Data science is one of the most growing scientific disciplines today. This course will introduce various mathematics related to modern data science. The subjects that will be covered for the semester include Introduction to data science and big data, Preprocessing of data, Interpolating data, Classification of data, Generating scheme of data, Text, sound, and image data, Stochastic and chaotic characteristics of data, data-driven computing, Topological and geometrical understanding of data, Manifold representation of data, Artificial intelligence and data, Data analysis with Python and visualization, etc. Students will also implement the learned materials into Python programming.

It is recommended, but not required, to take linear algebra, differential equations.

Introduction to Mathematical Data Science - by example (Instructor's lecture note), 2021.
Course materials provided by the instructor during the semester

Tentative Course Schedule
1. Introduction to data science
A. What is data science?
B. History of data science
C. What do we do with data science
2. Mathematical modeling with data
A. Covid-19 data
B. Modeling with differential equations (continuous modeling)
C. Particle simulation (dynamical system)
D. Cellular automata (discrete modeling)
E. Causality detection
F. Parameter estimation/Neural ODE
3. Random versus chaotic data
A. Paramecium Aurelia data
B. Population data
C. Logistic map
D. Chaotic behavior of data - Butterfly effect
E. Random versus chaotic data
4. Machine Learning & data
A. Inductive versus deductive analysis
B. Nonclassical approximations
       C. Modeling with perceptron/Support vector machine
D. Universal approximation theorem
E. Graph neural network
5. Prediction with data
A. Data interpolation/extrapolation - interpolation theory
B. Least squares approximation (regression)
C. Regularization (overfitting, L1 regularization)
D. Feature extraction (dimensional reduction)
E. Data visualization
6. Classification with data
A. Data classification
B. Principal Component Analysis
C. Regression (Logistic regression, optimization)
D. Linear regression, Logistic regression, SVM
E. Deep learning - multiple layers
7. Image data
A. Data source: cat vs. dog (from Kaggle)
B. Intro - Edge detection (image data) Jump function, differential map
C. Introduction of convolution approach
D. Classification via convolutional neural network
8. Text data
A. Python nltk
B. Text analysis (word frequency, Zipf law)
C. Corpus analysis
9. Sound data
A. Fourier analysis
B. Filtering/denoising
C. Spectrogram analysis
D. Inverse analysis
10. Multi-modal data analysis
11. Sampling of data
A. Sampling from simulation data
B. Monte Carlo versus polynomial chaos
12. Topological data analysis (TDA)
A. Geometry and topology in data analysis
B. Data analysis with persistent homology
C. Data analysis with graph network (mapper analysis)
D. Classification problem (MNIST hand-writing data)
E. Mapper
F. AI music data generation with TDA

The class is composed of lecture and tutorial sessions. Each class will provide both the theoretical and practical (computational) aspects of data science with hands-on tutorials. Weekly homework will be assigned. Students are required to attend the class regularly and submit their homework on time.