Syllabus
Spring 2021
Analysis II
Math 704


The Basics

Course Homepage: http://people.math.sc.edu/girardi/w7034.html
This Course homepage contains much of the needed course information, eg: the homework problems along with due dates, class handouts, course announcements, etc..
Blackboard (Bb): The main use of Bb is for a secure location to post the Zoom link for virtual class and the collection & returning of homework Recall, the actual statements of homeworks (along with due dates) are posted on the course homepage.
Required: Check your (official) USC email regularly (at lease daily) for this is a form of class communication.
Required: Bring your textbook and all handouts to each class meeting.
Class Meeting Info.: T/TH   at 11:40am - 12:55pm.
Virtual classes will be on Zoom (link is posted on Bb).
In person classes are in LC405.
Pandemic related information. All exams will be in person. As for the lecture, at the start of the semester we will hold all lectures virturally. Later in the semester, if the pandemic situation permits, I will announce in class that some Thursdays lectures will be in person.
Required Textbook(s): For Complex Analysis: this linked Complex Script.
For Measure and Integration: Measure, Integration & Real Analysis by Sheldon Alxler. Springer Graduate Texts in Mathematics (282). 2020.

Instructor Info Office Office Hours
Prof. Girardi
girardi@math.sc.edu
Our class zoom link. For office visits, since you cannot physically knock on my office door, we just need to arrange a time so that I can open the Zoom link for our meeting. I will have the Zoom link open for Office Visits on Tuesday and Thursdays at 4:05 (since I use to teach another class T/Th 2:50 - 4:05 and use the same link for all my classes and office visits). Visits are also by proir appointment (just ask me after class or send me an email).

Dates

For important course dates, see our (linked, one page) Semester Schedule, which contains this semester's holidays, tentative Exam 1 and 2 dates, and the final exam date. Our course's Exams 1 and 2 dates are tenative and might change due to items as: hurricanes in SC, floods in LC, floods in Columbia, a majority of students are not ready for the exam yet, there is a more natural (earlier) break of material for an exam.

Grading Policies
(tentative)
Collaboration on homework is strongly encouraged! Thus you are encouraged to work together in groups on the homework and turn in one group collaborated solution on homework. One of the goals of this class is to help students form study groups for their upcoming studies for the qual exam. Recall you must take exams alone so learn from your group but be able to produce solutions yourselves. Homework assignments must be texed and submitted over Blackboard.

USC attendance policy is: Absence from more than 10 percent of the scheduled class sessions, whether excused or unexcused, is excessive and the instructor may choose to exact agrade penalty for such absences. (See USC Undergraduate Studies Bulletin → Policy and Regulations → Undergraduate Academic Regulations → Attendance Policy.) Note the words excused or unexcused. So basically an absence is an absence, with or without a note from a: doctor, coach, lawyer or professor. Keeping with USC's attendance policy, the lowest (approx./at least) 10% of your collected homework assignment scores will be dropped. So if you miss a homework assignment then the zero you receive on that work will be part of your dropped lowest 10%. Thus, late homework is not accepted.

There will be 2 hourly exams along with a final exam. All exams are closed books/notes. Calculators are not allowed (nor needed). No exam grade is dropped. There will be no make-up exams (except in extreme situations). If you feel you have an extreme situation, contact to Prof. Girardi before the exam time.

Prof. Girardi is glad to answer your questions over graded work but you must ask within 6 days from the date the graded work is returned to the class. Your PIN is a 3 digit number that helps Prof. Girardi correctly record your scores and is the same PIN number as in 703. In order to receive credit for an assignment, both your name and PIN must be on the paper.

Your goal is to gain a working knowledge of the material; you have some personal freedom in this pursuit. By the end of the semester, demonstrate that you have mastered the material and your grade will be agreeable. With this in mind, your course grade (tentatively) will be based on your (personal) higher percentage in the below 2 schemes.

 scheme 1scheme 2
Homework 25% 10%
Exam 1 25% 30%
Exams 2 25% 30%
Final Exam 25% 30%
total 100% 100%

The baseline score is indicated below. These thresholds may be lowered (at the end of the semester) if appropriate.

