General Elective

MATH 312 / MATHEMATICAL STATISTICS**Sınıf:** **Credit:** 3**Precondition:** MATH. 203 or consent of the instructor

Decision theory; estimation; confidence intervals; hypotheses testing; large-sample theory; efficiency of alternative statistical procedures.

MATH 320 / LINEAR ALGEBRA**Sınıf:** **Credit:** 3**Precondition:** MATH. 107 and (MATH 103 or MATH 205) or consent of the instructor

Finite-dimensional real and complex vector spaces, bases of a vector space, linear maps, dual spaces, quadratic forms, self-adjoint and unitary transformations, eigenvalue problem, canonical form of a linear transformation, tensors, and applications.

MATH 350 / SELECTED TOPICS IN MATHEMATICS I**Sınıf:** **Credit:** 3**Precondition:**

Detailed examination of current topics in Mathematics.

MATH 351 / SELECTED TOPICS IN MATHEMATICS II**Sınıf:** **Credit:** 3**Precondition:** MATH. 203 or consent of the instructor

Detailed examination of current topics in Mathematics.

MATH 390 / INDEPENDENT STUDY I**Sınıf:** **Credit:** 3**Precondition:**

Investigation of one or more topics of interest with the guidance of an instructor. Presentation of a research proposal at the end of the term.

MATH 395 / INDEPENDENT STUDY**Sınıf:** **Credit:** 1.5**Precondition:**

Investigation of one or more topics of interest with the guidance of an instructor. Presentation of a research proposal at the end of the term.

MATH 401 / COMPLEX ANALYSIS**Sınıf:** **Credit:** 3**Precondition:** MATH. 301 or consent of the instructor

Complex numbers and functions; exponential and trigonometric functions; infinite series and products; limits, continuity and derivatives of complex functions; Cauchys theorem; Taylor and Laurent series; conformal mapping.

MATH 402 / TOPOLOGY**Sınıf:** **Credit:** 3**Precondition:** MATH. 301 or consent of the instructor

Topological spaces, subspaces, continuous functions, base for a topology, separation axioms, compactness, locally compact spaces, connectedness, path connectedness, finite product spaces, set theory and Zorn?s lemma, infinite product spaces, quotient spaces, homotopic paths, the fundamental group,induced homomorphisms, covering spaces, applications of the index, homotopic maps, maps into the punctured plane, vector fields, the Jordan curve theorem.

MATH 403 / FUNCTIONAL ANALYSIS**Sınıf:** **Credit:** 3**Precondition:** MATH. 301 or consent of the instructor

Basic principles of normed spaces. Normed and Banach spaces; Hilbert spaces; linear operators; dual spaces. Basic principles of functional analysis: Hahn-Banach theorem; open mapping theorem; uniform boundedness theorem, Krein-Milman theorem. Applications.

MATH 404 / GRAPH THEORY**Sınıf:** **Credit:** 3**Precondition:** MATH 104 or Concent of the Instructor

Fundamental concepts in graph theory; trees; matchings in graphs; connectivity and planarity; the colorings of graphs and diagraphs; Hamilton cycles; matroids.

MATH 405 / DIFFERENTIAL GEOMETRY**Sınıf:** **Credit:** 3**Precondition:** MATH. 208

Differential geometry of curves and surfaces in three-dimensional space; intrinsic geometry; geodesics; curvature; Gauss-Bonnett theorem.

MATH 406 / ACTUARIAL MATHEMATICS**Sınıf:** **Credit:** 3**Precondition:**

Contingency mathematics in the areas of life and health insurance, annuities, and pensions from both the probabilistic and deterministic approaches. Survival distribution and life tables; life insurance; life annuities; net premiums; net premium reserves; multiple life functions; multiple decrement models; valuation theory for pension plans; the expense factor; and non-forfeiture benefits and dividends.

MATH 407 / COMBINATORIAL ANALYSIS**Sınıf:** **Credit:** 3**Precondition:**

Problems of enumeration, structure, and optimization in such finite or discrete systems as graphs, matroids, partially ordered sets, lattices, partitions, codes and block designs.

