Spring 2026 : Math 380 Introduction to Mathematical Modeling

MATH 380 Introduction to Mathematical Modeling

Instructor: Hemanshu Kaul
Office: 125C Rettaliata Engg Center.
E-mail: kaul [at] iit.edu

Time: 3:15-4:30pm Tuesday and Thursday.
Place: Hermann Hall 007

Office Hours: Tuesday at 12:30pm-1:30pm, and Thursday at 4:30pm-5:30pm. And by appointment in-person or through Zoom (send email).
TA Office Hours: There is no TA assigned to this course. However, you are allowed and encouraged to ask for help at the Math Tutoring Center in RE 129. In particular, Alaittin Kirtisoglu, Wednesday 2pm-5pm.

ARC Tutoring Service: Mathematics tutoring at the Academic Resource Center.

Course Information:

The Course Information Handout has extensive description of the course - topics, textbook, student evaluation policy, rules for HW, as well as other relevant information. Read it carefully!

The official MATH 380 course topics.

Advice for students:

Excellent advice by Francis Su on good mathematical writing.

On a more abstract note, here is a discussion by Tim Gowers on Language and Grammar of Mathematics - which is part of what you are learning in a course like this.

Excellent advice for math majors, especially those planning to go on to graduate school, by Terry Tao, 2006 Fields medallist. Excellent reading.

Read this book on a variety of experiences in the journey to learn mathematics: Living Proof

Some of the primary sources of information/discussion for careers in Mathematical and Data Sciences:
MAA - Careers
SIAM - Careers
INFORMS - Careers
AMS - Careers

Class Announcements:

Thursday, 1/29 : Dates for Mid-term Exams #1 and #2 have been announced. Look below in the appropriate section. Mark your calendars.

Thursday, 1/15 : This webpage will be updated every Thursday unless otherwise announced.
Tuesday, 1/13 : Check this webpage regularly for homework assignments, announcements, etc.

Examinations:

Exam #1 : Tuesday, 2/24. [Note the due time, 3pm, for HW#5] Syllabus: Based on topics, examples, applications corresponding to HWs #1-#5.
Exam #2 : Tuesday, 4/14. [Note the due time, 3pm, for HW#10] Syllabus: Based on topics, examples, applications corresponding to HWs #6-#10.
Final Exam : TBA by the university. Syllabus: All topics covered during the semester.

Project:

Instructions: The project is an important part of this course - not just in terms of the grade, but for the sake of comprehensive, practical understanding of how to apply the modeling framework to an open-ended real-life problem. This is why the problem statements that I have given to you are just short and open-ended descriptions of the certain real-life situations. You have complete freedom in mathematical interpretation of the problem and how you "solve" it. The only requirement is that you use the mathematical modeling process, and justify your model and its conclusions as they apply to the problem. Its a test of your creativity in formulation of models and solution methods, and your ability to find and understand relevant mathematical knowledge.

Read carefully through this list of instructions and advice for your project.

Look through the example project report given at the end of this SIAM report in Appendix B on page 50 of the pdf file for an example on how to format and write your project report. The pages 1-50 of this pdf file are also useful as a detailed overview of how to approach the modeling process for a project. Also, look through SIAM Computing and Communicating Handbook for further technical suggestions for working on your project and the report.

Deadlines for the semester project (unless announced otherwise in class):
Third week of February: I will send you the list of project topics by email.
2/23: Email me with project team members (2 per team)
2/28: Email me with your choice of project topics among the project descriptions sent to you by email.
3/10: Each project team shares with me a 1-2 page document of your project plan to get started.
3/31: Discuss the initial draft and model(s) of your project with me in person. All members of your team must be present.
4/15: Email me the status of your project report and the current draft of your Project report.
5/1: Final submission of Project report and associated materials/programs/etc. Email me the PDF file of the report and other related programs/files before 10pm, Friday, 5/1. This Email should list in detail the contribution of each member of the team.

