Core Courses
The Core Courses in RTA 1 consist of two-semester sequences in differential geometry and in algebraic topology. These introduce the basic concepts of these fields, including manifolds, differential forms, Riemannian metrics, fundamental groups and covering spaces, and homology and cohomology. These courses form the basis for research in all areas of geometry and topology, including in fields like symplectic topology, algebraic K-theory, topological combinatorics, or mathematical physics.
Analysis and Geometry on Manifolds
Differentiable manifolds, tangent bundles, tensor fields
Vector fields and their flows
Differential forms, Stokes’ theorem, de Rham cohomology
Vector bundles, connections, parallel transport, curvature
Plus perhaps:
Introduction to Riemannian geometry – Riemannian metrics, geodesics, exponential map
Introduction to symplectic or complex geometry Lie groups
Riemannian metrics, geodesics, exponential map
Jacobi fields, variation of length and energy
Relations between curvature and topology – theorems of Gauß–Bonnet, Hadamard–Cartan, Bonnet–Myers, Synge, etc.
Geometry of Riemannian immersions and submersions
Geometry of homogeneous and symmetric spaces
Plus perhaps:
Geometric differential equations
Symplectic and complex geometry (Riemann–Roch theorem, Hamiltonian dynamics)
Lie Groups
Algebraic Topology I
Point-set topology – quotients, (path-)connectedness, compactness
Gluing constructions and CW complexes
Fundamental group, covering spaces, Van Kampen theorem, functoriality
Plus perhaps:
Homology
Mapping degree
Algebraic Topology II
Chain complexes, homological algebra
Singular homology and cohomology
Excision and Mayer–Vietoris
Eilenberg–Steenrod axioms and (co-)homology of CW complexes
Tor, Ext and universal coefficients
Cup products
Plus perhaps:
Fundamental class and Poincaré duality
Higher homotopy groups
Spectrum of Advanced Courses
Geometric Analysis; Geometric Measure Theory; Quantum Field Theory; The Renormalization Group; Gauge Theories; Group Theory in Physics; Integrable Systems; Scattering Theory; Spectral Geometry; Pseudo-Riemannian Geometry; Symplectic Geometry and Topology; Advanced Surface Theory; Riemann Surfaces; Computational Topology; Homotopy Theory; Stable Homotopy Theory; Equivariant Topology; Geometric Knot Theory