MATH 408 / GAME THEORY**Sınıf:** **Credit:** 3**Precondition:**

Games in extensive form; pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations and solution concepts; related topics and applications.

MATH 409 / OPTIMIZATION**Sınıf:** **Credit:** 3**Precondition:**

Convergence of sequences in Rn, multivariate Taylor's theorem. Optimality conditions for unconstrained optimization. Newton's and quasi-Newton methods for unconstrained optimization. Equality-constrained optimization, Karush-Kuhn-Tucker theorem for constrained optimization. Inequality-constrained optimization. Interior point methods for constrained optimization. Linear and quadratic programs, their numerical solution.

MATH 410 / NUMBER THEORY**Sınıf:** **Credit:** 3**Precondition:** MATH. 205 or consent of the instructor

Quadratic Reciprocity, Quadratic Forms, Gauss' Composition Law and Genus Theory, Cubic and Biquadratic Reciprocity, Number Fields, Hilbert Class Field, Orders in Imaginary Quadratic Fields, The Class Number, Class Field Theory and Cebatorev Density Theorem, Norms and Ideles, Elliptic Functions and Theory of Complex Multiplication. Divisibility, Primes, Congruences, Prime Modules and Primitive Roots, Groups, a review of Rings and Fields, Arithmetic Functions, Diophantine Problems, Farey Fractions ad Geometry of Numbers, Continued Fractions, Multiplicative Number Theory and Dirichlet Series.

MATH 411 / STOCHASTIC CALCULUS AND STOCHASTIC SYSTEMS I**Sınıf:** **Credit:** 3**Precondition:**

Modeling of stochastic systems. Introduction to Markov chains, renewal processes, queuing theory, reliability and time series models; Ito Calculus, Fokker-Planck and Kolmogorov differential equations; applications to the problems of environmental as well as physical systems such as allocation of resources, inventory control, transportation and finance.

MATH 412 / STOCHASTIC CALCULUS AND STOCHASTIC SYSTEMS II**Sınıf:** **Credit:** 3**Precondition:**

Modeling of stochastic systems. Introduction to Markov chains, renewal processes, queuing theory, reliability and time series models; Ito Calculus, Fokker-Planck and Kolmogorov differential equations; applications to the problems of environmental as well as physical systems such as allocation of resources, inventory control, transportation and finance.

MATH 413 / PROBABILITY THEORY**Sınıf:** **Credit:** 3**Precondition:** MATH Math 203 and Math 211 or consent of the instructor

Review of elementary probability; multivariate random variables and their functions; conditional distribution and expectation; generating functions and transforms; order statistics; multivariate normal distribution; types of convergence; laws of large numbers; central limit theorem.

MATH 414 / ALGEBRAIC GEOMETRY**Sınıf:** **Credit:** 3**Precondition:** MATH. 206 or consent of the instructor

Basic notions of commutative algebra and homological algebra: category of modules over a ring, flatness, Ext and Tor. General properties of schemes: affine schemes. projective schemes, dimension, projective and proper morphisms. Normal and regular schemes. Flat and smooth morphisms. Zariski's main theorem and applications. Coherent sheaves and Cech Cohomology.

MATH 450 / SELECTED TOPICS IN MATHEMATICS I**Sınıf:** **Credit:** 3**Precondition:**

Detailed examination of current topics in Mathematics.

MATH 451 / SELECTED TOPICS IN MATHEMATICS II**Sınıf:** **Credit:** 3**Precondition:** MATH. 302 or consent of the instructor

Detailed examination of current topics in Mathematics.

MATH 490 / INDEPENDENT STUDY II**Sınıf:** **Credit:** 3**Precondition:**

Work on the research proposal resulting from MATH 390 with the guidance of an instructor, culminating in a research paper suitable for presentation or publication.

MATH 491 / HONORS PROJECT**Sınıf:** **Credit:** 3**Precondition:**

Available to students with a GPA equal to or greater than 3.00 and with consent of the instructor.