Course Syllabus

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Content: This course provides a rigorous introduction to abstract algebra, including group theory and linear algebra.  The formal prerequisites for Math 55 are minimal, but this class does require a commitment to a demanding course, strong interest in mathematics, and some familiarity with proofs and abstract reasoning. 

Lectures: will be held in person Mondays, Wednesdays, Fridays from 10:30 to 11:45, starting Wednesday September 1st, in Science Center 507.

Important: if you are unwell or have been told to isolate/quarantine for any reason, please do not come to class. Instead, let Prof. Auroux know ahead of time (and/or contact another student who will be attending) if you'd like to watch the lecture via the backup Zoom meeting link (see under Zoom).  (This is strictly a backup, and will be set up manually and only when circumstances require it; sound and video quality may be low).

What can I do ahead of the semester to be ready for the class?

  • Read the detailed course information below!
  • Brush up on your math skills. Review basics about sets, functions, etc. (e.g. the beginning of Halmos' Naive Set Theory, sections 4 to 10) or getting familiar with the style of writing in the textbooks (Axler and Artin) by reading the beginning of the first chapter of each. (Don't worry yet about mastering any of the content, we'll restart completely from scratch).
  • Optional review/warm-up problems can be found here, in case you want to review background on sets and functions, assess your readiness, develop your problem-solving skills, or just can't wait.  Attempt them on your own, or even more fun, wait till mid-August, join the Slack workspace, and meet up with other prospective students to work on these collaboratively.
  • Take a moment to watch last year's course presentation video (but note one very important change in policy: this year's lectures will be in person, so there won't be video recordings and you will be expected to attend the lectures.)
  • If you have questions, e-mail Prof. Auroux and/or attend the registration week office hours, held Monday August 23, from 12 to 1:30pm (Eastern time), on Zoom.
  • If time permits, start learning how to use LaTeX (for example, using Overleaf, which saves you the trouble of installing it on your own computer), so that you can submit beautifully typeset assignments -- a useful skill for your subsequent math classes!  (Typesetting Math 55 assignments is strongly recommended, but not mandatory; we will accept scanned handwritten work as long as it is readable). See this excellent tutorial made by former CA Richard Xu for last year's Math 55 class.

Textbooks: there are two required texts, Axler's Linear Algebra Done Right and Artin's Algebra.  (We will aim to cover most of Axler, and most of chapters 2-10 in Artin).
Other books that may  (or may not) be helpful at various points in the course include Dummit and Foote's Abstract Algebra, Halmos' Naive Set Theory , Halmos' Finite-Dimensional Vector Spaces , Fulton and Harris' Representation Theory: A First Course , and Serre's  Linear Representations of Finite Groups .   Electronic versions of all of these books except for Artin and Dummit & Foote are freely available to Harvard students from the Springer-Verlag website (by clicking on the links above).

Course staff: (see this page for contact information and office hours)

  • Prof. Denis Auroux (auroux@math.harvard.edu)
  • Oliver Cheng (Course Assistant)
  • Leo Fried (Course Assistant)
  • Gaurav Goel (Course Assistant) [Section: Saturdays at noon starting 09/04 in SC 310. OH: Math Night, TBD.]
  • Dora Woodruff (Course Assistant)
  • Eric Yan (Course Assistant)

Lectures will be held MWF 10:30-11:45am.  Students are expected to attend all lectures. Handwritten lecture notes will be provided.

Discussion sections will be held by the CAs weekly at various times, in order to go over material from lecture and homework.  These are highly recommended for everyone.  Even if you don't need help with specific problems on the current assignment, section is a chance to review and ask general questions about course material (or related issues that may not be dealt with in the lecture), including any content not thoroughly covered in the lecture.  It's also a chance to get to know the CAs and your fellow students in smaller groups.

Homework will be assigned weekly, and is due on Wednesday each week.  Assignments will be posted on this site, and should be submitted electronically via this website.
Doing the homework in a timely fashion is essential to learning the material properly; given the pace, it is extremely hard to catch up if you fall behind in this class.  Extensions will be granted when circumstances genuinely warrant it (poor time management on your part is not normally a sufficient circumstance), but should be requested ahead of the deadline.

Exams: there will be a midterm (take-home, during the week of Sept. 27-Oct. 1) and a take-home final exam.

Course grades will be based on your homework (65%), the midterm (10%), and the final exam (25%).
One homework score will be dropped for everyone, so you may miss one assignment without penalty, but you are still responsible for working through the material.

Community:  One of the best features of Math 55 is the sense of community that most students get out of it.  Getting to know the CAs and your fellow students early in the semester, and forming study groups,  is an important part of the experience -- they'll be your support network when the math gets rough. Drop by office hours to introduce yourself and meet others (even if you don't have any specific questions). Participate in the Slack discussions. And please remember: it is up to you to make this community inclusive, welcoming, and supportive of all of its members -- and of all the other people around you, including those who aren't in Math 55.

