In addition to the above 14 basic courses, the BMS offers three one-semester courses that will provide students the opportunity to fill potential gaps in their general mathematical background. These courses will not be exclusively aimed at BMS students, but will be part of the master programs of the universities.
Outline of the contents:
Algebraic topology
◦ Review of point-set topology
◦ Fundamental group, covering spaces, van Kampen theorem
◦ Homology and cohomology
◦ Cell complexes and manifolds
◦ Exact sequences, Mayer–Vietoris, Poincaré duality
◦ Euler and Lefschetz numbers, fixed point theorems
Complex analysis
◦ Holomorphic functions, power series
◦ Path integrals and Cauchy’s theorem
◦ Singularities
◦ Analytic continuation and homotopy, residue calculus
◦ Riemann mapping theorem
◦ Elliptic functions
Functional analysis
◦ Banach and Hilbert spaces
◦ Bounded linear operators
◦ Basic principles (closed-graph, open mapping, Banach–Steinhaus)
◦ Dual spaces and convexity
◦ Weak and weak-star convergence
◦ Spectral theory for compact operators
Nonlinear optimization
◦ Nonlinear optimization problems, modeling
◦ Optimality conditions
◦ Numerical methods (Newton-type, conjugate gradient, trust-region)
◦ Nonsmooth optimization
◦ Interior point methods
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