Instructor: David A. Meyer
Office hours (Fall Quarter): AP&M 7256, W 10:00am-11:00am, or by appointment
Lecture: AP&M B412, MWF 9:00am-9:50am
Email: dmeyer "at" math "dot" ucsd "dot" edu
TA: Gautam Wilkins
Office hours (Fall Quarter): AP&M 5760, Tu 11:00am-12:00nn, F 1:00pm-2:00pm, or by appointment
Section: AP&M B412, Tu 8:00am-8:50am
Email: gwilkins "at" math "dot" ucsd "dot" edu
This course is a focused introduction to mathematical modelling. In 2014 I plan to discuss mathematical models drawn from a wide range of topics (but mostly outside the familiar contexts of the physical science and engineering) including biology, economics, political science, and culture. (For inspiration see [1,2].) The relevant mathematical methods include: (systems of) ordinary differential equations, graphs/networks, probability, partial differential equations, eigenvalues/eigenvectors, permutations, and dimension theory.
The goals of this course are: (1) to explain what it means to construct a mathematical model of some real-world phenomenon, (2) to introduce some of the mathematical ideas that are used in many such models, (3) to apply these methods to analyze one or more real problems, and (4) to understand how new mathematical ideas are motivated by such modelling.
The prerequisites are the lower-division math sequence through differential equations (20D) and linear algebra (20F or 31A), or consent of the instructor. Please contact me if you are interested but unsure if your mathematics background will suffice.
The textbook is E. A. Bender, An Introduction to Mathematical Modeling (Mineola, NY: Dover 2000).
I expect interest and enthusiasm from the students in this class. 30% of the grade is class participation, which includes occasional homework assignments, often for class discussion. 70% of the grade is based upon a mathematical modelling project for which each student writes a proposal (15%), writes a preliminary report (10%), gives a final presentation (20%), and writes a final report (25%). Some titles of projects from previous years are listed at the bottom of the page.
I recommend, but do not require, that you prepare your written materials using some dialect of TeX [3]. In any case, please do not send me Word documents; convert them to pdf first.
| 1 Feb 15 |
Application deadline for an Eric & Wendy Schmidt Data Science for Social Good Summer Fellowship (Summer 2015) at the University of Chicago |
| 31 Jan 15 |
Application deadline for the Mathematical and Theoretical Biology Institute Summer Program (3 June to 25 July 2015) at Arizona State University |
| 12 Jan 15 |
Application deadline for an Undergraduate Research Award (Summer 2015) and the Undergraduate School on Experimental Quantum Information Processing (25 May to 5 June 2015) at the Institute for Quantum Computing, University of Waterloo |
| 13 Oct 14 |
Nina Fefferman (Rutgers),
"How dynamic networks affect disease transmission" BioCircuits Institute Special Seminar, Natural Sciences Building 4211, 2:00pm |
| 7 Oct 14 |
Pak-Wing Fok (Delaware),
"Mathematical modeling of artherosclerosis" Informal Seminar on Mathematics and Biochemistry-Biophysics, AP&M 5829, 2:00pm |
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3 Oct 14 DM lecture |
administrative details overview/motivation population growth model y' = (M - y)y see "World population stabilization unlikely this century" for current assessment HWK (for M 6 Oct 14). Read Bender, Chap. 1. Find something in the news or elsewhere that suggests a system that could be modeled (or not); email me link (if there is one) and be prepared to discuss in class. |
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6 Oct 14 discussion |
discussion of news items or other sources suggesting mathematical models (or not) spread of Ebola supply of iPhone 6 ISIS recruitment via social media |
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8 Oct 14 discussion |
price of oil numbers of heat waves buttonless elevators US Supreme Court decisions HWK (for F 10 Oct 14). Read Bender, Appendix A and Chap. 5. |
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10 Oct 14 DM lecture |
a microscale probabilistic model for population change [notes] E. coli cells total population at time t is a random variable which can be simulated [Mathematica notebook] expectation value of which can be computed ODE model y' = k y describes expectation value of population in probabilistic model |
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13 Oct 14 DM lecture |
simulations show fluctuations away from expectation value [notes] describe range of results with cumulative probabilities [Mathematica notebook] can also compute variance analytically HWK (for W 15 Oct 14). Read M. Yglesias, "The real problem with Nate Silver's model is the hazy metaphysics of probability", Vox (11 Oct 2014) and be prepared to discuss. |
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15 Oct 14 DM lecture |
election prediction background on US Senate elections 2014 "models" based on aggregating polls, and other factors Yglesias' assertion that a single election result doesn't tell us which model is right statement and proof of Bayes' Theorem application of Bayes' rule to comparison of election models [notes] |
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17 Oct 14 DM lecture |
graph theory definitions of simple directed and undirected graphs how to represent graphs with drawings definition of (binary) relation definition of symmetric relation definition of multiset definition of (not necessarily simple) graph Seven Bridges of Königsberg problem Eulerian paths HWK. Read Varian's article on how to build a model [4]. Read Gray's article on how to write an abstract [5]. Look at some proposals/videos on Kickstarter, e.g., GoldieBlox or Indiegogo, e.g., Axent Wear. Draft project proposal (due by F 24 Oct 14): Describe the system for which you propose to construct a mathematical model. What question will the model answer? Why is that important/interesting? Has anything relevant been done to model this system previously? Give references. What features/variables will the model include? What features/variables may be relevant but will be exogenous to your model? What kind of mathematics will you use? If you intend to use real data, describe them and explain how you will get them. Give an approximate timeline for accomplishing the various pieces of your project. If you will be working with someone else, explain how the work will be allocated and coordinated. Should be 2-4 pages. I prefer that you submit an electronic version, ideally a pdf file of a TeX [3] document. If you want to try making a video à la Kickstarter or Indiegogo, I'll be pleased to watch it. |
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20 Oct 14 DM lecture |
more systems that can be represented as graphs Facebook: multiple types of entities and relations; ego+friends and connections Erdős and his collaborators Flickrverse internet: Lumeta 1998; CAIDA 1999 blogosphere Six degreees of Francis Bacon philosophical influences arXiv citation network degree distribution empirical results power law, for human networks, as above binomial, for percolation networks Erdős-Rényi model of percolation expected number of edges probability distribution for degree |
| [1] | I. Asimov, The Foundation Trilogy (New York: Gnome Press 1951). |
| [2] | P. R. Krugman, "Introduction to The Foundation Trilogy" (Folio Society 2012). |
| [3] | D. E. Knuth, The TeXbook, Computers and Typesetting, Volume A (Reading, Massachusetts: Addison-Wesley 1984). |
| [4] | H. R. Varian, "How to build an economic model in your spare time", The American Economist 41 (1997) 3—10. |
| [5] | N. Gray, "Abstract science", The Huffington Post (2012). |