Academic integrity policy: You are encouraged to discuss and collaborate with each other on the homework assignments. However,  make sure that you can work through the problems yourself, and write up your answers on your own. This is not only a matter of academic integrity, but also crucial for properly learning the material and the problem-solving skills that this course aims to cover.  For exams, collaboration or consultation of sources other than those explicitly permitted is not allowed.

Homework assignments

Direct links to the homework assignments will be posted here as PDF files.  Go to "Assignments" to see the LaTeX source, and to submit your solutions.

 

Lecture Notes

Here are direct links to the lecture notes as PDF files. Go to "Files" to see them all in a folder. 

  • Lecture 1 (Wed Sept 1:  Course logistics; groups; examples)
  • Lecture 2 (Fri Sept 3: Set theory; examples of groups continued; subgroups; homomorphisms)
  • Lecture 3 (Wed Sept 8: subgroups of Z; cyclic groups; equivalence relations; cosets)
  • Lecture 4 (Fri Sept 10: normal subgroups, quotient groups; exact sequences; the symmetric group)
  • Lecture 5 (Mon Sept 13: the symmetric group; center, commutators; free groups; rings and fields)
  • Lecture 6 (Wed Sept 15: more about fields; vector spaces; independence, span, basis, dimension)
  • Lecture 7 (Fri Sept 17: bases, dimension; linear maps and matrices; direct sums; rank, dimension formula)
  • Lecture 8 (Mon Sept 20: change of basis; quotient spaces; duals, transpose; linear operators)
  • Lecture 9 (Wed Sept 22: transpose; linear operators; invariant subspaces, eigenvectors, eigenvalues)
  • Lecture 10 (Fri Sept 24: eigenvectors; triangular matrices; generalized kernel, generalized eigenspaces)
  • Lecture 11 (Mon Sept 27: generalized eigenspaces; nilpotent operators; Jordan normal form; characteristic polynomial; real operators)
  • Lecture 12 (Wed Sept 29: real operators; categories and functors; bilinear forms) (handout)
  • Lecture 13 (Fri Oct 1: functors; bilinear forms; orthogonality; inner products; orthonormal bases)
  • Lecture 14 (Mon Oct 4: inner products; orthonormal bases; orthogonal and self-adjoint operators)
  • Lecture 15 (Wed Oct 6: self-adjoint and orthogonal operators, spectral theorem; Hermitian inner products)
  • Lecture 16 (Fri Oct 8: Hermitian inner products; complex spectral theorem; classifying bilinear forms)
  • Lecture 17 (Wed Oct 13: tensor product: definition and properties; trace) (handout)
  • Lecture 18 (Fri Oct 15: trace; tensor algebra; symmetric and exterior algebras)
  • Lecture 19 (Mon Oct 18: exterior algebra, determinant; modules over rings)
  • Lecture 20 (Wed Oct 20: modules; classification of finitely generated abelian groups; group actions)
  • Lecture 21 (Fri Oct 22: group actions, orbits and stabilizers; Burnside's lemma; conjugacy classes)
  • Lecture 22 (Mon Oct 25: finite subgroups of SO(3) and regular polyhedra)
  • Lecture 23 (Wed Oct 27: conjugacy classes; groups of order p^2; the symmetric group)
  • Lecture 24 (Fri Oct 29: the alternating group A_n; statement of Sylow theorems)
  • Lecture 25 (Mon Nov 1: statement of the Sylow theorems; examples, semi-direct products)
  • Lecture 26 (Wed Nov 3: Sylow theorems; proof; groups of order 12)
  • Lecture 27 (Fri Nov 5: groups of order 12; group presentations, Cayley graph, normal forms)
  • Lecture 28 (Mon Nov 8: presentations and normal forms: S_n, braid group, SL(2,Z))
  • Lecture 29 (Wed Nov 10: representations; sub-representations; irreducibility)
  • Lecture 30 (Fri Nov 12: irreducibility; Schur's lemma; representations of S_3)
  • Lecture 31 (Mon Nov 15: symmetric polynomials; the character of a representation)
  • Lecture 32 (Wed Nov 17: characters, orthogonality, and consequences; characters of S_4 and A_4)
  • Lecture 33 (Fri Nov 19: A_4; irreducible characters form a basis; the representation ring)
  • Lecture 34 (Mon Nov 22: irreducible characters of S_5 and A_5; restriction and induction)
  • Lecture 35 (Mon Nov 29: restriction and induction, Frobenius reciprocity; real representations)
  • Lecture 36 (Wed Dec 1: real and quaternionic representations; wrap-up, final info and advice)
  • Review part 1: Linear Algebra
  • Review part 2: Group Theory
  • Review problems from the textbooks

Course Summary:

Date Details Due