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

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

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.

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.

Detailed examination of current topics in Mathematics.

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

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.

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.

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.

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.