Basic techniques and methods for solving ordinary differential equations. Topics: linear, separable, and exact equations, existence and uniqueness theorems, difference equations, basic theory of higher order equations, variation of parameters, undetermined coefficients, series solutions, Laplace transform, systems of equations.

An introduction to the two pillars of modern physics. Special relativity and an introduction to general relativity, experimental basis of quantum theory, quantization of light, Hilbert space structure of qubits, multiparticle states, entangled states, incompatible observables, Bell inequalities, introduction to quantum information and quantum computing, Schrödinger equation.

Intermediate Mechanics covers much of what is generally covered in an upper division university mechanics course. The course is a sophisticated treatment of the mechanics of particles and systems. Rigid body dynamics, oscillating systems, Hamilton's principle for continuous systems, waves, general wave equation, and fluid dynamics.

Theory of differentiation and integration of complex functions. Topics: algebra of complex numbers, complex functions, multi-valued functions, exponentials, logarithms, analyticity, integrals, power series, Laurent series, residues, isolated singularities, poles and zeros.

Intermediate Mechanics covers much of what is generally covered in an upper division university mechanics course. The course is a sophisticated treatment of the mechanics of particles and systems. Tensors, vector calculus, Newtonian laws of mechanics and gravitation, oscillations, Green's functions, calculus of variations, Lagrangian mechanics, orbits, and physics in rotating frames of reference.

Differential calculus for functions of two or more variables. Topics: vectors and vector-valued functions in 2-space and 3-space, tangent and normal vectors, curvature, functions of two or more variables, partial derivatives and differentiability, directional derivatives and gradients, maxima and minima, optimization using Lagrange multipliers.

An introductory course in Linear Algebra. Topics: matrices, linear equations, vector spaces, bases, coordinates, linear transformations, eigenvectors, eigenvalues, and diagonalization.

Theory of differential equations involving functions of more than one variable. Topics: first order equations, classification of second order equations, initial-boundary value problems for heat equation, wave and related equations, separation of variables, eigenvalue problems, Fourier series, existence and uniqueness questions.

Integral calculus for functions of two or more variables. Topics: double and triple integrals, change of variables and the Jacobian, vector fields, line integrals, independence of path and the fundamental theorem of line integrals, Green's theorem, divergence theorem, and Stokes' theorem.

Theory of abstract algebra, with particular emphasis on applications involving symmetry. Topics: groups, quotient groups, symmetry groups in three dimensions, Pólya-Burnside method, rings and fields, polynomial and Euclidean rings, quotient rings, field extensions, and geometrical constructions.

Introduction to number theory and its applications. Topics: Euclid's algorithm, divisibility, prime numbers, congruence of numbers, theorems of Fermat, Euler, Wilson, Lagrange's theorem; residues of power, quadratic residues; magic squares; continued fractions; Diophantine equations; quadratic fields and quadratic integers.

Theory of functions of a real variable. Topics: sequences, series, limits, continuity, uniform continuity, sequences and series of functions, uniform convergence, differentiation.

An introduction to optics and thermodynamics. Topics: temperature, properties of matter, introduction to the kinetic theory of matter, light and electromagnetic waves, reflection and refraction of light, lens systems, interference and diffraction.