Weekly Class Log with HW:

Week #1 : 2 Lectures.
- Topics:Discussion of class structure and purpose. The process of math modeling - discussion with examples, Principle of proportionality, difference equations - examples from accounting/ finance/ science, discrete time vs. continuous time, limiting behavior of DDS(discrete dynamical system) and example of modeling births/deaths/resources through non-linear discrete dynamical systems. (From Sections 1.0, 1.1, 1.2, and elsewhere)
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#1.
- Homework:
  The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
  
  You are required to follow the detailed instructions and rules for HWs given in the Course Information Handout and through email comments.
  Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours on Tuesday, Wednesday, and Thursday.
  - Reading HW:
    Read and understand all Examples from Sections 1.1 and 1.2.
  - HW#1 for Submission. Due Thursday, 1/22, by 11pm in Canvas. Submit a single PDF file through Canvas Assignment.
    Section 1.1: #3bc, #10, (#12a and #13a).
    Section 1.2: Submit any two of the following three sets of problems: #2, (#6 and #7), #9.
    HW Solutions distributed in class.
    Questions? Ask on Canvas Discussion Forum. Or, Ask for help during the instructor and TA office hours, or through email to the instructor.

Week #2 : 2 Lectures.
- Topics: (stable and unstable) Equilibrium values/fixed points and solutions of DDS, Solutions methods and stability of equilibrium values of homogenous and nonhomogenous linear DDS, Interacting discrete dynamical systems via a interacting species population model (Competitive Hunter model), discussion of real-life conservation strategy in terms of the model. Proportionality in non-linear or translated linear systems -examples from physics. (From Sections 1.3, 1.4)
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#2.
- Homework:
  The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
  
  You are required to follow the detailed instructions and rules for HWs given in the Course Information Handout and through email comments.
  Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours on Tuesday, Wednesday, and Thursday.
  - Reading HW:
    Read and understand Examples from Sections 1.3 and 1.4.
  - HW#2 for Submission. Due Thursday, 1/29, by 11pm in Canvas. Submit a single PDF file through Canvas Assignment.
    Section 1.3: #1f, #2e, #3a, #6, #14ad.
    Section 1.4: #2, #4.
    HW Solutions distributed in class.
    Questions? Ask on Canvas Discussion Forum. Or, Ask for help during the instructor and TA office hours, or through email to the instructor.

Week #3 : 2 Lectures.
- Topics: Geometric similarity - relationship between geometric notions like volume, surface area, etc. in terms of a fundamental dimension; Geometric similarity - Examples from physics (raindrops) and biology (fish); (Real-life evidence for Geometric similarity based modeling!!); short discussion of SIR epidemic model; Strategy analysis using System of DDS in an Astronaut spacecraft docking procedure and its analysis with interpretation. Model fitting vs. data fitting/ Interpolation (to be discussed in next lecture; read Section 3.1). (From Sections 2.2, 2.3, 1.4 and elsewhere)
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#3.
- Homework:
  The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
  
  You are required to follow the detailed instructions and rules for HWs given in the Course Information Handout and through email comments.
  Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours on Tuesday, Wednesday, and Thursday.
  - Reading HW:
    Read and understand Examples from Sections 2.2, 2.3, 2.4, and 3.1. Think about Section 2.3 Problem #9 (main question, not parts ab).
    Real-life evidence for Geometric similarity based modeling!!
  - HW#3 for Submission. Due Thursday, 2/5, by 11pm in Canvas. Submit a single PDF file through Canvas Assignment.
    Section 2.2: submit one of (#6 OR #12).
    Section 2.3: #4, Project#2.
    Section 3.1: submit one of (#5 OR #7). [Comment: Look at Table 3.1 and 3.2 and the discussion in-between them (we will also discuss this in the next lecture). Explain the use of appropriate data transformation, and just roughly estimate (using your computational software if you wish) the values of the parameters from the plots.]
    HW Solutions distributed in class.
    Questions? Ask on Canvas Discussion Forum. Or, Ask for help during the instructor and TA office hours, or through email to the instructor.

