Academics / Courses / DescriptionsES_APPM 395: Selected Topics in Applied Mathematics
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Description
Mathematics of Life
Mathematical thinking is playing an increasingly dominant role in experimental design, data analysis, and the conceptual understanding of Life. Through reading a diversity of papers at the interface of mathematical and physical ways of thinking, and biological phenomena, we hope to expose young scientists to this field.
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Simple Discrete Models
This class will survey several paradigmatic "simple discrete models" from a number of fields. Through computational exploration of these models, students will familiarize themselves with concepts such as phase transitions, critical phenomena, fractal geometry, scaling theory, finite-size effects, and universality.
Prerequisites: CS 150 (or equivalent) or demonstration of Python proficiency
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Approximations
The goal of this course is to give a modern introduction to mathematical methods for solving hard mathematics problems that arise in the sciences. The main focus will be to explain the process of applied mathematics, namely how to take a hard problem, of the type ordinarily encountered in applications, and gain insight into its important features. Applied Mathematics is a no-holds-barred competition, in which one uses all available tools to understand a problem as much as possible. The approach requires a combination of (a) “real” mathematics, comprised of theorems and exact results; (b) courage and skill in making legitimate approximations; and (c) intelligent use of computers to both verify and extend the validity of the approximations. Theory, Approximate techniques, and Numerical methods will be taught as needed to solve the problems at hand. We will discuss these methods in the context of mathematics problems that arise in a variety of fields, ranging from pure mathematics (e.g... the zeros of the Riemann zeta function), to optics (e.g.. the colors of the rainbow), to quantum mechanics (e.g.. the semi-classical limit), to fluid mechanics. We will start with simple problems (polynomial equations, simple integrals and simple differential equations) and end the quarter with a study of nonlinear partial differential equations. We will try to convince you that one can understand quantitative features of arbitrarily hard mathematics problems, by intelligently combining all of the resources (computational and analytical) that you have at your disposal.
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From Physics to data-driven modeling: With applications in genomic, cellular, developmental, ecological and neuro biology
I teach what I wish to learn. The most important and powerful models in history have not been data-driven. Perhaps they have been data-constrained or data-inspired. But, there does seem to be a new brand of models that are more data-driven than in the past. My knee-jerk reaction is that the overwhelming majority of these more data-driven (insert AI/ML blah blah) modeling is of little general scientific or even engineering value in the long run. All that being said, how can we argue with the potential power of these approaches. So perhaps theory needs to evolve. Perhaps we go through a historical phase where theory and modeling is more data-driven. And,, following that, there will be a middle ground where the two philosophical paradigms will find a (un?)happy marriage? The book we will be following provides a starting point, a first map, that I hope helps us organize the various threads of this new approach to modeling. As is clear from this brief description, this course is not a course in biology. Instead, examples from biology will be the basis of tutorials and assignments. I will introduce the biological background needed in class, assuming that you have none of it. We will be assuming that you know how to code, and how to code in python. We will also be assuming that you have seen, and perhaps even have some level of mastery, over the basic building blocks of mathematics: calculus, linear algebra, and probability. Since this is the first time we are teaching this course we ask for your patience and help in building it. What I would like is for people to read sections of the book/pdf ahead of class and come ready to lectures for my attempt in trying to present the ideas. I hope we can collectively converge towards having insights into the material covered in this book. I will be adding additional sections on generative deep learning and unsupervised dimensionality reduction.
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Introduction to Applied Partial Differential Equations
This one-quarter course is an accelerated version of a two-quarter sequence ES_APPM 311-1 and ES_APPM 311-2. It is not recommended for students who took 311-1 and 311-2. The topics include: Ordinary differential equations; Sturm-Liouville theory, properties of special functions, solution methods including Laplace transforms. Fourier series: eigenvalue problems and expansions in orthogonal functions. Partial differential equations: classification, separation of variables, solution by series and transform methods.
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Modeling Experiments & Data (Also BioSci 354-0-1)
We are at the beginning of the quantitative era of biology. While the molecular era has endowed us with an experimental paradigm and a molecular parts list for many living systems, we are still far from understanding how a cell computes or how an organism puts itself together. High resolution imaging and sequencing are technologies that give access to dynamics and possess the ability to drive new discoveries. We must learn to quantify, analyze, model, and interpret them.
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Physical Mathematics
An undergraduate version of asymptotic methods including numerical methods.
Recommended text: Advanced Mathematical Methods for Scientists and Engineers; Asymptotic Methods and Perturbation Theory, by Bender and Orszag.