The lectures are at a beginning graduate level and assume only basic familiarity with Functional Analysis and Probability Theory. Topics covered include: Random variables in Banach spaces: Gaussian random variables, contraction principles, Kahane-Khintchine inequality, Anderson’s inequality. Stochastic integration in Banach spaces I: γ-Radonifying operators, γ-boundedness, Brownian motion, Wiener stochastic integral. Stochastic evolution equations I: Linear stochastic evolution equations: existence and uniqueness, Hölder regularity. Stochastic integral in Banach spaces II: UMD spaces, decoupling inequalities, Itô stochastic integral. Stochastic evolution equations II: Nonlinear stochastic evolution equations: existence and uniqueness, Hölder regularity.
How do populations grow? How do viruses spread? What is the trajectory of a glider? Many real-life problems can be described and solved by mathematical models. In this course, you will form a team with another student and work in a project to solve a real-life problem. You will learn to analyze your chosen problem, formulate it as a mathematical model (containing ordinary differential equations), solve the equations in the model, and validate your results. You will learn how to implement Euler’s method in a Python program. If needed, you can refine or improve your model, based on your first results. Finally, you will learn how to report your findings in a scientific way. This course is mainly aimed at Bachelor students from Mathematics, Engineering and Science disciplines. However it will suit anyone who would like to learn how mathematical modeling can solve real-world problems.