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5 I II I II I II I III I III I IV I II I II I II Garrigue, Jacques I III Garrigue, Jacques I III I ( ) I ( - ) I ( -,- ) III III IV IV G30 Linear Algebra II Richard, Serge Calculus II Herbig Anne-Katrin Math tutorial II Herbig Anne-Katrin & Richard, Serge iii
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106 2015 K-theory for C*-algebras, and beyond K-theory for C*-algebras, and beyond Serge Richard 2 3 Lecture notes have been made available TA () M1 M2 D ( ) ( ) A few students came only for the first couple of lectures, and then disappeared. Three master students (M1) and four PhD students came on a regular basis. One additional PhD student from France attended the second half of the course. The aim of this course was to develop K-theory for C -algebras and to provide enough information on cyclic cohomology in order to understand the notion of pairing between these two theories. The course was divided into 10 chapters, namely 1) C -algebras, 2) Projections and unitary elements, 3) K 0 -group for a unital C -algebra, 4) K 0 -group for an arbitrary C -algebra, 5) The functor K 1, 6) The index map, 7) Higher K-functors and Bott periodicity, 8) The six-term exact sequence, 9) Cyclic cohomology, 10) Application: Levinson s theorem. In order to narrow the language gap, lecture notes have been provided to the students. All material is available on the website: richard/ktheorie.html and also on the corresponding page of NU OCW. 95
107 K-theory for C*-algebras, and beyond 2015 The course consisted in 15 lectures, all done on the blackboard. A summary of the previous lecture was written on the blackboard before the beginning of each lecture, and recalled for a couple of minutes at the beginning of each lecture. The students were encouraged to ask questions before, during and after the lectures, but they rarely took these opportunities. The final grade was computed on the basis of a report written by the students either on the solutions of some exercises, or on the written presentation on some subjects which were skipped during the lectures. For the evaluation, it has been taken into account that attending a course in English and writing a document in this language were already a big effort for the students. The expected amount of material in the reports was evaluated accordingly. Grade Total A 3 3 B 0 0 C 0 0 F 0 0 Absent 0 0 Total 3 3 The students had a positive attitude towards this course, even if they were not really keen in speaking during the lectures. One of the doctoral students ask interesting questions. In addition, the students seemed happy when corrections or improvements were suggested on their reports. They all did their best for improving their reports. As expected it is more easy for them to communicate though s than in front of a person. 96
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152 Linear algebra II (G30) 2015 Linear algebra II (G30) Serge Richard Lecture notes available on richard/spring2015.html S. Lang, Introduction to Linear Algebra, second edition, Undergraduate texts in mathematics, Springer-Verlag, 1986 TA 1 () M1 M2 D ( ) ( ) Most of the 1 st year students attended all the lectures. The 2 nd and 3 rd year students did not attend the lectures on a regular basis. At least one 2 nd year student had another course at the same time, but has been successful for linear algebra II. During this semester we first continued the study of linear maps, especially the notions of composition of linear maps and of the inverse of a linear map. Then, we concentrated on the notions of scalar product and of orthogonality in an abstract framework. We introduced the concept of multi-linear maps and developed the theory of determinant from it. Cramer s rule and some formulas for the inverse of a matrix have been given. We went on with the notions of eigenvalues, eigenvectors, characteristic polynomial, and so on. At the same time, we spent a couple of hours on C, and discussed briefly complex vectors spaces, matrices with complex entries and hermitian matrices. As an application of the content of this course, we discussed some discrete dynamical systems and studied part of a paper dealing with the eigenvector behind Google page s ranking. The book of Lang has still been used for the basic subjects, but additional material has been provided for the introduction to C and for the applications. 142
