 ### Curriculum

Credits

24

32

32

8

8

104

Credits

4

4
• #### Calculus I

4

Introduction to differential calculus of functions of one variable. Review of elementary functions (including trigonometric, inverse trigonometric, exponentials and logarithms), limits, rates of change, the derivative and its properties, applications of the derivative, Differentiation and its various rules, Implicit differentiation, derivatives of elementary functions, Maxima and mimima, Linear approximations and differentials, Mean-value theorem, L'Hopital rule, Antiderivatives, Approximating area under the curve, Definite integrals, Fundamental Theorem of Calculus, Working with integrals, Vectors in plane, Dot products.

• #### Algebra I

4

This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

• 16

Credits

4

4
• #### Calculus II

4

Review of Definite Integrals, Review of the Fundamental Theorem of Calculus, Substitution, Velocity, Regions Between Curves, Volumes By Slicing, Length of Curves, Physical Applications, Integration by Parts, Trigonometric Integrals, Partial Fractions, Improper Integrals, Numerical Integration, Sequences, Infinite Seriies, Divergence and p-test, Ratio Test, Comparison Test, Alternating Series, Approximating Functions with Polynomials, Properties of Power Series, Taylor Series, Parametric Equations, Polar Coordinates, Linear Systems, Matrices, Linear map.

• #### Probability theory and Statistics

4

This course covers covers topics such as sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales, Counting Random variables, distributions, quantiles, mean variance Conditional probability, Bayes' theorem, base rate fallacy Joint distributions, covariance, correlation, independence Central limit theorem. The statistics part of the course contains Bayesian inference with known priors, probability intervals Conjugate priors, Bayesian inference with unknown priors Frequentist significance tests and confidence intervals Resampling methods: bootstrapping Linear regression, Sampling distributions; Point estimation - estimators, sufficiency, completeness, minimum variance unbiased estimation, maximum likelihood estimation, method of moments, Cramer-Rao inequality, consistency; Interval estimation; Testing of hypotheses - tests and critical regions, Neymann-Pearson lemma, uniformly most powerful tests, likelihood ratio tests; Basic non-parametric tests.

• 16

Credits

4
• #### Classical geometry

4

This course covers classical aspects of (largely plane) geometry: Euclidean geometry, vector geometry, projective geometry, transformation groups, and a tiny bit of hyperbolic geometry.

• #### Differential equations I

4

Study of differential equations, including modeling physical systems. Solution of first-order ODEs by analytical, graphical, and numerical methods. Linear ODEs with constant coefficients. Complex numbers and exponentials. Inhomogeneous equations: polynomial, sinusoidal, and exponential inputs. Oscillations, damping, resonance. Fourier series. Matrices, eigenvalues, eigenvectors, diagonalization. First order linear systems: normal modes, matrix exponentials, variation of parameters. Heat equation, wave equation. Nonlinear autonomous systems: critical point analysis, phase plane diagrams.

4

4
• 20

Credits

4
• #### Algebra II

4

This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices.

• #### Complex analysis

4

The topics spanned Cauchy's theorem, Taylor, Fourier and Laurent series and Laplace transforms. They also covered residues, evaluation of integrals, multivalued functions and potential theory in two dimensions.

4

4
• 20

Credits
• #### Differential equations II

4

The idea of the course was to give a solid introduction to PDE for advanced undergraduate students. We required only advanced calculus. The course went quite rapidly through a lot of material, but our focus was linear second order uniformly elliptic and parabolic equations. Some of the topics included the Laplace equation, harmonic functions, second order elliptic equations in divergence for, L-harmonic functions, heat equations, Green's function and heat kernels, maximum principles, Hopf's maximum principle, Harnack inequalities and gradient estimates for L-harmonic functions and more generally for solutions of heat equations. Morrey's and Capanato's lemmas, regularity of general solutions of second order elliptic equations in divergence form, the De Giorgi-Nash-Moser iteration argument, boundary regularity were also covered.

• #### Matrix algebra

4

The goals for this course are using matrices and also understanding them. The course include Solving Ax = b for square systems by elimination (pivots, multipliers, back substitution, invertibility of A, factorization into A = LU), Complete solution to Ax = b (column space containing b, rank of A, nullspace of A and special solutions to Ax = 0 from row reduced R), Basis and dimension (bases for the four fundamental subspaces), Least squares solutions (closest line by understanding projections), Orthogonalization by Gram-Schmidt (factorization into A = QR), Properties of determinants (leading to the cofactor formula and the sum over all n! permutations, applications to inv(A) and volume), Eigenvalues and eigenvectors (diagonalizing A, computing powers A^k and matrix exponentials to solve difference and differential equations), Symmetric matrices and positive definite matrices (real eigenvalues and orthogonal eigenvectors, tests for x'Ax > 0, applications), Linear transformations and change of basis (connected to the Singular Value Decomposition - orthonormal bases that diagonalize A), Linear algebra in engineering (graphs and networks, Markov matrices, Fourier matrix, Fast Fourier Transform, linear programming).

• #### Calculus III

4

This course includes vector algebra, dot product, matrices, determinant. Functions of several variables, continuity, differentiability, derivative. Parametrized curves, arc length, curvature, torsion. Vector fields, gradient, curl, divergence. Multiple integrals, change of variables, line integrals, surface integrals. Stokes' theorem in one, two, and three dimensions.

• #### Geomerty of curves and surfaces

4

Curves - Vector in R3, Parameterised curves, Unit-speed curve; Surfaces - The first fundamental form, Geodesics, The second fundamental form, Gaussian curvature, Plane curves, Simple closed curves, Global surfaces.

• 16

Credits
• #### Analysis of one and several variables

4

The real and complex numbers, the limit of a sequence of a number, functions and continuity, differentiation and integration, Integration over smooth curve on a plane, fundamental theorem of algebra, fundamental theorem of analysis, Riemann integral in n-variables, surface and surface integral, classical Gauss, Green and Stokes formulae, holomorphic and harmonic functions, Fourier series and transform.

• #### Algebra III

4

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

• #### Number theory

4

One of the central themes of modern number theory is the intimate connection between various arithmetic and analytic objects; these connections lie at the heart of many of recent breakthroughs and current programs of research, including the modularity theorem, the Sato-Tate theorem, the Riemann hypothesis, the Birch and Swinnerton-Dyer conjecture, and the Langlands program. Having said that, number theory is, after all, the study of numbers, thus our starting point is the ring Z, its field of fractions Q, and the various completions and algebraic extensions of Q. This means we will begin with some standard topics in algebraic number theory, including: Dedekind domains, decomposition of prime ideals, local fields, ramification, the discriminant and different, ideal class groups, and Dirichlet's unit theorem. We will spend most of the first half of the semester on these topics, and then move on to some closely related analytic topics, including zeta functions and L-functions, the prime number theorem, primes in arithmetic progressions, the analytic class number formula, and the Chebotarev density theorem. I also plan to cover at least the statement of the main theorems of local and global class field theory, as well as a preliminary discussion of Artin representations, Hecke characters, and their associated L-functions.

• #### Mathematical Finance

4

This course is ideal for students who want a rigorous introduction to finance. The course covers the following fundamental topics in finance: the time value of money, portfolio theory, capital market theory, security price modeling, and financial derivatives. We shall dissect financial models by isolating their central assumptions and conceptual building blocks, showing rigorously how their governing equations and relations are derived, and weighing critically their strengths and weaknesses.

• 16
• #### Other Courses not covered above

• Linear Programming and Optimization
• Discrete Mathematics
• Real Analysis
• Mechanics I
• Mechanics II
• Statistical mechanics
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