A B C D F
90 - 100 % 80 - 89 % 70 - 79 % 60 - 69 % below 60 %

Learning Outcomes

The learning outcomes are to cover many of topics listed on the Analysis Qual Syllabus (see list below) so that, after completetion of Math 703 and 704, students are prepared to start a summer of intensive studying for the Analysis Qual as to be able to complete Anaylsis Qual problem. Recall, several old quals are posted on the department's homepage (direct link is provided on our course homepage).

Metric Spaces (covered in 703)

  • metric spaces, continuous functions
  • separability, completeness
  • compactness, Heine-Borel
  • connectedness
Complex Analysis (started in 703 and finish in 704)
  • analytic functions: complex derivatives and Cauchy-Riemann equations, analyticity
  • special functions: $\log\left( z\right)$, $e^z$, trig functions
  • line integrals, Cauchy's theorem and its consequences: Cauchy integral formula, maximum modulus, power series, Fundamental Theorem of Algebra
  • classification of zeros and singularities, Laurent series
  • residue theorem, evaluation of integrals and series
Lebesgue theory of Measure and Integration (covered in 704)
  • outer measure, measurable sets, measure spaces, complete and regular measures
  • integration, Fatou's lemma and convergence theorems
  • extension theorem, product measure and Fubini's theorem, Lebesgue-Stieltjes integral
  • absolute continuity, Vitali's lemma, differentiation theory for monotone functions and integrals, functions of bounded variation
  • Egorov and Lusin Theorems
  • definition of $L^p$, H\"older and Minkowski inequalities, completeness of $L^p$, approximation by step and continuous functions

Common Sense and Courtesy

Academic Honesty. According to the USC Student Handbook code of student academic responsibility, the first law of academic life is intellectual honesty. We expect this of all of you. If you ever have the least bit of uncertainty about the ground rules, ask for clarification. Cheating and plagiarism will not be tolerated (such are grounds for an F in the course). No electronic posting of the course class notes nor other course material without prior written permission from Prof. Girardi. No improper emailing of students via Blackboard (it is totally fine, in fact encouraged, to use BB to set up Study Sessions). No improper use nor sharing with others information from the Student Contact Information Sheet. Violations of this policy will be dealt with according to University guidelines. Cheating on any assignment is grounds for an F in this course. A Statement of Academic Integrity can be found at http://www.sc.edu/academicintegrity.

Attendence/Participation. If attendence and/or class participation becomes a problem then, upon announcement in class, these will also factor into your grade. Upon such an announcement, absence or lack of participation exceeding 10% of class days (from the announced start day to the last day of class) will result in the lose of 2 full letter grades.

Timeliness. USC policy states: Enrollment in a course obligates the student not only for prompt completion of all work assigned but also for punctual and regular attendance and for participation in whatever class discussion may occur. It is the student's responsibility to keep informed concerning all assignments made. Absences, whether excused or unexcused, do not absolve the student from this responsibility. (See http://bulletin.sc.edu → Policy and Regulations → Undergraduate Academic Regulations → Grading Policies.) Thus students are expected to attend the entire class meeting: late arrivals and early depatures disrupt the class. Repeated late arrivals and/or early departures will not be tolerated. If you must leave class early, inform the instructor before the class begins, sit near the door, and gracefully exit. If you must arrive late, discretely find an open seat (and inform the instructor before hand if you know this will happen). This procedure, which produces a conducive learning environment, is out of fairness to all students in the course.

Cell Phones and company. In order to create a classroom environment conducive to learning, as well as by University policy, the use of cell phones during class is prohibited. During class, cell phones (as well as other electronic devices such as electronic tablets and smart watches) are to be turned off and stored in a closed (e.g., zipped or buttoned) bag (e.g., backpack or purse); they may not be stored in the person's clothing (e.g., jacket pocket). If you do not have proper storage with you, you may give the device to the instructor to babysit it during class. Improper use or storage of cell phones (as well as other electronic devices) could result in the device being confiscated (and not returned).

Electronic Device. If you want to use an electronic device to take class notes then you need to speak with Prof. Girardi before doing so.

ADA. If you have special needs as addressed by the Americans with Dissabilities Act and you have registered with the USC Office of Student Disability Services (OSDS), you need to speak with Prof. Girardi before using an approved accommodation.

Recording of Class. Recording (of any type, including but not limited to: audio, visual) of any part of the class is prohibited without the prior written permission the Prof. Girardi.


Findable from URL:    http://people.math.sc.edu/girardi/