153 2015 Linear algebra II (G30) The course consisted in 15 lectures, mid-term and final exam included. Three additional quizzes of 15 minutes have been organized during the semester. The exercises corresponding to this course have been done during the associated tutorial (math tutorial II). The feedback obtained through two questionnaires were quite positive, with a few complains about the difficulty or the degree of abstraction. The lecture notes were welcome and extensively used. Some students ask if they could attend a course entitled Linear algebra III! Most of the students were enthusiastic with the applications. Overall, the content of the course has been appreciated by most of the students. At the end of the last class, many students asked additional references or lecture notes for various topics related to mathematics. The students were encouraged to ask questions before, during and after the lectures. Some of them took this opportunity to get a better understanding of the course, and few of them asked lots of precise and interesting questions. The final grade was computed as follows: 30 %: midterm exam; 40 %: final exam; 30 %: quizzes. Grade Total S A B C F Absent Total The students had a very positive attitude towards this course. The degree of abstraction has been kept constant, and most of the proofs have been given in details. It is interesting to observe that with only 1 unfortunate exception, all students in year 1 have been successful for this course. The students in year 2 and 3 are less incline to work hard for this course. 143
154 Linear algebra II (G30) 2015 The exercises sessions (math tutorial II) are really useful and complementary to the lectures. The evaluation method was clearly announced in the syllabus and respected during the semester. No student complains about the method or about the evaluations. 144
155 2015 Calculus II (G30) Calculus II (G30) Anne-Katrin Herbig 2 / 1 0 Vector Calculus, 3rd Edition, Susan J. Colley, Prentice Hall, 2011 (576 pp., ISBN: ) TA 1 () M1 M2 D ( ) ( ) Throughout the semester about 39 students attended the class regularly. The main topics of this course are vector calculus, differentiation in several variables, extrema of functions of several variables, and double integrals. This course mainly consisted of lectures. Quizzes were given at the end of 4 lectures. Homework problems were defered to the calculus part of the Mathematics Tutorial II class. Feedback by students obtained through questionary in the middle and at the end of the semester was throughout very positive. 145
156 Calculus II (G30) 2015 The final grade was computed as follows: 30 %: midterm exam; 40 %: final exam; 30 %: quizzes. The main grading criterion was the understanding of the concepts, 60 % of the exams being very elementary. Grade UG1 UG2 Total S A B C F Absent Total The computational methods were correctly understood. Some students had difficulties with more abstract arguments. Most of them were asking questions during the lectures. They studied regularly. The criteria for grading were broadly announced in advance, and their actual application was slightly adapted after the exams. 146
157 2015 Math tutorial II (G30) Math tutorial II (G30) Anne-Katrin Herbig & Serge Richard TA 2 () M1 M2 D ( ) ( ) All first year students attended the tutorial on a regular basis and have been successful. Two second year students did not appear at all, and two second year students came very irregularly. These four students either failed or have been considered as absent. This tutorial is based on the lectures provided in Calculus II and Linear algebra II. The students have been divided into two groups, and each week the students have 45 minutes of tutorial with the professor in charge of Calculus II, and 45 minutes with the professor in charge of Linear algebra II. During the tutorial sessions, we discussed about a list of several problems and sketched part of their solutions. The students were expected to give back the full solutions of all (or part of) these exercises during the next session; solutions of the exercises were then posted on the course website. Additionally, four quizzes for linear algebra and four quizzes for calculus have been organized during the semester. 147
158 Math tutorial II (G30) 2015 The final grade was computed as follows: 40 %: homework (half from calculus and half from linear algebra homework) 60 %: quizzes (half from calculus and half from linear algebra quizzes) Grade 1 2 Total S A B C F Absent Total The students had a very positive attitude during the tutorial. They understood that these sessions were very useful for getting a better understanding of the courses Calculus II and Linear algebra II. On the other hand, some students certainly just copied the solutions to the exercises from some other students without really understanding these solutions. Such an attitude is easily detected by comparing the grades obtained for the homework with the grades obtained for the quizzes. The evaluation method was clearly announced in the syllabus and respected during the semester. No student complains about the method or about the evaluations. 148
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JAPANESE SHIP-OWNERS AND WORLD BULK MARKET BETWEEN THE TWO WORLD WARS Mariko Tatsuki Keisen University After World War I, Japanese shipping suffered for a long time from an excess of tonnage and severe
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