A13: Geometry (2025-26)
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- Lecturer: Profile: Jason Lotay
Geometry is a fundamental topic in mathematics with links to algebra, analysis, number theory and topology, as well as applied mathematics and theoretical physics, including classical mechanics, mathematical biology and General Relativity. This course will introduce students to foundational geometric concepts, such as submanifolds and manifolds, transversality and degree, which provide the essential tools for further study in many aspects of algebraic and differential geometry and topology. The course will discuss key examples of relevance both within and beyond geometry and give various applications of the theory. Finally, the course will look at important examples of geometric structures and their symmetries, namely projective and hyperbolic space. Aspects of these geometries are of central importance in many areas of mathematics, and we will discuss several of these links, including to Möbius transformations from complex analysis.
This course will provide an understanding of fundamental concepts underlying both differential and algebraic geometry, as well as important topics relevant to advanced algebra, applied mathematics, mathematical physics (both classical and more modern), number theory and topology. As such, the course will provide a firm foundation for pursuing further study in geometry and topology.
Review of total derivative for functions between Euclidian spaces, including relation between holomorphic and conformal maps.
Inverse and Implicit Function Theorems. Manifolds arising as submanifolds of Euclidean space and from matrix groups. Lagrange multipliers.
Sard's Theorem. Transversality and intersections of submanifolds. Degree of a map with applications, including Fundamental Theorem of Algebra.
Abstract definition of manifold. Projective space and projective transformations, including intersections and Möbius transformations. Hyperbolic space and transformations, with link to Möbius transformations.