Optimization Theory and Applications is a generic name for COMM 616. A more precise name for this edition of COMM 616 is "Integer and Linear Programming".

Linear programs (LPs) and integer linear programs (IPs) have been used for decades to solve problems in mathematics, computer science, and operations research. In this class, we explore mathematical concepts used in general purpose IP and LP algorithms. For LPs, we dive into the theory of polyhedra. For IPs, we study integer points in polyhedra; this will lead us to topics such as cutting planes and perfect formulations. We discuss current open questions in IP theory, e.g., distance between the IP and LP, sparsity of IP solutions, and the structure of IPs with bounded determinants.

This course will be proof based, but students will have the opportunity to model and solve problems using software.

Three-dimensional geometric models are the base data for applications in computer graphics, computer aided design, visualization, multimedia, and many other related fields. This course will address computerized modeling of 3D geometry, and focus on polygonal meshes, the default 3D shape representation. We will study data structures and algorithms for creating, manipulating, editing ans analyzing 3D models. We will also touch briefly on alternative representations, such as implicits, point clouds, and so on.

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 is a graduate-level introductory course to cryptography. The first half will be lectures on foundational topics and the second half will be student- or instructor-led presentations of research papers. The first half will focus on symmetric and asymmetric cryptography, which concerns the secret sharing of information.

The main references will be "Introduction to Modern Cryptography" by Katz and Lindell, and "An Introduction to Mathematical Cryptography" by Hoffstein, Pipher, and Silverman.

This is a graduate-level topics course in quantum computation but assumes no prior knowledge of quantum information. The main focus will be on quantum algorithms and introducing open research problems. More details can be found at the course website: https://wdaochen.com/teaching_2025w2

This course introduces fundamental principles and algorithms for optimization, which plays a central role in a variety of engineering problems.

The main focus is on convex programs, a class of optimization problems with a special, yet commonly-encountered structure. Students will gain thorough knowledge about how to formulate, recognize, solve (both analytically when appropriate and computationally using optimization libraries), and interpret solutions of convex programs. The course will also demonstrate how convex optimization, via relaxations, often provides a principled approach to understanding and solving even non-convex programs.

Representative topics include: convex analysis (convex sets and functions), first/second-order optimality conditions, convex programming hierarchy (linear/quadratic/semidefinite/cone programs), duality theory and KKT conditions, optimization algorithms including first-order methods (such as steepest descent, proximal methods, ADMM), second-order methods, and interior-point methods. General concepts will be illustrated through modern applications in machine learning, statistics, signal processing and controls.

Students entering the class should have a solid background in linear algebra and basic real analysis and should expect to build an even stronger foundation on these topics. A working knowledge of basic statistics and probability is also encouraged, although not necessary.

Meteorology and climatology at the micro-, local, and meso-scales. Interactions between land surfaces and atmosphere. Basics of atmospheric turbulence and transfer processes. Microclimates on scales of a leaf to those of a large valley.

The course gives an introduction to the mathematical theory and basic practice of numerical methods for partial differential equations, including algorithms, implementation, and empirical validation of theory, balancing intuition with "hard theory". The concepts and techniques covered in this course should be in every numerical analyst's arsenal. It is taught as two half courses:
Part 1 (Finite Element Methods): elliptic PDE, weak form, function spaces, Galerkin projection, conforming finite elements, implementation, variational crimes.
Part 2 (spectral methods): approximation with global trigonometric and algebraic polynomials in moderate dimension, fast algorithms (FFT), Galerkin, collocation and pseudo-spectral methods.
Additional topics will be covered through student projects and presentations. These can be flexible; examples from previous years include computational fluid dynamics, non-conforming and mixed method, PDEs on surfaces, high-dimensional problems.
The course is suitable for applied mathematics students who wish to deepen their training in numerical analysis and applied analysis techniques, as well as for science and applied science students who use numerical methods as a tool in their research but require an introduction to their theory to read relevant papers in the field and select the correct methods for their work.
There is no exam; assessment is via assignments, presentation(s), and an essay.

