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Advanced Courses
Advanced Courses
MATH 4586 - Algebraic Geometry, Fall 2018, (MWR, 10:30-11:35am, International Village 018). Instructor: Mboyo Esole.
MATH 4586 - Algebraic Geometry, Fall 2018, (MWR, 10:30-11:35am, International Village 018). Instructor: Mboyo Esole.
Concentrates on the basics of algebraic geometry, which is the study of geometric objects, such as curves and surfaces, defined by solutions of polynomial equations. Algebraic geometry has links to many other areas of mathematics—number theory, differential geometry, topology, mathematical physics—and has important applications in such fields as engineering, computer science, statistics, and computational biology. Emphasizes examples and indicates along the way interesting problems that can be studied using algebraic geometry.
MATH 3275 - Advanced Group Theory, Fall 2019, (MR, 11:45-1:25pm, Hastings 118). Instructor: Rob Silversmith
MATH 3275 - Advanced Group Theory, Fall 2019, (MR, 11:45-1:25pm, Hastings 118). Instructor: Rob Silversmith
Introduces homomorphisms, subgroups, normal subgroups, quotient groups, and group actions, illustrated with a variety of examples. Subsequent topics include the class equation, simple groups, the Sylow theorems, and their applications to the classification of finite simple groups. Discusses classical matrix groups, with an emphasis on SU(2) and SO(3) as fundamental examples, and introduces the notion of a Lie algebra. Develops representation theory of finite groups and its correspondence to the representation theory of compact Lie groups sketched, again using SU(2) as an example.
Introduces the basics of commutative algebra. Emphasizes rigorously building the mathematical background needed for studying this subject in more depth. Seeks to prepare students for more advanced classes in algebraic geometry, robotics, invariant theory of finite groups, and cryptography. Covers geometry, algebra, and algorithms; Grobner bases; elimination theory; the algebra-geometry dictionary; robotics and automatic geometric theorem proving.
Introduces students to the geometry of smooth manifolds. Topics include transversality, oriented intersection theory, Lefschetz fixed-point theory, Poincare-Hopf theorem, Hopf degree theorem, differential forms, and integration. Explores concepts and techniques of smooth geometry for understanding significant topological characteristics of manifolds.
Introduces the mathematical study of knots and links in space. Knot theory provides a concrete application for fundamental ideas in topology. Topics include knot diagrams and Reidemeister moves; connected sum and prime decomposition; satellites and companionship; Seifert surfaces and knot genus; Seifert matrices; knot signature and determinant; the Alexander polynomial; the Kauffman bracket and Jones polynomial; and braid presentations. Also discusses examples of knotting phenomena in physical systems.
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