Lisans Ders Kataloğu

Vector calculus, functions of several variables, directional derivatives, gradient, Lagrange multipliers, multiple integrals and applications, change of variables, coordinate systems, line integrals, Green's theorem and its applications.

Functions of one variable, properties of quadratic, cubic, exponential and logarithmic functions, compound interest and annuities, limits, continuity and differentiation, applied maximum and minimum problems, basic integration techniques, sequences and series.

Sequences, limits and continuity, differentiation and its applications, integration and its applications, fundamental theorem of calculus, transcendental functions, improper integrals.

Counting, the pigeonhole principle, permutations, combinations, binomial coefficients, generalized permutations and combinations, discrete probability, linear recurrence relations, generating functions, inclusion-exclusion, relations, closures of relations, equivalence relations, construction of integers and rationals, partial orderings, graphs.

First-order differential equations, linear equations, homogeneous and non-homogeneous, series solutions, the Laplace transform, systems of first-order linear equations, boundary value problems, Fourier series.

Groups, subgroups, cyclic groups, generating sets, permutations, orbits, cycles, alternating groups, cosets, Lagrange's Theorem, direct products, finite abelian groups, homomorphisms, normal subgroups, factor groups, simple groups, group actions, isomorphism theorems, Sylow's theorems.

Infinite series, conditionally convergent series, double series, uniform convergence, series and sequences of functions, power series, improper integrals with parameters, differentiation of transformations, linear functions, differentials and inverses of transformations, inverse and implicit function theorems.

Representations, irreducibility, Maschke's theorem, semisimplicity, characters, character tables, orthogonality relations, induction and restriction of characters, Mackey decomposition theorem, algebraic integers, Burnside's p^aq^b-theorem, Frobenius' normal complement theorem.

Divisibility theory, Euclidean algorithm, congruences, solutions of polynomial congruences, primitive roots, power residues, quadratic reciprocity law, arithmetical functions, distribution of prime numbers, Pell's equation, quadratic forms, some diophantine equations.

Elementary measure theory, sets of measure zero, Lebesgue measure, Lebesgue measurable sets and functions, Lebesgue integral, convergence theorems, the space L^1, absolutely continuous functions, functions of bounded variation, Hilbert space L^2, Fourier series.

Solutions of nonlinear equations, bisection, Newton, and fixed point iterations, direct solutions of linear systems, Gaussian elimination with partial pivoting, LU and Cholesky factorizations, iterative solutions of linear systems, vector and matrix norms, Neumann series, Jacobi, Gauss-Seidel and SOR iterations, projection methods, steepest descents, conjugate-gradient and GMRES methods, matrix eigenvalue problem, power method, Givens rotations, Jacobi iteration, Hessenberg form, QRiteration, polynomial interpolation, Lagrange polynomials, Newton’s divided differences, Chebyshev polynomials, least squares, spline interpolation.

Probability, conditional probability, Bayes’ theorem, independence, discrete and continuous probability distributions, expected value, estimation, confidence intervals, tests of hypothesis for one parameter, goodness of fit test, linear regression, analysis of variance.

Existence and uniqueness theorems, phase portraits in the plane, linear systems and canonical forms, non-linear systems, linearization, stability of fixed points, limit cycles, Poincaré-Bendixson theorem.

Sieve methods, lattices, distributive lattices, incidence algebra, Mobius inversion formula, Mobius algebras, generating functions, exponential formula, Lagrange inversion formula, matrix tree theorem.

Selected topics in the history of mathematics and related fields.

Propositional and quantificational logic, formal grammar, semantical interpretation, formal deduction, completeness theorems, selected topics from model theory and proof theory.

Language and structure, theory, definable sets and interpretability, compactnees theorem, complete theories, Löwenheim-Skolem theorems, quantifier elimination, algebraic examples.

Introduction to algebraic number theory and algebraic curves, geometric introduction to function fields of curves, affine and projective varieties, divisors on curves, Riemann-Roch theorem, basics of elliptic curves.

Convergent series of meromorphic functions, entire functions, Weierstrass' product theorem, partial fraction expansion theorem of Mittag-Leffler, gamma function, normal families, theorems of Montel and Vitali, Riemann mapping theorem, conformal mapping of simply connected domains, Schwarz-Christoffel formula, applications.

Review of vector spaces, normed vector spaces, lP and LP spaces, Banach and Hilbert spaces, duality, bounded linear operators and functionals.

Numerical solutions of initial value problems for ordinary differential equations (ODE), Picard-Lindelof theorem, single step methods including Runge-Kutta methods, examples and consistency, stability and convergence of multistep methods, numerical solution of boundary value problems for ODE’s, shooting, finite difference, and collocation methods, finite element methods, Riesz and Lax-Milgram lemmas, weak solutions, numerical solutions of partial differential equations, examples of finite difference methods and their consistency, stability, and convergence including Lax-Richtmeyer equivalence theorem, Courant-Friedrichs-Lewy condition, and von Neumann analysis, Galerkin methods, Galerkin orthogonality, Cea’s lemma, finite element methods for elliptic, parabolic and hyperbolic equations.

First variation of a functional, necessary conditions for an extremum of a functional, Euler's equation, fixed and moving endpoint problems, isoperimetric problems, problems with constraints, Legendre transformation, Noether's theorem, Jacobi's theorem, second variation of a functional, weak and strong extremum, sufficient conditions for an extremum, direct methods in calculus of variations, the principle of least action, conservation laws, Hamilton-Jacobi equation.

Topological spaces, compactness, connectedness, continuity, separation axioms, homotopy, fundamental group.

Review of special relativity, differentiable manifolds, tensors, Lie derivative, covariant derivative, parallel transport, geodesics, curvature, Einstein's field equations, Schwarzschild black hole, Cauchy problem, maximally symmetric spacetimes, singularity theorems.

Smooth functions and smooth manifolds embedded in Euclidean space, tangent spaces, immersions, submersions, transversality, applications of the implicit function theorem, Morse functions, Sard's theorem, Whitney embedding theorem, intersection theory mod 2, Brouwer fixed point theorem, Borsuk-Ulam Theorem, and other related results.

Plane Euclidean geometry and its group of isometries, affine transformations in the Euclidean plane, fundamental theorem of affine geometry, finite group of isometries of R, Leonardo da Vinci's theorem, geometry on the sphere S, motions of S, orthogonal transformations of R, Euler's theorem, right triangles in S, projective plane, Desargues' theorem the fundamental theorem of projective geometry.

Individual research supervised by a member of the department.