／Last updated on 2019/02/07 Although complex functions, in the wider sense, are mappings from complex numbers to themselves, i.e. complex-valued functions of a complex variable, the main objects of Complex Analysis are functions satisfying analyticity. The analyticity is a property of local representability of a function as convergent power series, which is equivalent to the condition that a function is holomorphic (differentiable with respect to a complex variable) in the corresponding domain.In this course, we introduce complex functions, and learn holomorphy of functions and the Cauchy--Riemann equations. Next we define complex integration along a curve on the complex plane, and learn Cauchy's integral theorem/formula, etc. In virtue of this result, we have the Taylor series expansion and the Laurent series expansion of functions. The former is power series expansion of functions, which is the most fundamental result in Complex Analysis. The latter is applied to the study of singularities and the residue theorem. In addition, we derive many techniques useful for direct applications such as the maximum modulus principle, calculation of solutions to differential equations using the method of power series, and determination of the number of zeros of functions by Rouche's theorem.When we study Complex Analysis, we are impressed that all of the needed theorems are derived very naturally one after another. Hence it is said that this theory has a beautiful system of mathematics. In particular, the most amazing fact is that every analytic function is completely determined by the behavior in a very small domain. This is similar to the fact that every life can be completely regenerated from a cell.Analyticity is a property that our familiar many real functions have -- polynomials, rational functions, exponent functions, logarithm functions, trigonometric functions, and all combinations of them have analyticity. Therefore, Complex Analysis is easy to apply to many areas. The knowledge of Complex Analysis is very important for various application areas such as electromagnetism, fluid mechanics, heat transfer, computer system theory, signal processing, etc. Objectives: Students understand what is a "holomorphic function", and can apply Cauchy's integral theorem/formula to several problems. They learn analytic functions are expanded by Taylor/Laurent series. Also, they use residue theorem to some integral calculation.Attainment goals: holomorphic functions, Cauchy--Riemann equations, complex integrals, Cauchy's integral theorem, Cauchy's integral formula, Taylor series, Laurent series, singularities, residue theorem. 1. complex plane, point at infinity2. holomorphic functions, Cauchy-Riemann equations3. harmonic functions4. exponent functions, trigonometric functions, logarithm functions, roots, complex powers of complex numbers5. complex integrals6. Cauchy's integral theorem, integrals of holomorphic functions7. Cauchy's integral formula, Liouville's theorem, maximum modulus principle8. complex sequence and series9. sequence and series of functions, uniform convergence10. power series and its convergence domain11. Taylor series expansion12. Laurent series expansion, zero points, singularities13. residue theorem14. application to several (real) definite integrals(Details depend on each class.) Handouts by each instructor and the following:Sugiyama's class: Nattokusuru Fukusokansu, Yoshitaka Onodera, Kodansha, 2000.Asai's class: Handout is a main textbook. As a side reader, Nattokusuru Fukusokansu, Yoshitaka Onodera, Kodansha, 2000.Li's class: A first course in Complex Analysis with application, Dennis G. Zill and Patrick D. Shanahan, Jones and Bartlett Publishers, Inc, 2003. Sugiyama's class: Determine after discussion with students (Last academic year: Mid-term Exam. 30%, Final Exam. 40%, Quiz 10%, Homework 20%)Asai's class: Final Exam. 100%. (More than 80% of homework assignments should be submitted.) Full score of Final Exam. is approx. 125 points. The raw score p is converted to a scaled score s by the formula: s=80+(p-80)/2 (p>80), s=p (p<80) (in principle).Li's class: Homework 26% (2 x 13 Assignments, Attendance > 2/3), Final Exam 74%. Preferably prerequisite courses: Differential and Integral Calculus I, Differential and Integral Calculus II.Other related courses: Fourier analysis, Electromagnetism. http://web-int.u-aizu.ac.jp/~sugiyama/Lecture/CA/2019/welcome.html (Home page for Sugiyama's class)~k-asai/classes/holm/ (Directory for Asai's class)http://web-ext.u-aizu.ac.jp/~k-asai/classes/class-texts.html (Handouts and Exercises for Asai's class) https://github.com/uoaworks/complex-analysis (Lecture Notes and Assignments for Xiang Li's Class)