Week #4 : 2 Lectures.
- Topics: Model fitting vs. data fitting/ Interpolation; Sources of qualitative and quantitative errors in the modeling process, Transforming data to fit linear systems. What is error for a collection of data points vs. a model? Fitting a model to data as minimizing appropriate l_p distance between vector of observations and vector of predictions. Chebyshev Approximation Criterion - min max absolute deviation as a linear optimization problem. Minimizing average deviation. Least squares criterion - min sum of squares of deviations, Applying calculus to find best fit model in LSC, Normal equations and critical points in LSC plus application of second derivative test, Using LSC for fitting a straight line, Fitting a power curve with fixed exponent, fitting a power curve with unknown exponent, Transforming non-linear data. Comparing various models for the same set of observed data with LSC and with CAC, Error calculations. (From Sections 3.1, 3.2, 3.3, 3.4, and elsewhere)
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#4.
- Homework:
  The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
  
  You are required to follow the detailed instructions and rules for HWs given in the Course Information Handout and through email comments.
  Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours on Tuesday, Wednesday, and Thursday.
  - Reading HW:
    Read and understand Examples 1 and 2 from Section 3.4.
  - HW#4 for Submission. Due Thursday, 2/12, by 11pm in Canvas. Submit a single PDF file through Canvas Assignment.
    Section 3.2: #2b, #3.
    Section 3.3: Submit one of (#4 OR #8).
    Section 3.4: #7ab, #8(compare all 4 models from #7 and #8 here, as we did in the class lecture).
    [Comment: When solving the problems for Chebyshev Approximation Criterion, it is expected that your solution explicitly includes the linear optimization problem that has to be solved to apply CAC. After writing the linear optimization problem, you should solve it using any solver. For example, you can use: MATLAB : LinProg; MATHEMATICA - I; MATHEMATICA - II; Python; OTHER SOLVERS.]
    HW Solutions distributed in class.
    Questions? Ask on Canvas Discussion Forum. Or, Ask for help during the instructor and TA office hours, or through email to the instructor.

Week #5 : 2 Lectures.
- Topics: Optimization based Decision making, general form of an optimization problem, Introduction to Linear Optimization through variants of a model, Integer, Mixed Integer and binary optimization problems, multiobjective problems and how to convert them into single objective through three approaches, Binary programs and using 0-1 variables to make decisions with various constraints, Knapsack Problem and budgeting problems, modeling dependent "yes/no" decisions, approximating a non-linear function by a piecewise linear function using 0-1 variables. Geometric solutions of Linear programs, Geometric and linear algebraic intuition behind simplex algorithm, Local search algorithms - underlying concepts like neighborhood and step-size; thinking of simplex algorithm, optimization from Calculus as local search algorithms. (From Sections 7.1, 7.2, 7.3, and elsewhere)
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#5.
- Homework:
  The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
  
  You are required to follow the detailed instructions and rules for HWs given in the Course Information Handout and through email comments.
  Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours on Tuesday, Wednesday, and Thursday.
  - Reading HW:
    Read and understand Examples 1 and 3 from Section 7.1. And Examples 1 and 2 from Section 7.2.
  - Homework #5 for submission [PDF].. Due Thursday, 1/22, by 3pm in Canvas. NOTE the submission time! Submit a single PDF file through Canvas Assignment.
    HW Solutions distributed in class.
    Questions? Ask on Canvas Discussion Forum. Or, Ask for help during the instructor and TA office hours, or through email to the instructor.

Weeks #6 and #7 : 3 Lectures and 1 Mid-term Exam.
- Topics: Local search algorithms - underlying concepts like neighborhood and step-size; thinking of simplex algorithm, optimization from Calculus as local search algorithms. Graphs and networks - basic concepts such as visual and algebraic representations, examples of vertices and edge relations from Math, Computer Science, Operations Research, Network Science, Data Science, Biology, Epidemiology, etc. Conflict-free allocation of scarce resources as Graph Coloring and related examples of scheduling/ allocation of resources, Proper coloring and chromatic number of a graph, Chromatic number of a graph as 0-1 Linear optimization problem, Greedy Algorithms, Greedy coloring for proper coloring of a graph, Monitoring a network model as a vertex cover problem, Greedy algorithm for vertex cover, Vertex Cover problem modeled as a 0-1 linear optimization problem.
  Maximum Flow problem on networks as a linear optimization problem, Modeling Matching problem as a max flow problem, Min cost transportation problem for logistics as a linear program, Eulerian graphs and Eulerian tours as a model for Konigsberg bridge problem. (From Sections 8.1, 8.2, 8.3, 8.5 and elsewhere)
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#6 and Notes#7 and Notes#8.
- Homework:
  The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
  
