Mathematics

This course teaches data-based model inference and predictive model generation. It covers the core principles of the question structure, data collection and organization, statistical inference, predictive modeling, and decision-making process. The course also studies basic theories about intermediate-level data conversion, data refinement, model fit, model selection, model diagnosis, etc., and learn them by data practice.

This course offers a study of topology. Topics include: metric spaces; topological spaces; continuous application; separation properties; compactness; locally compact spaces and compactifications; connection and paths.

This course provides an introduction to artificial neural networks and deep learning, with both theoretical and practical aspects. This course gives a basic knowledge of artificial neural networks and deep learning: both the theoretical background and how to practically use these methods for typical problems in machine learning and data mining. The course covers the most common models in artificial neural networks, with a focus on the multi-layer perceptron. The course contains three computer exercises where the student train and evaluate different ANN models.

 This course offers an introduction to Markov processes in discrete and continuous time. Topics include Markov chains, Poisson process, Markov processes, and an introduction to renewal theory and regenerative processes.

This course examines the fundamentals of Bayesian inference, including the specification of prior and posterior distributions, Bayesian decision theoretic concepts, the ideas behind Bayesian hypothesis tests, model choice and model averaging, the capabilities of several common model types, such as hierarchical and mixture models. It also looks at the ideas behind Monte Carlo integration, importance sampling, rejection sampling, Markov chain Monte Carlo samplers such as the Gibbs sampler and the Metropolis-Hastings algorithm, and use of the WinBuGS posterior simulation software.

Set theory is a beautiful and magical subject stemming from transparent and easy observations leading us to a surprising and somewhat unbelievable logical world on which contemporary mathematics is based. Its controversial and contrasting history attracts our attentions as well. In this beginning course, we focus on set operations, orderings, cardinal and ordinal arithmetics which, as primitive notions, are absolutely necessary in learning almost every subject of mathematics. The course also introduces more mysterious and advanced parts of the subject whose full clarifications can be pursued by interested students in their senior or graduate level courses. We often touch on set theory itself, overview the axiom of foundation, the consistency and independence problems, the theory of large cardinals, descriptive set theory, etc. 

This course is an introduction to topological spaces. It deals with constructions like subspaces, product spaces, and quotient spaces, and properties like compactness and connectedness. The course concludes with an introduction to fundamental groups and covering spaces. The course discusses topics including sets and functions, images and preimages, and finite, countable, and uncountable sets; how the topology on a space is determined by the collection of open sets, by the collection of closed sets, or by a basis of neighborhoods at each point, and what it means for a function to be continuous; the definition and basic properties of connected spaces, path connected spaces, compact spaces, and locally compact spaces; what it means for a metric space to be complete, and characterizing compact metric spaces; the Urysohn lemma and the Tietze extension theorem, and characterizing metrizable spaces; and the construction of the fundamental group of a topological space and applications to covering spaces and homotopy theory.