Mathematics

This course offers a comprehensive exploration of scientific computing, covering essential topics crucial for solving mathematical problems encountered in scientific and engineering fields. Beginning with an introduction to the fundamentals of numerical methods, including error analysis and computational complexity, students explore solving systems of linear equations using various techniques such as direct and iterative methods. The course further aims to eigenvalue computation methods and approaches for solving nonlinear equations. Interpolation techniques for approximating functions from discrete data points are also covered in detail. Through hands-on exercises and computational assignments, students develop practical skills in numerical analysis, enabling them to tackle diverse mathematical challenges in scientific computing effectively.

The course reviews number theory including the fundamental theorem of arithmetic, modular, and arithmetic; groups including definition, basic examples of groups, subgroups, normal subgroups, factor groups, isomorphisms, and homomorphisms, Lagrange's theorem, permutation groups, symmetric and alternating groups, finitely generated Abelian groups; rings including definition, basic examples of rings, isomorphisms and homomorphisms, ideals, factor rings, polynomial rings, factorization of polynomials as products of irreducible polynomials; and fields including characteristic, simple field extensions, finite fields.

The main theme of the course is the interplay between Number Theory and rings. Students need to be familiar with the basics of prime numbers, unique factorization of integers and modular arithmetic. This is an advanced course with Fundamentals of Pure Mathematics as a prerequisite. 

This course examines group theory and ring theory, with a view towards commutative algebra, algebraic number theory and representation theory.

This course offers students a grounding in the language of modern machine learning, with a focus on particular topics in linear algebra, differential calculus, probability, and statistics. Rather than focusing on theorems and their proofs, the course covers the key tools (and theorems) within the topic areas, and to illustrate these with exemplars drawn from machine learning. The course is delivered through a mixture of lectures and classes, and involves a mix of traditional lecture delivery, interactive notebooks, and problem sets.

This course provides an introduction to programming within the statistical package R. Various computer-intensive statistical algorithms are discussed and their implementation in R is investigated. Topics to include basic commands of R (including plotting graphics); data structures and data manipulation; writing functions and scripts; optimizing functions in R; and programming statistical techniques and interpreting the results (including bootstrap algorithms).

The course covers sufficient statistics, factorization criteria, exponential families, Rao-Blackwells theorem, ancillary statistics, Cramér-Rao's bound, Neyman-Pearson's lemma, permutation test, and connection between hypothesis testing and confidence intervals. Asymptotic methods: maximum likelihood estimation, profile, conditional and penalized likelihood as well as hypothesis testing with likelihood ratio-, Wald- and score-method. Bayesian inference: estimation, hypothesis testing, and confidence interval and the difference compared to frequentist interpretation.

This course examines analytical functions; cauchy-riemann equations; complex mappings; cauchy's integral formulas; morera's, liouville's & rouche's theorems; taylor & laurent series; analytic continuation, residues & applications to integration; and boundary-value problems.

Dynamic programming is a neat way of solving sequential decision optimization problems. Integer Programming provides a general method of solving problems with logical constraints. Game theory is concerned with mathematical modelling of behavior in competitive strategic situations in which the success of strategic choices of one individual (person, company, server, ...) depends on the choices of others. By the end of this course, students have gained: ability to formulate and solve a sequential decision optimization problem; ability to formulate and solve optimization problems with logical constraints; ability to find optimal and equilibrium strategies for zero- and nonzero-sum 2x2 matrix games; and mastery of the theory underlying the solution methods.

This course covers basic notions of information theory. Entropy as measure of uncertainty. Constrained optimization with Lagrange multipliers. Maximum entropy inference with constraints. Partition function, free energy as generating function. Collective behavior in spin systems: from independent voters to the tight-knit model (or Curie-Weiss ferromagnet); phase transitions and spontaneous symmetry breaking.  Distributions of functions of random variables using Kronecker delta.  Laplace's approximation for integrals. Bolzmann distribution and 1d Ising chain: exact calculation for free energy. Variational approximations and trial (factorized) distributions. Time permitting: multi-party voters, stochastic dynamics and Markov Chains, models on social networks, traffic flow and epidemic models.