Math 612: Topics in Mathematical Biology (Single Cell Analysis)

INSTRUCTOR INFORMATION

Instructor: Geoffrey Schiebinger
Email: geoff@math.ubc.ca
Office: Math 118
Office Hours: Tues 11-12pm in Math 118, and by appointment.

ANNOUNCEMENTS

SEP 5 Lecture notes will be posted below.
SEP 10 This week's office hours will be rescheduled because I am giving the Statistics Seminar at that time.
SEP 13 Please sign up to scribe one of the up-coming lectures at the Google Docs link here.
Oct 8 Homework 1 posted (see Assignments below).
Oct 15 Class Projects topics list posted here.
Oct 17 Typo corrected in Homework 1! And Hint added! (see Assignments below).
Dec 3 I will hold Office Hours From 10 to 12 on Thursday December 5.

LECTURE NOTES

Week 1	Sep 3 no class	Sep 5 Course Overview
Week 2	Sep 10 Biology Review	Sep 12 Measurement Technologies I
Week 3	Sep 17 Measurement Technologies II	Sep 19 Empirical distributions and dimensionality reduction
Week 4	Sep 24 Mixture models and clustering	Sep 26 Developmental Stochastic Processes I
Week 5	Oct 1 Developmental Stochastic Processes II	Oct 3 Inferring developmental couplings
Week 6	Oct 8 Convex Optimization	Oct 10 Lagrangian Duality
Week 7	Oct 15 Entropy and Second Law	Oct 17 Scaling Algorithm for Entropic OT
Week 8	Oct 22 Wasserstein Space	Oct 24 Wasserstein Curves
Week 9	Oct 29 Gradient Flows	Oct 31 Entropic Interpolation: Schrodinger Bridges
Week 10	Nov 5 Data Science Lab	Nov 7 Data Science Lab
Week 11	Nov 12 Student Presentations	Nov 14 Student Presentations

ASSIGNMENTS

Homework 1 can be downloaded here. Hint added to problem 4, and M decreased to ease computation. The Jupyter notebook can be downloaded here. If your browser adds ".txt" to the end of the filename, please remove this to get a ".ipynb" file. Typo correction: Delete the words "convex hull" from Exercise 3 on spectral clustering.

Homework 2 can be downloaded here.

COURSE INFORMATION

Class times and location:

Day	Start Time	End Time	Building	Room
TTh	9:30 AM	11:00 AM	MATH	126

Course web page: http://www.math.ubc.ca/~geoff/courses/W2019T1/Math612.html.

Pre-requisite: Linear algebra as in Math 307.

COURSE OUTLINE

New measurement technologies like single-cell RNA sequencing are bringing "big data" to biology. This course introduces a mathematical framework for thinking about questions like: How does a stem cell transform into a muscle cell, a skin cell, or a neuron? How do cell types destabilize in diseases like cancer? Can we reprogram a skin cell into a stem cell? We will learn how to model developing organisms as stochastic processes in gene expression space. We will cover random matrices, stochastic processes, entropy, optimal transport, convex optimization, duality, gradient flows, geodesic interpolation, and developmental genetics.

The course will be organized into modules as follows:

Biology review (Lectures 2-3)
Measurement Technologies (Lectures 3-4)
Primer on probability (Lectures 5-7)
Developmental trajectories (Lectures 8-10)
Optimization (Lectures 11-13)
Computational optimal transport (Lectures 13-14)
Topics in optimal transport (Lectures 15-17)
Data science lab (Lectures 18 - 19)
Student presentations (Lectures 20-25)

Text: We will cover some material from Computational Optimal Transport by Marco Cuturi and Gabriel Peyré. This book is available for free on arXiv.

Grades

The course grade will mainly be determined by homework and a final project. A small portion of the grade will be based on scribing of lecture notes.

Homework (45%): There will be three homework assignments and students will be given two weeks to complete each one. The homework will involve a combination of theoretical exercises and programming challenges (in python).

Final Project (50%): The final project consists of three stages for a total of 50%. The first stage is a short written proposal of a research idea, which will be due in late October and count for 10%. The second stage is an oral presentation to the class in mid-November. This presentation will count for 20%. The third is a written report (20%).

Grading of project: The grade for the presentation and write-up will be based on the content and the exposition (i.e. clarity of communication). These two components will have roughly equal weight. Communication is important in interdisciplinary fields like this because

Scribing (5%): Each lecture will have an assigned scribe who will be responsible for taking notes and writing them up nicely in latex using the latex template. Students will sign up for their desired dates, beginning on the second lecture. The scribed notes will be due at the beginning of the next class.