**Instructor:**

**David Bortz**

**Pre-req.**APPM 2360 and 3310 or equivalent or instructor permission

It is a common refrain to hear how the 21st century is The Century of Biology. But, what does that mean for mathematicians who want to work on real biological problems and biologists who want to employ sophisticated mathematical tools to help understand a particular phenomena? And, most importantly, what does that mean for students who aspire tpo bridge the gap? The goals for this class are for students to:

(1) Develop a fundamental understanding of how mathematical models are created from data in the biological sciences

(2) Illustrate how the study of these models can be used to gain novel insights in complex biological systems. We will thoroughly investigate case studies in several elds including: Infectious Disease Epidemiology (including modeling of the spread of COVID-19), Molecular Systems Biology, Sea Turtle Ecology, Glucose Metabolism, Neuron Action Potential Propagation, HIV Pathogenesis, and Microbial Community Dynamics. This class is designed for advanced undergraduate and beginning graduate students.

Note: In the last year, Prof. Bortz has served as a member of the State of Colorado's COVID-19 modeling team. Accordingly, where feasible in the course, we will make use of data and examples from the pandemic and its impact locally and around the world.

**Instructor:**Manuel Lladser

**Pre-req.**APPM 3570 or equivalent.

**Description.**Random graphs, also called random networks, have been used to understand the robustness of the Internet, study food webs in predatory interactions, and predict unknown metabolic interactions, among countless other applications. This course introduces and analyzes various key random graph models, including the Erdös-Rényi and the Stochastic Block models. It presents these and other topics related to discrete random structures in a coherent and self-contained manner to facilitate their use to model and analyze more general random networks. The course should be especially appealing to undergrad and grad students who seek intuition as well as a mathematical exposition of random graph theory.

**Note.**This course is different but complementary to Dynamics on Networks (APPM 4/5720) given in Fall 2019 semester by J. Restrepo.

**Applied Deep Learning (APPM 4720/5720)**

**Instructor:**Maziar Raissi

**Description:**This course is primarily designed for graduate students. However, undergraduate students with demonstrated strong backgrounds in probability, statistics (e.g., linear & logistic regressions, numerical linear algebra and optimization are also welcome to register. We will be pursuing the objective of familiarizing the students with state-of-the-art deep learning techniques employed in the industry. Deep learning is a field that has been witnessing a mini-revolution every few months. It is therefore very important that the students registering for this course are eager to learn new concepts. So much of deep learning is just software engineering. Consequently, the students should be able to write clean code while doing their assignments. Python will be the programming language used in this course. Familiarity with TensorFlow and PyTorch is a plus but is not a requirement. However, it is very important that the students are willing to do the hard work to learn and use these two frameworks as the course progresses.

**Specific topics to be covered in Spring 2021:**

- Unconditional and conditional Generative Networks
- Multimodal Learning
- Natural Language Processing, including word representations, text classification, neural machine translation, language modeling
- Reinforcement Learning, off-policy and on-policy methods
- Graph Neural Networks
- Recommender Systems
- Speech and Music

**Instructor:**Stephen Becker

Linear methods are used throughout STEM fields because linear methods/problems are so much easier to work with, and because many nonlinear problems have linear approximations. In optimization, the analog to "linear" is "convex". This course APPM 5630 studies convex optimization because it is relatively well understood, and because many nonconvex problems have convex approximations. In particular, we investigate landmark convex optimization algorithms and their complexity results. We develop theory while also surveying current practical state-of-the-art methods. Topics may include Fenchel-Rockafellar duality, KKT conditions, proximal methods, and Nesterov acceleration. Homeworks involve both theoretical problems and programming components (Matlab or Python are suggested). Class involves a final project (see https://amath.colorado.edu/faculty/becker/teaching.html

**Prereqs:**APPM 4440 “Real Analysis” or equivalent, and either APPM 4650 or APPM 5600 "Numerical Analysis" or equivalent. Some knowledge of linear programming is useful.

**Stochastic Differential Equations (APPM 6570)**

**Instructor:**Yu-Jui Huang

This course provides a comprehensive investigation of stochastic differential equations (SDEs). Applications in stochastic control theory, mathematical finance, and machine learning will also be introduced. Weeks 1-3 will cover preparatory materials, such as Brownian motion, martingales, stochastic integrals, and Ito's formula. Weeks 4-9 will focus on the theory of SDEs, including strong and weak solutions (existence, uniqueness, comparison results, the martingale problem) and connections to partial differential equations (Dirichlet problem, Cauchy problem). Weeks 10-15 are left for applications, such as the stochastic control theory and its applications to mathematical finance and stochastic gradient flows driven by SDEs with applications to unsupervised and reinforcement learning.

Text: Brownian Motion and Stochastic Calculus

(Second edition, by Ioannis Karatzas and Steven Shreve, Springer).

(Second edition, by Ioannis Karatzas and Steven Shreve, Springer).

Recommended prerequisite: APPM 6560 or MATH/APPM 6550.

**Radial Basis Functions with Applications to the Geosciences (APPM 7400-005)**

**Instructor: Bengt Fornberg**

Radial Basis Functions (RBF) and RBF-generated finite differences (RBF-FD) are powerful methodologies for solving PDEs to high accuracy in non-trivial geometries, with irregularly shaped material interfaces, and/or when spatially variable resolution is required. In several recent large-scale applications to the geosciences and for fluid mechanics, these methods have been shown to outperform all previous alternatives. While this course is listed as an upper level graduate course, it will not require prerequisites beyond undergraduate calculus and differential equations; some previous numerics experience can be helpful. The course is designed for students who want to solve problems in different application areas and would like to see this topic surveyed. Our focus will be on how, when, and why RBF-based approaches work, more by means of examples and heuristic explanations than by rigorous analysis. Course assignments will mostly take the form of projects and presentations. The course text book (B. Fornberg and N. Flyer:

*A Primer on Radial Basis Functions with Applications to the Geosciences*, SIAM 2015) will be made available electronically.