Math 612D: Single Cell Analysis

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?

• How 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. Along the way, we will cover topics in probability, stochastic processes, convex optimization, duality, optimal transport, gradient flows, geodesic interpolation, and developmental genetics.

See the Course Outline below for a tentative schedule of topics.

Course Information

Instructor: Geoffrey Schiebinger

Time: Tuesdays and Thursdays from 9:30 to 11:00.

Location: Math 225 and Zoom. Sign up here to get on the email list and get the Zoom links.

Prerequisites: Linear algebra and probability at an undergraduate level.

This is a PIMS Network Course (OT + Bio)

Announcements

• The first day of class is on Thursday, September 8th.

• Class project sign-up sheet: here

Outline

Lecture 1 Course Overview September 8 r7A+ir1^

Lecture 2 Biology Review September 13 UwYg9V1@

Lecture 3 Measurement Technologies I September 15 3=es.8eC

Lecture 4 Measurement Technologies II September 20 fkev2#JG

Lecture 5 Empirical distributions and dimensionality reduction September 22 3dD0&h7.

Lecture 6 Developmental stochastic processes I September 27 %kMF3=xG

Lecture 7 Developmental stochastic processes II September 29 6sv.Q#gV

Lecture 8 Inferring developmental couplings October 4 n^0zOqh^

Lecture 9 No class October 6

Lecture 10 Convex optimization October 11 0IVXf%D9

Lecture 11 Lagrangian duality October 13

Lecture 12 Entropy and the second law October 18 AEJ5V!CU

Lecture 13 Scaling algorithms for entropic OT October 20 17%rey^!

Lecture 14 Wasserstein space October 25 6w5+upSV

Lecture 15 Wasserstein curves October 27 w*D2!Kzm

Lecture 16 Entropic interpolation: Schrodinger bridges November 1 b+AW!RQ9

Lecture 17 Discussion November 3

Lecture 18 Gradient flows November 8

The remainder of the course will consist of student presentations on selected topics.

Homework

• Homework 1 will be assigned in near the end of September.

Lecture recordings from 2020

Lecture 1 Course Overview September 10

Lecture 2 Biology Review September 15 Notes

Lecture 3 Measurement Technologies I September 17 Notes

Lecture 4 Measurement Technologies II September 22

Lecture 5 Empirical distributions and dimensionality reduction September 24 Notes

Lecture 6 Mixture models and clustering September 29 Notes

Lecture 7 Developmental stochastic processes I October 1

Lecture 8 Developmental stochastic processes II October 6

Lecture 9 Inferring developmental couplings October 8

Lecture 10 Convex optimization October 13

Lecture 11 Lagrangian duality October 15

Lecture 12 Entropy and the second law October 20

Lecture 13 Scaling algorithms for entropic OT October 22

Lecture 14 Wasserstein space October 27

Lecture 15 Wasserstein curves October 29

Lecture 16 Gradient flows November 3

Lecture 17 Entropic interpolation: Schrodinger bridges November 5

Lecture 18 Data science lab November 10

The remainder of the course will consist of student presentations on selected topics.

Homework from 2020

• Homework 1 is due on Tuesday October 20. The starter code for Problem 3 is here.

• Homework 2 is due on December 9th.