This course offers a comprehensive exploration of modern optimization theory and algorithms, with applications in machine learning and operations research. It begins with an examination of fundamental concepts and problem properties in optimization, such as convexity, duality, smoothness, and subdifferentials. Students will then learn to design first-order optimization algorithms tailored to various problem characteristics, with a strong focus on how these structures impact convergence analysis.

The course also covers lower complexity bounds for different function classes, providing students with a critical understanding of the computational limits inherent in optimization algorithms. In the final part of this class, students will apply these advanced optimization techniques to tackle real-world challenges in machine learning, data science, and operations research, with particular attention to topics like optimal transport and distributionally robust optimization.

CPSC 424 is an advanced computer graphics course focussed on geometric modeling, i.e. creation and manipulation of shapes. The topics covered in this course are:

Introduction to curves and surfaces, in particular splines, subdivision surfaces, polygonal meshes. Principles and mathematical foundations for representing complex geometry for computer graphics and numerical simulations. Practical applications of different modeling techniques.

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This course covers recent advances in building digital representations of humans for a variety of applications, such as product design, character animation, and medicine. The focus is on building realistic computational models of real humans interacting with real objects, like clothing. This is year we will specifically focus on differentiable computational models to support machine learning and accurate real-time predictions. We will use state-of-the-art software frameworks (e.g., NVIDIA warp) for high performance computing for some assignments. Basic knowledge of Python is required.

Topics are organized into 6 modules, building up levels of realism in digital human models. The focus is on computational models, but we will also learn basic biomechanics of real human bodies.
(1) Introduction. Review of background for computing with 3D objects, in the simplest setting. Geometry, kinematics, dynamics, and numerical integration.
(2) Human Shape. 3D scanning. 3D Meshes. Registration.
(3) Human Motion. Kinematic representations. Skeletons: real and animated. Motion capture.
(4) Clothing. A first look at simulating physics. Hyperelasticity. Finite element models.
(5) Soft Tissue Simulation. Contact. Eulerian and Lagrangian discretizations.
(6) The rest of the story. Discuss papers based on student interest. Examples: Machine Learning applied to any of the above.

Convex optimization is a key tool for analyzing and modeling a range of computational problems that arise in machine learning, signal and image processing, theoretical computer science, operations and logistics, and other fields. It’s also the backbone for other areas of optimization, including algorithms for nonconvex problems. This course aims to provide a self-contained introduction to a few of the many geometric and intuitive ideas in convex analysis and their usefulness for understanding and developing computationally-efficient algorithms for a range of scientific and engineering problems.

This course provides students with computational and modeling skills that can be employed to analyze macroeconomic phenomena (including scaling-up policy effects) and to answer microeconomic questions relating to the optimal choices of individuals, households, firms, and groups. The main purpose of the course is to introduce methods that allow the mapping of different data into computational models: such techniques can be used for the quantitative evaluation of government policies, to examine historical inequality patterns, to study individual and aggregate wage dynamics, identify individual and households' responses to shocks, to rationalize firms' growth patterns and for many other problems. At the end of the course, students can apply such methods in their PhD work and pursue independent quantitative analysis using the methods learned in class.
The course provides (i) an overview of general equilibrium analysis and its existing (and potential) applications to topics in various fields, with a special focus on applications that entail the use of economies where agents are heterogeneous and markets are incomplete; (ii) an overview of computational methods to numerically solve for the individual decisions of economic agents as well as for equilibria of the model economies discussed in class; (iii) an in-depth discussion of specific topics that are of interest to both teacher and class (the choice reflects students' suggestions in any given year).
Students are incentivized to reproduce results from one or more applied and computational papers (including in their field). These exercises are facultative and collaboration among students is strongly encouraged when solving problems, which often involve sharing information and dividing tasks in the spirit of co-authorship. Based on past years' experience, by the end of the
course, students will be able to set up, analyze, and numerically compute equilibrium models with heterogeneity. Many students end up making significant use of these methods in their dissertation research and field.

This course introduces fundamental principles and methods for optimization, which plays a central role in a variety of engineering problems. Our main focus is on convex programs, a class of optimization problems with a special, yet commonly-encountered structure, that everyone who uses computational mathematics will benefit from knowing about.

Specifically, the course is designed to give the graduate student thorough knowledge about how to formulate, recognize, solve and interpret the solution of convex programs. Representative list of topics includes: convexity, first/second -order optimality conditions, linear/quadratic/cone programs, duality and KKT conditions, first/second -order methods, ADMM. These concepts will be illustrated through applications in modern machine learning, statistics, and signal processing, covering topics such as compressive sensing, matrix completion, over-parameterization, and double descent.

Students entering the class should have a solid background in linear algebra and basic real analysis. A working knowledge of basic statistics and probability is also encouraged, although not necessary.

The purpose of this course is to a) introduce the student to the dynamical principles governing the large- scale, low-frequency motions in strongly rotating fluid systems (like the ocean, atmosphere, and liquid planetary core) and their consequences; and b) to develop the skills required to manipulate and use these principles to solve problems.

There are no formal prerequisites. This course is mathematical and assumes a working knowledge of vector calculus (e.g. div, grad, curl), partial differential equations (i.e. you can solve at least some of them), and some exposure to complex analysis (e.g. you know that if z = x + iy, then ez = ex cos(y) + iex sin(y)). A background in fluid dynamics, geophysics, atmospheric sciences, and/or oceanography is *not* required.

This research-oriented course will explore the theoretical underpinnings of graphical modeling and causality. A graphical model is a mathematical structure that describes complex dependencies between random variables. More precisely, given a (directed or undirected) graph, we envision the random variables as sitting at each of its vertices, and the conditional independence statements that hold among them can be read off from the graph structure. We will study both undirected and directed graphical models. Given samples from a graphical model, we will discuss model selection: the problem of finding the graph that the data arose from, and inference: the problem of estimating the distribution assuming we know the graph. We will explore different types of algorithms used to solve these questions as well as the mathematical theory involved.

Building on the theory of graphical models, we will study causal discovery. Here, we are interested in finding a directed graph that depicts the causal relationships among the observed random variables (e.g., X --> Y if X causes Y). We will discuss how to solve this problem in both the observational and interventional (e.g. randomized control trials) settings. We will conclude with theory and algorithms for the case of hidden variables as well as directed cycles in the graph.

Advances in imaging techniques have enabled the access to 3D shapes present in a variety of biological structures: organs, cells, organelles, and proteins. Since biological shapes are related to physiological functions, biological studies are poised to leverage such data, asking a common statistical question: how can we build mathematical and statistical descriptions of biological morphologies and their variations?

In this course, we will review recent attempts to use advanced mathematical concepts to formalize and study shape heterogeneity, covering a wide range of imaging methods and applications. The main mathematical focus will be on introducing the theoretical frameworks of Riemaniann geometry, diffeomorphisms and metrics over shape space, manifold learning, optimal transport theory with application for image analysis, with some other concepts covered in specific applications (e.g. quasiconformal mapping theory for shape representation, 3D reconstruction in Fourier space…). Specific biological imaging techniques, including atomic force microscopy, cryo-EM and cryo-ET, as well as computational methods and softwares will also be covered with guest lectures and tutorials from expert practitioners.

To pass the course, students will be required to write a blog post summarizing some course material of their choice for a dedicated website (https://bioshape-analysis.github.io/blog/), and to write a literature review or do a small project.