開講学期／Semester 2019年度／Academic Year 　1学期 ／First Quarter 2nd year 2.0 Takahiro Tsuchiya Takahiro Tsuchiya, Chunhua Su, Igor Lubashevskiy － －

更新日／Last updated on 2019/01/16 (Prof. Lubashevskiy, I. is in charge of ICTG class.)Probability and Statistics is the most useful area in mathematics. We present an introduction to Probability and Statistics for 2nd year students. As a study of random variation and statistical inference, Probability and Statistics are important in computer science and other wide areas of Mathematical sciences. The words of error, mean, variance, correlation, estimation are used most often. But, their definitions are not well known andbadly understood. In this course, we explicitly explain these words and concepts of Probability and Statistics. They are useful knowledge for students. Moreover, statistical analysis is the basis on solve statistic problems in research and business. 1. basis of statistics2. 1-dim data3. 2-dim data4. probability5. random variable6. probability distribution7. multi-dimensional probability distribution8. Law of large number9. sample distribution10. sample from Gaussian distribution11. estimation 1(mean)12. estimation 2(variance)13. hypothesis test 1 (mean)14. hypothesis test 2(variance) Tokei kaiseki Nyumon ( Tokyo Univ Press ) Test 80 and reports 20 Formal prerequisites:M3 Calculus I or M4 Calculus II An Introductiion to Probability Theory and its Application, Vol 1Tokeigaku Nyumon (Tokyo Univ Press)

開講学期／Semester 2019年度／Academic Year 　2学期 ／Second Quarter 3rd year 2.0 Nobuyoshi Asai Nobuyoshi Asai, Xiang Li － －

更新日／Last updated on 2019/02/15 In this course students study properties of sets which have algebraic operations on two elements. According to the condition of operations, the set is called as group, ring and field. Treating abstract algebraic operations derives general properties for these algebraic objects. As fundamentals in computer science students have already studied ``Algorithms and Data Structures''and ``Discrete Systems''. In these courses abstract algorithms, data structures and discrete systems have been introduced and these abstract thinking can be applied to solve many practical problems. These esprits of algebraic systems are the same, and one of basis of the powerful tools to describe, think and solve many practical problems. Using abstract algebraic discussions, coding theory, public key encryption, random number generation and other interesting applications will be introduced. In this lectures proofs of the theorems will be explained simply and the meaning of the theorems and their applications will be enhanced. Furthermore, abstract thinking and deductive thinking will be taught. In order to make students understand the paper exercises and programming exercises will be given. Also quizzes and homework will be carried out to confirm students understanding. Understand algebraic structures and their applications In this course, we will mainly study the following topics: algebraic operation and structure, semi-group, group, normal subgroup, quotient group, homomorphism theorem, finite group, direct product(direct sum) decomposition, symmetric group, general linear group, ring, matrix ring, ideal and quotient ring, Chinese Remainder theorem, prime and maximal ideal, localization, principal ideal ring, unique factorization ring, Euclidean domain, polynomial ring, field, field extension, algebraic extension, minimal decomposition field, finite field, constructable, M random sequence, coding theory. Class schedules and topics might be different by lecturers. The following is a tentative example. For detailed information, please refer to the web page of each class.01 promenade to algebraic system 02 remainder of integer and polynomial03 group(1): Lagrange theorem04 group(2): quotient group and homomorphism theorem 05 group(3): analysis of group structure06 applications of group07 Mid-exam08 ring and field(1): ideal, quotient ring09 ring and field(2): polynomial ring10 ring and field(3): reversible11 application(1): quotient field and operator theory12 ring and field(4): extension of field 13 application(2): M-sequence random number generation14 application(3): error correct coding N. Asai class杉原，今井，工学のための応用代数，共立出版 (1999).S. Ding classMainly uses hand out. Mid-term Exam. 30%, Final Exam. 40%, Quiz 10%, Homework 20% Kowledge and skill of the following classes are required:Linear algebras I, & II, Discrete Systems N.Asai class homepage:http://hare.u-aizu.ac.jp/AppA/2019S.DIng class homepage:Will be announced  in the classN. Asai's practical work experience:1997-2000  Researcher, WaveFront Co. Ltd. 2002-2003 Guest researcher, National Institute of Environmental Study2001-2010 Collaborative Research with Asahi Glass CompanyHe has practical work experience at Wave Front Co. Ltd for numerical simulations, modelings and high performance computings from 1997 to 2000. After joined to U. of Aizu, he has continued with companies on the mentioned topics. These experiences of designing models, data structure and algorithms are deeply related with this class topics especially on designing model and algorithms.

