Instructor: David A. MEYER
Email: dmeyer "at" math "dot" ucsd "dot" edu
Office hours (Winter Quarter): AP&M 7256 F 12:00pm- 1:00pm, or by appointment
Lectures: Center Hall 217B MWF 11:00am-11:50am
TA: Linbo LIU
Email: linbo "at" ucsd "dot" edu
Office hours (Winter Quarter): AP&M 6414 M 7:00pm-8:00pm, or by appointment
Recitations: AP&M 2301 M 8:00pm-8:50pm
This is a second course in mathematical modelling. In 2019 I plan to focus on mathematical models drawn from a range of topics coordinated with students' projects, which are often outside the more familiarly mathematical sciences. (For inspiration see [1,2].) I may also, however, discuss some models of weather and of climate change. The relevant mathematical methods will 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 (18 or 31A), and a first course in mathematical modeling (111A), 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 below.
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.
| Mar 11, 2019 |
Application deadline for UCSD Physical Sciences Undergraduate Summer Research Award |
| Feb 1, 2019 |
Application deadline for Perimeter Scholars International Masters Program |
| Jan 31, 2019 |
Application deadline for Mathematical and Theoretical Biology Institute Summer Program |
| Jan 7, 2019 |
Application deadline for Perimeter Institute's Undergraduate Theoretical Physics Summer Program |
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Jan 7, 2019 DM lecture |
administrative details overview/motivation models, metaphors & imagination [slides] discussion of project ideas E. Agliari et al., "Efficiency of information spreading in a population of diffusing agents", Physical Review E 73 (2006) 046138 A. Apolloni et al., "A study of information diffusion over a realistic social network model", in Proceedings of IEEE International Conference on Computational Science and Engineering (2009). F. Baker, The Basics of Item Response Theory (College Park, MD: ERIC Clearinghouse on Assessment and Evaluation 2001). HWK (for W Jan 9). (Re)read Bender, Chap. 1. What is modeling?; Varian [4]; Gray [5]; Goldin [6] |
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Jan 9, 2019 DM lecture |
discussion of project ideas X. Diego et al., "Key features of Turing systems are determined purely by network topology", Physical Review X 8 (2018) 021071 T. O'Donoghue and J. Somerville, "Modeling risk aversion in economics", Journal of Economic Perspectives 32 (2018) 91-114. J. Peck et al., "Lower bounds to the robustness to adversarial perturbations", in Proceedings of Advances in Neural Information Processing Systems 30 (2017). F. Angelini and M. Castellani, "Culture and economic value: A critical review", Journal of Cultural Economics (2 November 2018) 1-16. |
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Jan 11, 2019 DM lecture |
discussion of project ideas R. F. Engle et al., "A dymimic model of housing price determination", Journal of Econometrics 28 (1985) 307-326. HWK (for F Jan 18). Draft project proposal: 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. Should be 2-4 pages. Please submit a pdf file electronically, ideally from a TeX [3] document. |
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Jan 14, 2019 DM lecture |
stochastic models states of a system why make transitions between states probabilistic the Markov property Example: tic-tac-toe against an opponent who plays randomly Example: random walk transition matrix and probability distribution vector over states Example: multiple random walkers with a dynamical network of relations |
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Jan 16, 2019 DM lecture |
graduate admissions data basic ideas of data analysis: compile data; anonymize data; clean data explore data construct a model; fit the model ratings have large variance |
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Jan 18, 2019 DM lecture |
more basic ideas of data analysis: try to improve model, possibly by changing completely fit the new model iterate total and partial orders low rank approximation alternating minimization |
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Jan 21, 2019 |
No lecture; Martin Luther King, Jr. day. California "King" tide: 8:35am |
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Jan 23, 2019 DM lecture |
another basic idea of data analysis: overfitting fitting a degree n-1 polynomial through n points Newton's method Lagrange's theorem [code] |
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Jan 25, 2019 DM lecture |
an additive model, with the same number of parameters testing for overfitting with cross-validation semi-order as output of model |
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Jan 28, 2019 |
project discussions John Lam: insurance and reinsurance markets point processes and marked point processes Nick Roberts: adversarial examples for neural networks multivariable Taylor series error bounds Marina Torras: Fibonacci spirals symmetry breaking in PDEs |
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Jan 30, 2019 |
project discussions Samantha Ngan: optimizing game play dynamical network models Terry Le: evaluating student preparation item response theory models Andrew Chavez: valuing public art quantifying cultural value |
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Feb 1, 2019 |
project discussions Jessica Lee: predicting house prices data collection and feature selection Xiangyu Liu: optimizing quadcopter design aerodynamics Brian Nguyen: predicting Amazon pricing supply and demand |
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Feb 4, 2019 DM lecture |
Climate change: global warming and sea level rise What would happend to sea level if Greenland's ice sheet melted? (Part 1) [slides] |
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Feb 11, 2019 |
No lecture; DM at the Southwest Quantum Information and Computation Workshop. |
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Feb 13, 2019 |
project progress reports |
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Feb 25, 2019 DM lecture |
HWK (for weeks 9 & 10). 20+5 minute presentation describe system & state question [~4 minutes] describe model, but don't go into too many details [~8 minutes] explain results & answer to question [~7 minutes] how could model be improved/extended? [~1 minute] I recommend a slide presentation for efficiency. |
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Feb 27, 2019 DM lecture |
HWK (due Friday, Mar 15). Final project report Introduction: describe system & explain question and why it is interesting Describe model: what is being included/excluded; how do different pieces fit together; derive model/eqiuations [somewhere in the Introduction or Description, explain previous relevant models & why yours is different] Describe data: what are they? from where do they come? how reliable are they? Analyze model: explain math/computations; give results Conclusion: what is the answer to the question, from results? discuss answer; How might model be extended/improved? References: standard bibliographical format; citations in text; not wikipedia Approximately 10 pages; pdf, not Word.. You can include data/code as separate files, or links. Not a diary ("First I did this, then this, ..."); it should read like a scientific paper. Write sentences, paragraphs, sections, in the best English you know; not bullet points like on a slide presentation. |
| [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). |
| [6] | A. Bleicher interview with R. Goldin, "Why math is the best way to make sense of the world", Quanta magazine (2017). |