Math 456: Mathematical modeling (archived course page from Spring 2018)
Topics to be covered include systems of nonlinear differential equations and Markov chains, with emphasis on ideas such as equilibria, stability, and long time behavior, to few. Most of the theory in the course will deal with Markov chains.
The textbook for this class is Richard Durrett's Essentials of Stochastic Process, 2nd or 3rd edition.
Classroom: LGRT 141 for both sections.
Lecture Time: Tue-Thur, Section 1 10:00-11:15 am, and Section 2 8:30-9:45 am.
Office Hours: Tuesday 1-3 pm, or by appointment.
Teaching Assistants Office Hours:
Lingchen Bu: Mondays 10 am-12 pm at LGRT 1323E.
John Lee: Wednesdays 10 am-12 pm at LGRT 1423O.
Virtual Office Hours: Friday: 10 am-11 am.
Problem sets (30%) -- Lowest grade is dropped.
Group modeling project (40%)
Important dates(Updated Feb. 24th)
Week 1. Friday January 27th day questionnaire is due (link sent via email on first day of classes).
Week 2. I will be out of town this week (there will be no lectures).
Week 6. Thursday March 1st Project Abstract due.
Week 10. Thursday April 5th Midterm.
Week 11. Thursday April 12th Project's preliminary reports due.
Finals week. May 8th. Final Reports due at midnight.
1st (due 2/08)
2nd (due 2/22)
3rd (due 3/08)
4th (due 3/29)
Practice Midterm (Midterm is Thursday April 5th)
Course schedule + Weekly Slides
- Week 1: (Slides) A case study in modeling: Newton's Law of Universal Gravitation. Modeling with differential equations. Challenges and pitfalls. The evolution operator associated to an ODE. Nonlinear versus linear systems. Review of a few tricks from differential equations: separation of variables, Gronwall's lemma, the matrix exponential, and Lyapunov functionals.
- Week 3: (Slides) Discrete time systems and some examples. A rapid probability course: states, events, probability distributions, random variables. Conditional probability and independence. The Total Probability Formula. Gambler's ruin.
- Week 4: (Slides) Gambler's ruin continued. Markov chains and the Markov property. Transition probabilities and basic examples of Markov chains. Multi-step transition probabilities and the Chapman-Kolmogorov equation.
- Week 5: (Slides) More examples of chains: the Ehrenfest chain, the Wright-Fisher model, random shuffling, random walks on a graph. Weighted graphs. Evolution of the probability distribution for the state. Stationary distributions. Irreducible and closed chains. Irreducible chains and their stationary distributions.
- Week 6: (Slides) Stationary distributions, detailed balance condition, and stopping times.
- Week 7: (Slides) Weighted Graphs, the Laplacian, and it's connection with exit distributions. The Strong Markov property (stated).
- Week 8: (Slides) Proof of the Strong Markov Property. Number of times a state is visited. Periodicity of a state. Statement of the convergence theorem.
- Week 9: (Slides) Computing limits of the n step transition probabilities and examples. Bounding probability for hitting times. Decomposition of the state space into recurrent and transient states.
- Week 10: (Slides) Limit theorems. Asymptotic frequency. The ergodic theorem.
- Week 11: (Slides) The ergodic theorem continued. Gibbs distributions. The Metropolis-Hastings algorithm, and simulated annealing.
- Week 12: (Slides) Mixing times for Markov chains. Spectral analysis for reversible chains.
- Weeks 13-14: Group modeling presentations.