開講学期／Semester 2019年度／Academic Year 　3学期 ／Third Quarter 3rd year 2.0 Lothar M. Schmitt Lothar M. Schmitt － －

更新日／Last updated on 2019/03/01 The course has three parts:(1) Review of Boolean Logic and Propositional Logic.=============================================Review to bring students with different prerequisites to the same level. Topics discussed include:Mathematical propositions and propositional logic (truth values in {0,1}); Set theory; Boolean algebras as generalizing principle of the latter two subjects; Stone's Theorem.Some computer applications of the material is discussed as well.(2) Fuzzy Sets and Applications.===========================Here the structure of (1) is generalized to truth values in the full interval [0,1] in IR. This allows to define fuzzy subsets of a given set, the embedding of regular sets into the set of fuzzy subsets. A special case of fuzzy subsets are fuzzy relations. The main application of this chapter is the rigorous development of the Tamura classification method based upon a given fuzzy relation. An indication how to use this method in image reconstruction is given.See: http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5408605(3) Model Theory.=================In mathematics, model theory is the study of (classes of) mathematical structures (e.g., groups, fields, graphs, universes of set theory) using tools from mathematical logic. A theory is built which allows to say in a proper calculus: One can verify a formula or theorem in any set interpretation, if and only if one can find a formal proof of that formula or theorem by means of strict textual manipulations according to a fixed set of proof-rules without referring to special properties of any example. This is known as Goedel's completeness theorem. To learn Logic and Applications from three perspectives: classical Boolean Logic, Fuzzy Logic, and Model Theory (logic of proofs, first order logic). Boolean Logic I-III.Fuzzy Logic and Relations IV-VII.Model Theory VIII-XIV. [1] Online Lecture Notes (email L@LMSchmitt.de for the link).[2] Pattern Classification Based on Fuzzy Relations, S. Tamura et al,: IEEE Transactions on Systems, Man and Cybernetics, 1971[3] Mathematical Logic (Springer Undergraduate Texts in Mathematics)H.-D. Ebbinghaus, J. Flum, W. Thomas Required for admission to final exam: [a] sufficient attendance (2/3), and [b] participation in online quizzes (2/3, this is for training only). The final exam determines the grade. Rather self-contained course. Understanding basic logic and real numbers is welcome.Formal prerequisites: None Online Lecture Notes (email L@LMSchmitt.de for the link).

開講学期／Semester 2019年度／Academic Year 　3学期 ／Third Quarter 3rd year 2.0 Shigeo Takahashi Shigeo Takahashi － －

