Applied Mathematics Toolkit

Modern scientific and engineering systems are increasingly defined by nonlinearity, high dimensionality, partial observability, and uncertainty. Addressing such complexity requires more than isolated techniques. It calls for an integrated applied mathematics perspective that connects modeling, data analysis, and computation within a coherent framework. This book develops such a unified toolkit. Rather than treating mathematical methods as separate disciplines, it emphasizes their complementary roles in understanding, analyzing, and predicting complex systems, with a focus on intuition, practical relevance, and transferability across scientific and engineering fields.

• Presents a broad collection of essential applied mathematics tools, organized to emphasize their interplay rather than treating them as isolated topics.
• Each method is introduced through simple yet illuminating examples that make advanced concepts accessible without sacrificing rigor.
• For every technique, the presentation highlights the underlying motivation, key assumptions, strengths, limitations, and the contexts in which it is most effective.
• Theory, data, and algorithms are consistently integrated to demonstrate how they work together in practical applications.
• Is accompanied by short ``one concept $\cdot$ one example'' videos, each 5-10 minutes long, focusing on a single idea and using deliberately simple examples to support initial exposure, intuition building, and efficient review.
• Example codes included to reinforce the concepts and facilitate hands-on learning.

This book is intended for advanced undergraduate students, graduate students, and researchers in applied mathematics, science, and engineering who seek practical mastery of modern mathematical tools. It is well suited to topic courses and self-study, particularly for readers working at the intersection of modeling, data, and computation. By combining conceptual clarity with hands-on examples and an integrated methodology, the book provides a foundation for tackling real-world problems in a wide range of interdisciplinary settings.

Nan Chen is an Associate Professor in the Department of Mathematics at the University of Wisconsin-Madison and is affiliated with the Institute for Foundations of Data Science. He received his PhD from the Courant Institute of Mathematical Sciences at New York University in Applied Mathematics and Atmospheric and Ocean Sciences, followed by two years as a Postdoctoral Research Associate at Courant. Dr. Chen's research spans applied mathematics, atmospheric and ocean sciences, materials science, and data science. He has received several major honors, including the Kurt O. Friedrichs Prize, the Silver Medal of the New World Mathematics Awards, the Office of Naval Research Young Investigator Award, and multiple teaching awards. He has published over 100 papers in leading journals, including Proceedings of the National Academy of Sciences, Nature Geoscience, Nature Communications, and Notices of the American Mathematical Society. Dr. Chen has delivered more than 100 invited presentations worldwide, and his work has been featured in SIAM News, SIAM DSWeb, Eos, and many other media outlets. He has organized nearly 30 workshops and mini-symposia at major conferences. Dr. Chen has mentored postdoctoral researchers, PhD students, and undergraduates, many of whom have received major research and presentation awards. He is committed to supporting early-career scientists and promoting interdisciplinary and international collaboration at the interface of applied mathematics, engineering, and geoscience. He currently serves as Secretary of SIAM Mathematics of Planet Earth and Chair of the AGU session ``Applied Math Perspectives on Modeling, Analyzing, and Predicting Complex Geophysical Systems'', playing an active role in shaping interdisciplinary efforts at the interface of applied mathematics and other disciplines.

Charlotte Moser is a PhD student in Mathematics at the University of Wisconsin-Madison, where she works as a research assistant at the interface of applied mathematics and environmental science. Her research focuses on mathematical modeling, uncertainty quantification, data assimilation, machine learning, Bayesian inference, and the analysis of complex dynamical systems, with applications to atmospheric and oceanic phenomena. She received her B.S. in Applied Mathematics, with minors in Environmental Science and Psychology. Ms. Moser has received multiple honors, including the DANOC Best Student Presentation Award, the Henry Schaerf Mathematics Graduate Award, and the AGU Outstanding Student Presentation Award, as well as the Dean’s Distinguished Fellowship, the Mathematics Honors Program distinction, the IBA Biomathematics Education with Applications and Methods Grant, and the JD and Marcia Wine Mathematics Award. She has presented her research at leading national and international conferences and is emerging as a young researcher at the intersection of applied mathematics and environmental science. In addition to her research, she is actively engaged in scientific communication and education. She develops accessible content on applied mathematics and Earth science through online platforms, aiming to broaden participation and foster interdisciplinary learning. She is also committed to broadening participation in STEM and fostering collaborative scientific communities. Her work is driven by a strong interest in connecting mathematical theory with real-world environmental challenges.