Mathematics

This course provides essential mathematical tools for students of Biology, reinforcing key concepts and introducing methods for modeling biological systems. Topics include matrix algebra, systems of equations, real functions, calculus (limits, derivatives, and integrals), and an introduction to ordinary differential equations. Emphasis is placed on practical applications relevant to biological phenomena and experimental sciences.

This course provides a solid foundation in essential mathematical concepts for students in economics and management. It ensures a consistent level of mathematical proficiency to prepare for further study of advanced quantitative techniques. The course covers linear functions, absolute values, square roots and inverses, second-degree polynomials, and derivatives.

Starting from the concept of limit of a sequence, learn how it is possible to give a precise meaning to the concepts of "infinite", "infinitely small", and "infinitely large". Students also learn how to work with series, and understand how these tools can be applied to define limits of functions. Students see some of the concepts that they have covered in school in a new light, and analyze them in great depth. Students learn how to give rigorous proofs of mathematical statements, and how to sketch the graph of a function. 

This course builds on Stochastic Processes I and introduces an array of stochastic models with biomedical and other real world applications. Topics include Poisson process, compound Poisson process, marked Poisson process, point process, epidemic models, continuous time Markov chain, birth and death processes, martingale. The course requires students to take prerequisites.

This upper division course applies concepts in symbolic methods and analysis to solve a variety of problems in combinatorics. Course content includes: 1. Combinatorial Structures and Ordinary Generating Functions: symbolic enumeration methods, integer compositions and partitions, words and regular languages, tree structures 2. Labelled Structures and Exponential Generating Functions: labelled classes, surjections, set partitions, words, alignments, permutations, labelled trees, mapping and graphs 3. Complex Analysis, Rational and Meromorphic Asymptotics: generating functions as analytic objects, analytic functions and meromorphic functions, singularities and exponential growth of coefficients 4. Singularity Analysis of Generating Functions: coefficient asymptotics, process of singularity analysis. The course requires students to take prerequisites.

This course introduces advanced mathematical knowledge used in quantitative finance, including differential equations, numerical partial differential equations, optimization and dynamic programming, advanced probability, and neural network. Motivating examples in finance will be given as well. The course requires students to take prerequisites.

This course addresses current needs for the statistical modeling of random patterns and structures in spatial contexts, which arise in multiple fields ranging from geophysical, life and earth sciences, to communication engineering and social network analysis. The course approach relies on computational and statistical tools from stochastic geometry. The course requires students to take prerequisites.

The objective of this course is to work on optimisation problems which can be formulated as linear and network optimisation problems. The course covers formulating linear programming (LP) problems and solving them by the simplex method (algorithm); looking at the geometrical aspect and developing the mathematical theory of the simplex method; studying problems which may be formulated using graphs and networks. These optimisation problems can be solved by using linear or integer programming approaches. However, due to its graphical structure, it is easier to handle these problems by using network algorithmic approaches. Applications of LP and network optimisation are demonstrated. Major topics: Introduction to LP: solving 2-variable LP via graphical methods. Geometry of LP: polyhedron, extreme points, existence of optimal solution at extreme point. Development of simplex method: basic solution, reduced costs and optimality condition, iterative steps in a simplex method, 2-phase method and Big-M method. Duality: dual LP, duality theory, dual simplex method. Sensitivity Analysis. Network optimisation problems: minimal spanning tree problems, shortest path problems, maximal flow problems, minimum cost flow problems, salesman problems and postman problems. The course requires students to take prerequisites.