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


Instructor:   Geoffrey Schiebinger
Office:   Math 118
Office Hours:   Tues 11-12pm in Math 118,   and by appointment.

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.


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       


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.

Class times and location: 
Day Start Time End Time Building Room
TTh 9:30 AM
11:00 AM
MATH 126

Course web page:

Pre-requisite: Linear algebra as in Math 307.


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.


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.