In this course we formulate and analyze continuum-based mathematical
models of phenomena in a wide range of areas of application. A list of
the topics for the course is given below. The mathematical modeling
typically leads to PDE/ODE models, which can be analyzed by dynamical
systems and asymptotic techniques. For each topic covered I plan to
give some insight into the formulation of appropriate mathematical
models from the underlying physics. Wherever possible, asymptotic
analysis is then used to systematically simplify the underlying model
into a more tractable form. Asymptotic and numerical methods, based
on MATLAB, will then be used to provide insight into the solution
behavior. Finally, for many of the topics considered, we will examine a recent
journal article (either in an Applied Math or Engineering journal)
that is closely related to the topic. One of the main mathematical themes
that links the different areas is stability and bifurcation theory.

Topics to be covered: (please see online syllabus for more details).
1) Nonlinear Oscillators: Forced Oscillators, Entrainment,
Stick-Slip Oscillations, Synchronization of Oscillators
2) Floquet Theory: stability of periodic solutions
3) Dynamical Hysteresis for ODE's; Slow passage problems
4) Delay-Differential Equations; Machine-Tool Vibrations, Car-Following
Models
5) Bifurcation problems in combustion theory and nonlinear heat transfer
6)Materials Science Modeling: MEMS modeling, theroelastic contact
7) Lubrication theory and slow viscous flow phenomena;
singularity formation

This course will give students an overview of Non-Newtonian Fluid Dynamics, and discuss two approaches to building constitutive models for complex fluids: continuum modeling and kinetic-microstructural modeling. In addition, it will provide an introduction to multiphase complex fluids and to numerical models and algorithms for computing complex fluid flows.

This course will focus on the analysis of algorithms for continuous optimization, widely used in data science and machine learning. We will go over the mathematical tools that will be needed, drawing from convex analysis and duality, monotone operator theory and fixed-point theory, as well as stochastic processes and illustrate how they are used for designing and analyzing widely used optimization algorithms.

We will cover both sequential convergence analyses and non-asymptotic complexity analyses for algorithms in a variety of settings, including nonconvex and convex problems, different classes of nonsmooth problems, online learning, problems involving functional constraints, and problems admitting finer structures encountered in applications, such as smoothness, strong convexity, restricted strong convexity or Lojasiewicz-type assumptions.

A representative list of algorithms we focus on will include, first-order methods and their accelerated, proximal or adaptive variants, stochastic first-order algorithms and their enhancements with variance reduction and adaptivity, Frank-Wolfe algorithm, second-order algorithms, mirror descent, augmented Lagrangian methods, operator splitting algorithms (for example, extragradient, proximal point) for solving generalized problems such as min-max games and variational inequalities, as well as their stochastic variants. Computational aspects of the methods will also be investigated as part of assignments.

The purpose of this graduate course is to equip graduate students with cutting-edge techniques in mathematical and computational modelling, analysis and simulations of semi-linear parabolic partial differential equations (PDEs) of reaction-diffusion type. The course covers (i) formulating models using first principles based on the mathematical translation of physical observations and the use of appropriate physical laws, (ii) mathematical analysis of the resulting models (linear stability analysis) and (iii) the development of novel numerical methods based on finite differences and finite elements. The finite element method is extended to deal with reaction-diffusion systems posed on domains and surfaces that continuously deform in space and time, leading to the concept of evolving bulk-surface finite elements. Students will learn hands on techniques in analysis and simulations using open-source finite element software packages such as FeNiCs.

This course will introduce the major tools in stochastic analysis from an applied mathematics perspective. Topics to be covered include Markov chains (both discrete and continuous), Gaussian processes, Ito calculus, stochastic differential equations (SDEs), numerical algorithms for solving SDEs, forward and backward Kolmogorov equations and their applications. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have seen a little analysis, particularly in the context of studying PDEs, but will generally avoid measure theory. The target audience is graduate students in applied mathematics or related fields, who wish to use these tools in their research for modelling or simulation.

Prerequisites include good upper-level undergraduate or early graduate knowledge of: probability, linear algebra, PDEs, and ODEs. Some prior experience with numerical analysis is helpful but not necessary.

Homework will be a critical part of the course, and will include some programming assignments.

Notes from previous versions of the course can be found here: https://personal.math.ubc.ca/~holmescerfon/teaching.html#asanotes

Topics in Quantum Field theory including renormalization, the renormalization group, Yang-Mills theory, an introduction to conformal field theory, anomalies, generalized symmetry

Introduction to group theory with applications in physics. Representation theory, discrete groups, Lie groups.

Introduction to the basics of quantum field theory, including functional techniques, perturbation theory, Feynman diagrams, renormalization and an introduction to quantum electrodynamics.