更新日／Last updated on 2019/08/16 The contents of this course include the fundamental concepts on topology, which serves as the foundation of modern mathematics, and its applications. In topology, two shapes can be considered as the same if we can transform one to the other smoothly while maintaining the underlying connectivity within the shapes. This idea leads to a new approach to characterizing the global features of shapes and has promoted the development of computer science and engineering so far. Specifically, its contribution ranges from the fundamental representations of graphs and surfaces to the state-of-the-art techniques for extracting features in data analysis. This course will facilitate to study various important concepts for classifying shapes from a topological viewpoint, by employing specific one and two-dimensional shapes as examples. The objectives of this course are to study several topological invariants or characteristics that describe the global structures of geometric objects, such as the Euler characteristic and the Betti number, together with the topological classification of closed surfaces using their opened ones along boundaries. This will be followed by understanding the relationship between the geometric objects (e.g., the topological space) and algebraic objects (e.g., the Homology group), which allows us to classify the geometric objects by calculating the associated Homology groups. 01) Guidance02) 1-dimensional topology: Seven Bridges of Königsberg and Eulerian paths03) 1-dimensional topology: Connectivity and Euler-Poincare theorem04) 1-dimensional topology: Embedding into Euclidean space05) 2-dimensional topology: Closed surfaces06) 2-dimensional topology: Surfaces with boundary07) 2-dimensional topology: Classification of closed surfaces08) 2-dimensional topology: Connectivity and Euler-Poincare theorem09) 2-dimensional topology: Embedding into Euclidean space10) Homology: Groups and Homomorphisim11) Homology: Chain complexes12) Homology: Homology groups13) Homology: 0-dimensional and 1-dimensional Homology groups14) Homology: Connectivity and Euler-Poincare theorem 瀬山士郎著「トポロジー：柔らかい幾何学」 増補版 日本評論社"Topology: Yawarakai Kikagaku" by Shiro SeyamaNippon-Hyouron Publisher(The text book is written in Japanese) Term-end exam (50%), quizzes in class (25%), reports on assignements (25%)Please submit your report before the session startsThose who take the term-end exam are only eligible for creditYou are requested to attend more than 2/3 of sessions to take the term-end examCheating (include proxy submission of quizaes reports) will be strinctly penalized(Make-up exam will be provided according to the guidelines of absense) No formal prerequisites.It is preferable to have earned credits of "Linear Algebra I" and "Applied Algebra." Stephen Barr 「Experiments in Topology」Dover Publications, INC

開講学期／Semester 2019年度／Academic Year 　3学期 ／Third Quarter 3rd year 2.0 Lothar M. Schmitt Lothar M. Schmitt － －

更新日／Last updated on 2019/03/01 We study the ideas of topology from the application perspective of analysis. For this purpose, the definition of the real numbers IR is reviewed and it is shown how the sup-axiom in IR relates to the existence of limits. In the major part of the course, the relationship between(1) limits in IR^n(2) the canonical topology in IR^n(3) continuous functions is discussed. The relationship between these structures is studied extensively. As an application, we show, e.g., that compactness and associated uniform continuity are the ingredients which make integration work.The second major part of this course introduces a rigorous treatment of the concept of continuous deformation (homotopy) and the fundamental groups which one can construct using homotopy equivalence classes of continuous functions on a set with a given topology. As an application we show the fundamental theorem of algebra at the end of the course. Learn topological concepts in the context of the geometry in finite dimensional vector spaces and the natural (Euclidean) norm including embedded objects such as the torus in IR^3. Relate the concepts of supremum, limit, topology and continuous functions. Show how these concepts and homotopy apply in other mathematical disciplines. Review of the Real Numbers IR, the Supremum, and the Vector Space IR^n (I-II).Limits in IR^n --- Definition, Estimates, Basic Properties and Theorems (III-IV).Topology (mathematical object) and relation to Limits (V-VI).Continuous Functions and relation to Limits and Topology (VII-IX).Applications to concepts in Analysis --- Differentiation and Integration (X).Homotopy (continuous deformation of functions) and Associated Groups (XI-XIII).Fundamental Theorem of Algebra and other Applications of Homotopy (XIV). [1] A Geometric Introduction to Topology (Dover Books on Mathematics) C. T. C. Wall[2] Analysis I (Addison-Wesley) S. Lang Required for admission to final exam: [a] sufficient attendance (2/3), and [b] participation in online quizzes (2/3, this is for training only). The final exam determines the grade only. This is a follow-up course on topology and the reader is supposed to be familiar with the introductory course on topology given in UoA.Formal prerequisites: None Lecture Notes can be obtained from L.M.Schmitt on CD. (email: L@LMSchmitt.de)