Prerequisites 54 or a course with equivalent linear algebra content. 55 or an equivalent exposure to proofs
Course description for Math 113 (fall 2021) and Math 191 (spring 2022) Professor Harrison will be teaching a two-semester series in the academic year 2021-22 on abstract algebra that will will develop basic concepts of algebra from the viewpoint of category theory. Students who complete both courses will have covered the main requirements of Math 113 and Math 110, but will go much further than these courses because of the unification and insights afforded by category theory. Students who complete the series will be ready to take advanced upper division courses and possibly some beginning graduate courses (after consultation with the professor.) Category theory is becoming the standard language of much of modern mathematics and the majority of wikipedia pages on advanced mathematics make use of it. Let’s get on board!
The only prerequisite is an elementary knowledge of set theory and a willingness to work steadily and carefully, often without immediate rewards. The text is “Natural Linear Algebra,” and is written by Jenny Harrison and Harrison Pugh. It has received excellent reviews from H110 students, but it has grown too large for a one semester course. (A pdf will be provided, but expect it to evolve during the course.)
Although this text is considered to be abstract, a good fraction of the students who take it are from the applied sciences, including CS, engineering, economics, and physics. The authors’ interests are towards applications of algebra to enlighten analysis, topology, geometry and physics.
The syllabus below is taken from the table of contents of the text and is divided into two parts. However, the dividing line is not written in stone. We will pace the course according to the interests, needs and abilities of the engaged students. Former students of Professor Harrison from H110 or 113 are welcome to enroll in Math 191.
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Syllabus:
Math 113 Fall 2021:
I. Sets and functions:
1.1 Sets and relations: power sets, ordered pairs and Cartesian product of two sets, partially ordered sets, partitions and equivalence relations
1.2 Functions: set functions and commutative diagrams, range, image and preimage, properties of functions
1.3 Lists, sequences and indexed families
1.4 Cardinality
Vector spaces
2.1 Groups and fields: binary operations, rings and polynomials
2.2 Vector spaces: examples and properties
2.3 Subspaces: span of a subset, intersection, addition and translation
2.4 Vector space homomorphisms: isomorphisms, images and preimages of subspaces, endomorphisms
2.5 Free vector spaces and formal sums: free space of a field over a set, universal property for free vector spaces, the fee space of a vector space over a set
2.6 Algebras: bilinear maps, algebra of endomorphisms.
Quotients, direct sums and the isomorphism theorems
3.1 Quotients of vector spaces: universal property for quotient spaces, equivalence relations
3.2 Vector space isomorphism theorems
3.3 Direct sums, direct products, direct sum decomposition
3.4 Combining quotients and direct sums: exact sequences of vector spaces, complementary subspaces, splitting lemma, Injective and projective limits
3.5 Quotients of algebras: Ideals, isomorphism theorems for algebras
Basis and dimension
4.1 Linear independence and dependence, existence of a basis
4.2 Dimension: well-defined, rank-nullity, nilpotent operators
4.3 Dual vector spaces: double duals, dual of a linear map, coimage, cokernel, codimension
4.4 Matrices and homomorphisms: change of basis, rectangular matrices, block diagonal matrices
Groups and Rings
5.1 Examples, the dihedral group
5.2 Group homomorphisms
5.3 Subgroups: Images and preimages, intersection, product and translation, normal subgroups
5.4 Quotient groups and their universal property
5.5 Isomorphism theorems for groups
5.6 Subgroups generated by sets: Words and generators, cyclic groups, Lagrange’s theorem
5.7 The symmetric group
5.8 Free groups: universal property for free groups, abelianization of a group
5.9 Direct products of groups
5.10 Rings and Ideals: Quotient rings, maximal ideals, modules over a ring.
Category theory
6.1 Objects and morphisms: commutative diagrams, opposite categories
6.2 Functors: product categories and bifunctors, functor categories and natural transformations
6.3 Universal properties: Initial and terminal properties, examples, corollaries
6.4 Equivalent categories, units and counits
Math 191 Spring 2022:
Tensor algebra
7.1 Bilinear map principle
7.2 Tensor product: universal property, two-tensors, bases for tensor products, complexification functor, tensor product functors, trace and divergence (coordinate-free)
7.3 k-fold tensor products: multilinear maps, tensor algebra, generating sets, universal property for tensor algebra, tensor product of modules
Some important quotients of the tensor algebra
8.1 Quotients of graded algebras
8.2 Symmetric algebra
8.3 Exterior algebra
8.4 Determinant (coordinate-free) pushforward and pullback
Jordan-Chevalley decomposition and applications
9.1 Factoring polynomials and endomorphisms
9.2 Eigenvalues and eigenvectors, characteristic polynomial, generalized eigenvectors
9.3 Complex Jordan-Chevalley decomposition: semi simple endomorphisms, simultaneously diagonalizable commuting endomorphisms, categorification of the characteristic polynomial, complex Jordan normal forms
9.4 Cayley Hamilton theorem and minimal polynomials
9.5 Real Jordan-Chevalley decomposition
9.6 Invariants of endomorphisms
Operators on inner product spaces
10.1 Inner products: dual of an inner product space, adjoints, self-adjoint endomorphisms, category of inner product spaces
10.2 Norms: seminorms, relations on normed spaces
10.3 Orthonormal bases: Gramm-Schmidt process, isometries and orthonormal bases, Schur decomposition
10.4 The spectral theorem
10.5 Positive endomorphisms
10.6 Polar decomposition and singular value decomposition
Koszul complex and discrete calculus
11.1 Chain complexes and chain maps
11.2 Creation and annihilation operators
11.3 Exterior product and its adjoint, prederivative and its adjoint
11.4 Clifford algebra
11.5 Boundary operator and the quantum chain complex
11.6 Infinitesimal versions of the main integral theorems of calculus
11.7 Differential forms and chains.
Various applications will be at our fingertips by this point. We can pick and choose!