  You are required to follow the detailed instructions and rules for HWs given in the Course Information Handout and through email comments.
  Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours on Tuesday, Wednesday, and Thursday.
  - Reading HW:
    Read and understand the examples in Sections 8.1, 8.3, and 8.5.
  - Homework #6 for submission [PDF] . [Based on 3+ lectures]. Due Thursday, 3/5, by 11pm in Canvas. Submit a single PDF file through Canvas Assignment.
    
    Questions? Ask on Canvas Discussion Forum. Or, Ask for help during the instructor and TA office hours, or through email to the instructor.

Week #8 : 2 Lectures.
- Topics: Maximum Flow problem on networks as a linear optimization problem, Min cost transportation problem for logistics as a linear program, Eulerian graphs and Eulerian tours as a model for Konigsberg bridge problem. (From Sections 8.1, 8.3, 8.4, 8.5, and elsewhere)
  Monte Carlo algorithm for calculating area under a curve, volume under a surface, etc., the concept of bounding box and its importance. Idea of using randomness in computation - MCMC, volume of high dimensional body, etc. Random point generation - middle square method, linear congruence method. (Monte Carlo) Modeling probabilistic behavior using random numbers - flipping fair/ unfair coin, roll of fair/unfair die, flipping pairs of coins, etc. Monte Carlo simulation modeling of rainy days and probability of three consecutive rainy days, Comparison with simulating a "rainy week", Simulating "nice week" with at least 4 days that have no rain and low winds, multiple weather conditions, etc. Underlying assumptions and criticism of these models. (From Section 5.1, 5.2, 5.3, 5.5 and elsewhere)
- Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#8 and Notes#9.
- Homework:
  The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
  
  You are required to follow the detailed instructions and rules for HWs given in the Course Information Handout and through email comments.
  Work on the HW problems before and over the weekend so that you ask for help on Canvas forums or in-person with the instructor or the TA during office hours on Tuesday, Wednesday, and Thursday.
  - Reading HW:
    Read and understand the Example from Sections 5.5.
    [Try these two problems but do not submit] Section 5.2: #1b, #2c.
  - HW#7 for Submission. Due Thursday, 3/12, by 11pm in Canvas. Submit a single PDF file through Canvas Assignment.
    - Section 5.1: Submit 2 out of the 3 problems listed below for this Section.
      [For each of these problems, first carefully write all steps of the algorithm as applied to the problem. Remember you have to find the appropriate bounding rectangle/box. Write the solution in the way I did such problems in class.]
      #3(run the simulation on Mathematica/Matlab for n=100, 200, 300, etc. to get approximate values of pi),
      #5,
      #7(compare your answer to actual value of the volume).
    - Section 5.3: Submit both problems listed below for this Section.
      [First write the step-by-step algorithm you will be using and then run it on computer for different values of n (number of trials). Write the solution in the way I did such problems in class.]
      #2,
      #4.
    - Final Problem. Use Monte Carlo Simulation to predict the outcome of the following game of chance over n rounds:
      In each round of the game, two fair coins are flipped simultaneously and Player 1 calls "Evens" or "Odds". "Evens" means that coins must match (both heads or both tails), and "Odds" means that coins will not match. If Player 1 calls correctly then he/she wins $1 from Player 2. If Player 1 calls incorrectly then Player 2 wins $1 from Player 1. (When you run the algorithm on computer, if you wish you may assume that player 1 always has a fixed strategy, e.g. always calls "even", or always calls "odd", etc., and compare the outcomes based on each strategy.)
    Questions? Ask on Canvas Discussion Forum. Or, Ask for help during the instructor and TA office hours, or through email to the instructor.