Mathematics

 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.

Many statistical models try to explain how one variable relates to others. In this course, how to analyze multiple variables simultaneously, multivariate analysis. Both how they depend upon other variables, but also how they depend upon each other. With the tremendous amount of data available nowadays, e.g. in genetics, it is often the case that the number of variables is far greater than the number of observations. This demands special techniques that are learned in this course. Course content includes matrices and multivariate normal distribution, singular value decomposition and its geometric interpretation, principal component analysis including its functional formulation, factor analysis, cluster analysis, prediction theory including prediction with high-dimensional predictors, penalized regression and prediction, sparse matrices, linear discriminant analysis, and large-scale inference.  

This course covers various proof techniques and provides practice proving sample propositions using these techniques. Students learn basic discrete mathematics and theoretical computer science topics such as sets and functions, and practice proving propositions related to these topics. The course also covers intermediate discrete mathematics topics, including trees and graphs, and provides practice proving related propositions. Students also learn additional discrete mathematics topics (e.g., counting, probability), and apply proof techniques to prove related propositions. While there is no specific prerequisite course required, students should have basic mathematical knowledge. 

This course introduces the fundamentals of regression modeling, providing essential knowledge for students pursuing advanced study in statistics or careers as professional statisticians. Topics include parameter estimation in linear models, hypothesis testing for model comparison, model selection techniques for predictive purposes, detection of assumption violations, and identification of influential observations.

This course begins with a study of the most classical objects in algebraic geometry: conics and plane curves. Students spend time examining these examples to develop a feeling for how algebraic equations and geometric shapes interact and prove an early version of Bezout's theorem. The central part of the course develops the theory of sheaves and schemes, which provide the natural framework in which to formulate and generalize classical results. The course introduces morphisms of schemes and their fundamental properties, and it studies divisors and line bundles as fundamental tools for encoding geometric information. Students examine the local structure of schemes, including objects such as differential forms. The class also introduces Čech cohomology, both as a computational method and as a bridge to more advanced cohomological techniques. The course concludes with the Riemann-Roch theorem.

This course covers fundamental stochastic models of probabilistic phenomena, including conditional probability, stochastic processes, Markov chains, properties and applications of Markov chains, Poisson processes, renewal processes, and martingales. Topics include Conditional Expectation, Martingales in Discrete Time, Optional Stopping Theorem, Martingale Inequalities, Convergence and Uniform Integrability, Markov Chains, Long-Time Behavior of Markov Chains, Poisson Process, Brownian Motion, and Stochastic Differential Equations.

This course covers analytic functions, special functions (gamma function, Bessel functions, Legendre polynomials and spherical harmonics), Fourier series and Fourier transforms, Laplace transforms, Ordinary differential equations, partial differential equations, and green functions.

This course examines groups, which are best understood as symmetries of mathematical objects. Students explore geometric group theory and the connection between the algebraic properties of a group and the geometric properties of the spaces it acts on.

This course examines the fundamental concepts of probability and statistics required for data analysis. Topics include sampling; introduction to experimental design; review of simple probability; estimation; confidence intervals; hypothesis testing including types of errors and power; inferences about means and proportions based on single and independent samples; matched pairs designs; introduction to nonparametric methods; contingency tables; regression; and analysis of variance.