\magnification=\magstep1
\centerline{\bf Alex Zarechnak's Oral Exam Syllabus}
%Alex Zarechnak's Oral Exam Syllabus
\centerline{May 13, 2003}
\medskip
\noindent{\bf
I.  Bohmian Mechanics
}

	A.  Quantum Mechanics:  Standard Formalism

	{\narrower
		1.  Postulates

		2.  Ehrenfest's Theorem

		3.  Uncertainty Principle

		4.  Propagator for free particle (1-d)

		5.  Standard examples of solving time ind. Schrodinger Eq.

		6.  Formulation of Spin

		7.  Formulation of Identical Particles without spin

		8.  Decoherence

	}


	B.  Quantum Mechanics:  Foundational Issues

	{\narrower
		1.  Measurement Problem

		2.  Interpretations of Quantum Mechanics (non-Bohmian)

			{\narrower%\sevenrm
			a.  Copenhagen Interpretation

			b.  Decoherent Histories

			c.  Spontaneous Localization

			d.  Many Worlds

			}

		3.  Hidden Variables

		{\narrower
			a.  Nonlocality

{\sevenrm	\indent\indent	i.    EPR paradox
	
		\indent\indent	ii.   Bell's Inequality
}

			b.  Other Impossibility Theorems

{\sevenrm	\indent\indent	i.    Hardy's Theorem

		\indent\indent	ii.   Von Neuman's Theorem

		\indent\indent	iii.  Kochen-Specker Theorem

}	

		}


\noindent
	C.  Bohmian Mechanics

%	{\narrower\noindent
		1.  Postulates

		2.  Formulation of Spin

		3.  Formulation of Identical Particles without spin

		4.  Bohmian Mechanics and the Measurement Problem

		5.  Global Existence and Uniqueness of Trajectories

		6.  The Quantum Potential

		7.  Emergence of Standard Formalism

			{\narrower%\sevenrm
			a.  Quantum Equilibrium Hypothesis (QEH)

			b.  Equivariance

			c.  Conditional Wave Function

			d.  Effective Wave Function

			e.  Emergence of Born Statistics

			f.  Absolute Uncertainty

			g.  The Foundation of QEH

			h.  The Role of Operators

{\sevenrm	\indent\indent	i.     Bohmian Experiments

		\indent\indent	ii.    Discrete Experiments

		\indent\indent	iii.   Self-Adjoint Operators

		\indent\indent	iv.    PVMs

		\indent\indent	v.     POVMs

		\indent\indent	vi.    Measure Valued Quadratic Maps on H

		\indent\indent	vii.   Weak Formal Measurements

		\indent\indent	viii.  Strong Formal Measurements

		\indent\indent	ix.    Weak Formal Experiments

		\indent\indent	x.     Strong Formal Experiments

		\indent\indent	xi.    Measurements of Commuting Families

		\indent\indent	xii.   Sequential Measurements

}}

		8.  Contextuality

		9.  Bohmian Mechanics and the Impossibility Theorems

		10. Specific experiments from a Bohmian perspective

	{\narrower%\sevenrm
		a.  Stern-Gerlach experiment

			b.  Double Slit Experiment

			c.  Spectral Lines of Hydrogen

}}

\vfil\break
\noindent{\bf
II. Differential Geometry
}

      A. Basic Definitions and examples

{\narrower
	     1. Definitions of Manifolds, tangent vectors, vector fields, vector bundles

            2. Examples: Surfaces, Lie groups-Matrix groups, submanifolds

            3. Various mappings such as immersions, induced maps

}

      B. Tensors and differential forms

{\narrower
            1. Tensors of all types, tensor fields, maps and tensors

            2. Exterior algebra, exterior derivative, differential forms

            3. Orientability and n-forms

            4. Symmetrizing, alternating, contracting, and multiplying tensors

            5. Tensor Derivations

            6. Lie Derivatives

            7. Poincare Lemma and its partial converse

}
      C. Vector fields

{\narrower
            1. Existence and Uniqueness Theorems for ODE

            2. One-Parameter groups

            3. Vector Fields as flows and as differential operators

            4. Lie algebra of vector fields

}

      D. Metrics and Connections

{\narrower
            1. Definition of Metrics and Connections

            2. Covariant Derivative

            3. The Levi-Civita connection

            4. Parallel Translation

            5. Variations

            6. Geodesics

            7. Frame fields

            8. Cartan Structural Equations

            9. Exponential Map

            10. Hopf-Rinow-De Rham Theorem

            11. Curvature

	{\narrower%\sevenrm
                  a. Riemann Curvature Tensor

                  b. Sectional Curvature

                  c. Bianchi Identities

}}

      E. Integration on Manifolds

{\narrower
            1. Definition of the integral

            2. Manifolds with boundary

            3. Stokes' Theorem

}

      F. Surface theory

{\narrower
            1. Fundamental Forms, Gauss Curvature, Principal Curvature

            2. The Gauss Equation and Codazzi-Mainardi Equations

            3. Gauss-Bonnet Theorem

}
\end