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Convexity and the Krein-Milman theorem. Additional topics sperm swallowing may include compact operators, spectral theory of compact operators, and applications to integral equations. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. Spectral theorem for bounded self-adjoint and normal operators (both forms: the spectral integral and the "multiplication operator" formulation). Riesz theory of compact operators. Positivity, spectrum, GNS construction.

Density theorems, topologies and normal maps, traces, comparison of projections, type classification, examples of factors. Additional topics, for example, Tomita Takasaki theory, subfactors, group actions, and noncommutative probability.

Sperm swallowing remainder of the course may treat either sheaf cohomology and Stein manifolds, or the theory of analytic subvarieties and spaces. Flows, Lie derivative, Lie groups and algebras.

Additional topics selected by peer reviewed. Homotopy theory, fibrations, relations between homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Measure theory concepts needed for probability. Laws of large numbers and central limit theorems for independent random variables.

Conditional expectations, martingales and martingale convergence theorems. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor. Six hours of Lecture per week for 8 weeks.

Terms offered: Fall 2021, Fall 2020, Fall 2019 The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Second-order elliptic equations, parabolic and hyperbolic equations, calculus sperm swallowing variations methods, additional topics selected by instructor. Advanced topics in sperm swallowing offered according to students demand and faculty availability.

Fourier and Laplace transforms. Completeness and sperm swallowing theorems. Interpolation theorem, definability, theory of models.

Relativization, degrees of unsolvability. Constructive ordinals, the hyperarithmetical and sperm swallowing hierarchies. Recursive objects of higher type.

Partial differential equations: stability, accuracy and convergence, Von Neumann and CFL conditions, finite difference solutions of sperm swallowing and parabolic equations. Finite sperm swallowing and finite element solution of elliptic equations. Ultraproducts and ultralimits, saturated models. Methods for establishing decidability and completeness. Model theory of various languages richer than first-order. Operations on sets and relations.

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