### Courses

#### Courses in Algebra

Students can earn 1 credit hour for attending the regular**Algebra Seminar**on

**Mondays at 4:30pm-5:20pm.**Graduate Students intending to attend the algebra seminar should include

**algebra seminar**in their course list to avoid being assigned teaching duties for that time.

#### Combined Undergraduate and Graduate Courses

**MATH 511 - Introduction to Algebraic Systems.** (3) I.

Properties of groups, rings, domains and fields. Examples selected from subsystems of the complex numbers, elementary number theory, and solving equations. Pr.: MATH 222.

**MATH 512 - Introduction to Modern Algebra.** (3) I, II.

Introduction to the basic algebraic systems, viz., groups, rings, integral domains, and fields, often drawing from elementary number theory. Special emphasis will be given to methods of theorem proving. Pr.: MATH 222 or consent of instructor.

**MATH 515 - Introduction to Linear Algebra.** (2-3) II.

Finite-dimensional vector spaces, linear transformations and their matrix representations, dual spaces, invariant subspaces, Euclidean and unitary spaces, solution spaces for systems of linear equations.

**MATH 730 - Abstract Algebra I.** (3)

Groups, rings, fields, vector spaces and their homomorphisms. Elementary Galois theory and decomposition theorems for linear transformations on a finite dimensional vector space. Pr.: MATH 512 or consent of instructor.

**MATH 731 - Abstract Algebra II.** (3) II.

Continuation of MATH 730. Pr.: MATH 730 or consent of instructor.

#### Graduate Courses

**MATH 810 - Higher Algebra I.** (3)

Theory of groups, theory of rings and ideals, polynomial domains, theory of fields and their extensions. Pr.: MATH 731.

**MATH 811 - Higher Algebra II.** (3)

Continuation of MATH 810. Pr.: MATH 810.

**MATH 812 - Homological Algebra I.** (3) I.

Introduction to the basics of homological and cohomological delta functors with applications to specific categories. Topics include Abelian categories, functors, exactness, chain complexes and maps, cochain complexes and maps, projective and injective resolutions, left and right derived functors, homological dimension, double complexes, spectral sequences, Tor and Ext functors, cohomology of groups, Lie algebras, and Hochschild homology. Pr.: MATH 811.

**MATH 813 - Homological Algebra II.** (3) II.

Introduction to special homology and cohomology theories. Topics include Galois cohomology, cyclic homology, local cohomology theories, localization, derived categories, and the calculus of fractions in Abelian categories. Pr.: MATH 812.

**MATH 814 - Lie Algebras and Representations I.** (3) I, in odd years.

Introduction to Lie algebras. Topics include ideals, homomorphisms, nilpotent and solvable algebras, radicals, killing forms, Cartan subalgebras, semisimple Lie algebras and root systems, classification of semisimple Lie algebras, conjugacy theorems, enveloping algebras and PBW theorems, Serre relations, and constructions of semisimple Lie algebras and their enveloping algebras. Pr.: MATH 811.

**MATH 815 - Lie Algebras and Representations II.** (3) II, in odd years.

Introduction to Kac-Moody algebras and their representations, Verma modules and BGG categories, and the Kac-Weyl character formula. In addition, special topics include quantum groups and their representations. Pr.: MATH 814.

**MATH 816 - Algebraic Geometry I.** (3) I.

Introduction to affine algebraic varieties over algebraically closed fields. Topics include Hilbert Nullstellensatz, Zarski topology, morphisms, differentials, smoothness, separability, and normality, algebraic and projective varieties, sheaf theory, sheaf cohomology, and vector bundles. Pr.: MATH 811.

**MATH 817 - Algebraic Geometry II.** (3) II.

Introduction to schemes. Topics include categories, representable functors, prime spectra, algebraic schemes, separable schemes, normal schemes, proper schemes, schemes with bases, completions, formal schemes, Zariski's Theorem, Frobenius morphisms, and etale morphisms. Pr.: MATH 816.

**MATH 818 - Introduction to Algebraic Groups I.** (3) I, in even years.

Introduction to algebraic groups. Topics include a review of algebraic geometry on varieties and morphisms, Lie algebras of algebraic groups, actions of algebraic groups over algebraic varieties, Jordan-Chevalley decompositions, solvable and unipotent algebraic groups, radicals and unipotent radicals, Borel subgroups, and parabolic subgroups. Pr.: MATH 811.

**MATH 819 - Introduction to Algebraic Groups II.** (3) II, in even years.

Introduction to reductive and semisimple algebraic groups, Bruhat decompositions, flag varieties, cohomology groups of line bundles over flag manifolds, Chevalley groups and their representations, Weyl modules and classification of irreducible modules, group functors and schemes, Hopf algebras and their representations. Pr.: MATH 818.

**MATH 890 - Riemann Surfaces.** (3) II.

Introduction to Riemann Surfaces. Topics will include complex charts, complex structures, holomorphic and meromorphic maps (and functions), order of poles and zeros, covering spaces, monodromy of holomorphic maps, differential forms, DeRham cohomology groups, integration on Riemann surfaces, Cech cohomology of sheaves, compact Riemann surfaces, finite theorems, divisors and sheaves of meromorphic forms, and the Riemann Roch Theorem. Pr.: MATH 702, MATH 811, and MATH 822.

**MATH 920 - Theory of Groups.** (3) I. Group representations and group characters, transfer, signalizer functors, theory of pushing-up, groups of Lie type, (B, N)-pairs, chamber systems and buildings, sporadic simple groups, amalgam methods, Bass-Serre theory. Pr.: MATH 811.

**MATH 925 - Group Representations and Character Theory I.** (3) I.

The basic topics in representation theory are covered: Schur's Lemma, irreducibility, class functions, characters, orthogonality relations, Frobenius-Schur theorem, induced characters and Frobenius reciprocity, Mackey's theorem, Clifford's theorem, exceptional characters and applications to group orders, generalized characters and Brauer's characterizations of characters. Pr.: MATH 811.

**MATH 926 - Group Representations and Character Theory II.**

Depending on the interests of the students, topics may be chosen from the following: modular representations, Brauer's theory of blocks, characters of the linear groups, homologically induced representations, representations of complex Lie algebras. Pr.: MATH 925.

**MATH 991 - Topics in Algebra.** (3)

On sufficient demand. Selected topics in modern algebra. May be repeated for credit. Pr.: Consent of instructor.