Normed and Banach spaces; linear operators; duality; inner product and Hilbert spaces; Riesz representation theorem; Hahn-Banach theorem; uniform boundedness principle; open mapping theorem; strong, weak and weak* convergence.

Completeness axiom for real numbers; convergent sequences; compactness; continuous functions; differentiation; linear and topological structure of Euclidean spaces; limit, compactness and connectedness in a Euclidean space; continuity and differentiation of functions of several variables; inverse and implicit function theorems.

Natural numbers; modular arithmetic; introduction to groups; cyclic and permutation groups; homomorphisms and isomorphisms; normal; factor, simple and free groups; introduction to rings, integral domains, and fields; factor rings and ideals; extension fields; outline of Galois theory.

A course in basic concepts and tools of statistics for students who will study social and Behavioral sciences. Topics to be covered are representation of quantitative information in social sciences, forms of numerical data, creating and interpreting graphical and tabular summaries of data, descriptive statistics, estimation of population parameters, confidence intervals, basic hypothesis testing, t-statistics, chi-squared tests and analysis of variance.

Limits and continuity; derivative and properties of differentiable functions; mean value theorems, Taylor's formula, extreme values; indefinite integral and integral rules; Riemann integral and fundamental theorem of calculus; L'Hospital's rule; improper integrals.

Metric spaces and their topology; continuity, compactness and connectedness in a metric space; completion of a metric space; differentiation and Riemann integration; sequences and series of functions; uniform convergence; Ascoli-Arzela theorem; Stone-Weierstrass theorem; Banach fixed-point theorem and its applications.

Review of Unique Factorization Domains and Principal IdeaI Domains, Maximal and prime ideals, Nilradical, Local rings, Modules, Cayley-Hamilton theorem, Nakayama's lemma, Exact and split exact sequences, Noetherian rings, Noetherian modules, Hilbert basis theorem, Integral extensions, Integral closure, Non-singularity, Normal rings, Noether normalization, Hilbert nullstellensatz, Spec(A), Localization, Support of a module and the associated primes, Discrete valuation rings, Trace and separability, Completion, Artin-Rees Lemma, An overview of further topics: Dimension theory, Regular rings, Connections with geometric notions.

First order differential equations. Second order linear equations. Series solutions of ODE?s. The Laplace transform and applications. Systems of first order linear equations. Nonlinear equations and systems:existence, uniqueness and stability of solutions. Fourier series and partial differential equations.

Descriptive statistics; measures of association, correlation, simple regression; probability theory, conditional probability, independence; random variables and probability distributions; sampling distributions; estimation; inference (confidence intervals and hypothesis testing). Topics are supported by computer applications.

Counting problems; combinatorial methods; integers, divisibility and primes; graphs and trees; combinatorics in geometry; introduction to complexity and cryptography.