Mathematics

Study the fundamentals of real analysis, including Axioms of the real numbers, supremum and infimum; Countable sets; Sequences and series; Open and closed sets, compactness; Limits, continuity, differentiability; Sequences and series of functions, uniform convergence, power series; Integration. Please note that lectures alternate during the week so that students can take any of MAM2012S, MAM2013S and MAM2014S concurrently.

The course covers basic theory of analytic functions including elementary properties of analytic functions in one variable. Complex differentiability and Cauchy-Riemann equations. Calculation rules. Elementary examples of analytic functions: power series expansions, exponential functions, branches of logarithms, and functions defined by these calculation rules. Contour integrals in the complex plane. Cauchy’s integral theorem and integral formula. Existence of a primitive function and local power series expansion of analytic functions. Cauchy estimates, Liouville’s theorem, and the fundamental theorem of algebra. Theory of meromorphic functions, Laurent series expansion, and the residue theorem. Residue calculus. Further elements of the theory of holomorphic functions such as argument principle, Rouché’s theorem, and open mapping property. Harmonic functions. Regularity, existence of harmonic conjugate, mean value property, maximum principle, Poisson integrals.

This course gives knowledge of and familiarity with concepts and methods from the theory of dynamical systems which are important in applications within almost all subjects in science and technology. In addition, the course should develop the student's general ability to assimilate and communicate mathematical theory, to express problems from science and technology in mathematical terms and to solve problems using the theory of dynamical systems.

This course introduces the language and methods of the area of Discrete Mathematics and show how discrete mathematics can be used in modern computer science (with the focus on algorithmic applications). Topics covered include (1) sets, relations and functions; (2) basic logic, including propositional logic, logical connectives, truth tables, propositional inference rules and predicate logic; (3) proof techniques, including the structure of mathematical proofs, direct proofs, disproving by counterexample, proof by contradiction; (4) basics of counting, including counting arguments, the pigeonhole principle, permutations and combinations, solving recurrence relation; (5) graphs and trees; (6) discrete probability, including finite probability space, axioms of probability, conditional probability; and, (7) linear algebra, including vectors, matrices and their applications. The course is offered in a blended-learning format. Students are provided with a set of video lectures that they can watch multiple times. Student contact time is in a tutorial format aimed at reinforcing the principles introduced in the online lectures and giving students time to do exercises under the supervision of tutors. Co-requisites: MAM1004S or MAM1005H (unless a pass has been obtained for MAM1004F or MAM1031F or equivalent).

This course covers homology, cohomology and applications, CW-complexes, and basic notions of homotopy theory.

The main content of this course consists of several kinds of polynomials of graphs, groups and graphs, and strongly regular graphs. It will enable the students know the algebric method to study combinatorial structures.

 Study the fundamentals of abstract algebra and number theory, including induction, strong induction and Well-Ordering axiom; Divisibility and prime factorization; Modular arithmetic; Permutations; Groups, Subgroups, Cyclic groups; Isomorphisms; Simple groups, Factor groups, Lagrange's Theorem; The First Isomorphism Theorem. Please note that lectures alternate during the week so that students can take any of MAM2012S, MAM2013S and MAM2014S concurrently. Course entry requirements: MAM1031F and MAM1032S or equivalent.

This course is part of the Laurea Magistrale degree program and is intended for advanced level students. Enrollment is by permission of the instructor. This course is a problem-based introduction to probability and stochastic processes. No previous knowledge of probability is assumed, but knowledge of calculus in one or more variables is required.

The course is divided into 6 parts:

1. Axiomatic definition of probability. Uniform probability spaces. Counting methods: replacement, ordering. Conditional probability. Independence for events. The law of total probability. Bayes' rule.
2. Discrete random variables. Independence for random variables. Joint, marginal, and conditional densities. Common random variables and their interpretation: Bernoulli, discrete uniform, binomial, hypergeometric, geometric, Poisson, Pascal.
3. Expectation of discrete random variables. Variance and its properties. Expectation and variance of common random variables. Covariance and correlation. Variance of a sum. Null correlation and independence. Linear prediction.
4. Conditional expectation and its properties. Conditional Variance. Sigma-algebras, Continuous Random variables. The Uniform and Exponential distributions. Distribution functions and densities.
5. Marginal, joint and conditional densities. Gamma, Normal and Cauchy distribution. Derived Distributions: monotonic and general case. Conditional Expectation. Law of total expectation. Markov and Chebishev Inequalities.
6. Convergence of Random Variables. The Weak and Strong Laws of Large Numbers. Characteristic Functions and their properties. CF of a sum. CF of common random variables. The Central Limit Theorem.

At the end of the course the student has good knowledge of probability theory of discrete and continuous random variables. Particular attention is paid to the theory of stochastic processes, both diffusive and with jumps. The student masters the main techniques of stochastic calculus applied to finance, such as stochastic differential and integral domain and change of measure techniques.