Mathematics

This course examines ways of solving the (usually partial) differential equations that arise in physical, biological, and engineering applications. Many of the methods covered, such as Fourier Transforms, also have applications beyond the solution of differential equations.

This course provides an introductory overview of probability theory, presented in a mathematically rigorous manner. Starting from the definitions of events, random variables, independence, and expectation, we also cover some basic applications such as weak convergence, the law of large numbers, characteristic functions, the central limit theorem, etc. 

Prerequisites: Elementary level of calculus (required), analysis (required), and linear algebra (optional). 

This course introduces the basic results and techniques of linear programming (LP) and its related topics in operations research. There is an equal emphasis on all three aspects of understanding, algorithms and applications. The course serves, together with a course on network models, as essential concept and background for more advanced studies in operations research. The topics include the simplex method, the dual simplex method, parametric programming, decomposition methods and interior point methods.

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.