MAT 536, Complex Analysis I

Spring 2024

Christopher Bishop

Distinguished Professor, Mathematics
Stony Brook University

Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631

Summary:

MAT 536 is an introduction to one complex variable and is one of the requried core courses for first year PhD students. It covers the basics results: definitions of holomorphic functions, the maximum principle, the Cauchy integral theorem, harmonic functions, the Perron method, Normal families, the Riemann mapping theorem. If time permits, we will given an introduction to Riemann surfaces and prove the famous uniformization theorem: every simply connected Riemann surface is conformally equivalent to either the disk, plane or sphere.

Textbook:

Complex Analysis, 1st edition, by Donald Marshall, Cambridge University Press. We will attempt to cover the whole book (minus a few sections).

Time and place

Currently, I expect the class to meet Mondays and Wednesdays from 10:00am to 11:20am. Room P-127 in Physics.

Grades

Grades will be determined by a combination of problem sets, a midterm and a final exam, each counted as one third of the total score. The midterm will be in class on Wed March 20. The final exam will be Friday, May 10 from 11:15am to 1:45pm, in Physics P-127, our usual room. Each exam will consist of a section involving true/false questions knowledge of the definitions and results from class, followed a several short proofs, usually with some choice given to the student.

Office hours:

Prof Bishop: Mon 9-10, Mon 2-3, Wed 9-10 (tentative, may change). You can always email me a question, or to set up an appointment for an in-person or Zoom meeting. If you just drop by my office, and I am in, I am usually happy to speak with you.

Alex Rodriguez (grader): Wednesday, 4-5pm.

Problem sets:

Will be assigned weekly from the textbook and posted here. Hand solutions in during class on Mondays; we will attempt to return on Wednesday. Hand written or LaTeX is acceptable. Handwritten problems sets should be very legible; I won't require the grader to try to decipher poor penmanship. If you use a non-textbook source,( e.g., a webpage or another textbook), you should cite this in your solution, although most problems will not require any research like this. I will generally try to post problem sets at least a week before they are due.
        Due Mon Jan 29: Problem Set 1
        Due Mon Feb 5: Problem Set 2
        Due Mon Feb 12: Problem Set 3
        Due Mon Feb 19: Problem Set 4
        Due Mon Feb 26: Problem Set 5
        Due Mon Mar 4: Problem Set 6
        No problem set for Mar 11 (Spring Break)
        No problem set for Mar 18 (Spring Break)
        No problem set for Mar 25 (Midterm week)
        Due Wed Apr 3: Problem Set 7
        No problem set for Apr 8 (Algebra II midterm):
        Due Mon Apr 15: Problem Set 8
        Due Mon Apr 22: Problem Set 9
        Sample final, solutions . (updated 5-6-24, correcting an error in Part 1, number 7)
        solutions to final exam

Lecture notes:

Slide for lectures will be posted here. These generally follow the textbook, with some additions and deletions. Slides will be updated throughout the semester.
        Slides for Course Introduction and Chapter 1. Used on Mon Jan 22.
        Slides for Chapter 2.
        Slides for Chapter 3.
        Slides for Chapter 4.
        Slides for Chapter 5.
        Slides for Chapter 7.
        Slides for Chapter 10.
        Slides for Chapter 12.
        Slides for Chapter 13.
        Slides for Chapter 14.
        Slides for Chapter 15.
        Slides for Chapter 9.
        Slides for Chapter 11.
        Slides for Trees, Triangles and Tracts, Part I
        Slides for Trees, Triangles and Tracts, Part II .
        Link to Video Recording . (one YouTube video covers both talks).
        Link to conference website: Transcendental Dynamics and Beyond: topics in complex dynamics 2021, (Centre de Recerca Matem├ática, Barcelona, April 19-23, 2021,

Lecture recordings:

        Class recordings are posted HERE.

Tentative lecture schedule (will be updated depending on progress in lectures)

        Mon Jan 21: Introduction to class, Chapter 1
        Wed Jan 23: Chapter 2, Polynomials 2.1, Fund. Thm. 2.2, Power series 2.3
        Mon Jan 29 Chapter 2, Analytic functions 2.3 Elementary oeprations 2.5, Maximum Prin 3.1
        Wed Jan 31: Chapter 3, Local behavior 3.2, Growth 3.3
        Mon Feb 5: Chapter 4, Integration on Curves 4.1, Equivalence of analytic and holomoprhic 4.2
        Wed Feb 7: Chapter 4, Morera's theorem, Runge's theorem
        Mon Feb 12: Finish chapter 4, uniform limits of analytic functions, applications of Runge's theore, Chapter 5, Cauchy's theorem 5.1, Winding numbers 5.2
        Wed Feb 14: Chapter 5, Removable singularities 5.3, Laurent Series 5.4, The argument prinicple 5.5
        Mon Feb 19: Finish Chapter 5, Laurant series, argument principle
        Wed Feb 21: No class
        Mon Feb 26: Chapter 7, harmonic functions, Cauchy-Riemann equattions
        Wed Feb 28: Chapter 7, Harmonic conjugates, Lindelof's maximum principle, Harnack's inequality
        Mon Mar 4: Chapter 10, normal families, spherical derivatives, Marty's theorem
        Wed Mar 6: Chapter 10, Hurtwitz's theorem, Rieman mapping theorem, Zalcman's lemma
        Mon Mar 11: Spring Break
        Wed Mar 13: Spring Break
        Mon Mar 18: Chapter 10, Montel's theorem, Picard's theorem, Julia and Fatou sets
        Wed Mar 20: Midterm
        Mon Mar 25: Chapter 12, Jordan curve theorem
        Wed Mar 27: Chapter 12 and Chapter 13, Caratheodory's therorem, Perron families
        Mon Apr 1: No class
        Wed Apr 3: Chapter 13, local barriers, 2nd proof of Riemann mapping theorem
        Mon Apr 8: Chapters 14, the monodromy theorem
        Wed Apr 10: Chapter 14, universal covers, deck transformations
        Mon Apr 15: Chapter 15, the moduluar function, Green's function
        Wed Apr 17: Chapter 15, the uniformization theorem
        Mon Apr 22: Chapter 9, Residue calculus, computing integrals
        Wed Apr 24: Chapter 11, Mittag-Leffer's theorem, Blaschke products, factorization
        Mon Apr 29: Brief introduction to some recent results: harmonic measures, dessins d'enfants, triangulations of Riemann surfaces
        Wed May 1: Last class
        Friday May 10: Final Exam 11:15am-1:45pm, Physics P-127 (usual lecture toom).

LaTex:

Although it is not required, you may wish to consider writing up your solution in TeX, since eventually you will probably use this to write your thesis and papers.

The not too short introduction to LaTex

Additional links:

Link to Mathematical Biographies

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Send me email at: bishop at math.sunysb.edu