Course Syllabus

Fall 2020 BIOINF 501

Mathematical Foundations for Bioinformatics

 

Overview

The course covers some of the fundamental mathematical techniques commonly used in bioinformatics and biomedical research. These include: 1) principles of multi-variable calculus, and complex numbers/functions, 2) foundations of linear algebra, such as linear spaces, eigen-values and vectors, singular value decomposition, spectral graph theory and Markov chains, 3) differential equations and their usage in biomedical system, which includes topic such as existence and uniqueness of solutions, two dimensional linear systems, bifurcations in one and two dimensional systems and cellular dynamics, and 4) optimization methods, such as free and constrained optimization, Lagrange multipliers, data denoising using optimization and heuristic methods. Demonstrations using MATLAB, R, and Python will be included throughout the course.

 

Instructors
Kayvan Najarian, Daniel Burns, Jonathan Gryak, Reza Soroushmehr, and Ivo Dinov

 

Start Time, End Time and Location

MW 1:30PM - 3:00PM, Room: USB 4153

Only the first session - Monday, August 31 - will be both in-person and online. All other sessions will be remote only.  Students are welcome to take the entire course online if they so choose.

 

Topics/Modules

Module 1: Review of Some Basic Methods in Mathematics

Probability functions

Review of complex variables and functions

Taught by: Kayvan Najarian

Duration: 2 lectures

            

Review of multi-variable calculus

Taught by: Reza Soroushmehr

Duration: 2 lectures

          

Module 2: Linear Algebra

Part I

Introduction to linear systems

Matrix products

The inverse of a linear transformation

Linear spaces

Orthogonality

Determinants, eigenvalues and eigenvectors

Symmetric matrices and diagonalization

Solving systems of linear equations

Taught by: Jonathan Gryak

Duration: 5 lectures

 

Part II

Singular value decomposition

Principal component analysis

Spectral graph theory

Taught by: Daniel Burns

Duration: 4 lectures

 

Module 3: Differential Equations

Part I

Introduction to differential equations

First and second order linear equations

Existence and uniqueness of solutions

Difference equations

Systems of linear equations

Phase plane and bifurcation: diagrams and analysis

First order nonlinear systems

Taught by: Jonathan Gryak

Duration: 4 lectures

 

Part II

Differential equations for compartmental modeling of biomedical systems

Taught by: Reza Soroushmehr

Duration: 2 lectures

 

Module 4: Optimization

Data Science and Predictive Analytics EBook (University of Michigan Library)

Free (unconstrained) optimization vs. Constrained Optimization

Foundations of R

Equality and Inequality constraints

Lagrange Multipliers

Linear and Quadratic Programming

Manual vs. Automated Lagrange Multiplier Optimization

Data Denoising: Application of computer optimization techniques in medicine and biology

Heuristic methods - Genetic algorithms, simulated annealing

Applications (supervised classification & unsupervised clustering)

Instructor: Ivo Dinov

Duration: 6 lectures

 

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Software tools: Matlab, R, Python, and open-science tools.