Topics:

1. Introduction to turbulent flows and transport in turbulent flows. This will cover the nature and scales of fluid motion in turbulence, statistical approaches to analysis of turbulent flows including Reynolds averaged Naiver-Stokes and Reynolds-averaged advection-diffusion equations, turbulent viscosity and turbulent diffusivity hypotheses and their modelling via one-equation and two-equation models.
Sources: Davidson, P.A., 2015. Turbulence, Oxford University Press. Pope, S.B., 2009. Turbulent Flows, Cambridge University Press.

2. Particles motion in (turbulent) fluid flows. This will cover the Maxey-Riley framework and its extensions to larger particle Reynolds number, particle-turbulence interactions in the dilute particle particle concentration limit, Source: Balachandar, S. 2024 Fundamentals of Dispersed Multiphase Flows. Cambridge University Press.

3. Applications of Topics 1&2. Depending on the interests of the group, this could include for example (1) sediment transport models e.g. continuum advection-diffusion type models, Euler-Lagrange etc.; (2) anisotropic particle transport models e.g. spheroids at low Re (Jeffery’s), rigid (Batchelor, Khayat-Cox) and flexible rods (Cosserat); (3) particle motion in density stratified environments (More & Andekani 2023 Annual Review)

Structure:

We will spend about 4 weeks on each topic. There is more material than we can probably cover, but that will allow us to adapt things a little based on the interests of those in the course.

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.

The course describes numerical algorithms for dealing with sparse linear algebra problems. Topics included are:

• Sparse matrices
• relevant applications: constrained optimization, fluid flow problems, electromagnetics, convection-diffusion,…
• data storage scheme

• Linear systems
• Krylov subspace solvers: minimum residual methods, conjugate gradients, bi-conjugate gradients
• preconditioning techniques: spectral analysis, field of values analysis
• multigrid
• direct solvers: matrix graphs, fill-in reducing orderings, bandwidth reduction

• Least squares and problems with constraints
• iterative solution methods for least squares: CGLS, LSQR
• techniques for solving saddle-point systems: block preconditioners, Schur complement approximations

• Eigenvalue problems
• Lanczos and Arnoldi
• Implicit restarts
• Jacobi-Davidson

This course will introduce the basic concepts and techniques in communication complexity, as well as its motivations and its applications—to learning theory, data structures, circuit complexity, combinatorics, and so on. Theorems in communication complexity often have implications for the structure of problems in learning theory, and for the resources required in various models of computation, like streaming algorithms and query complexity.

While teaching the basic concepts and applications of communication complexity, the course will also emphasize the open problems: what we don’t know about communication and why it matters.

We will cover the fundamental algorithms used for efficient convex optimization, with a focus on rigorous convergence analysis. At the end, you should have an understanding of why convex optimization algorithms work, and be able to effectively apply these tools to your own research. This course will be different from and complementary to CPSC 536M taught by Michael Friedlander in Term 1.

Convex optimization has been crucial to recent breakthroughs in theoretical computer science, machine learning, statistics, signal processing, and many other fields.
In this class we will cover the fundamental algorithms used for efficient convex optimization, with a focus on rigorous convergence analysis. We will also see how this framework can be used to give fast algorithms for various problems.
This course is especially well-suited to students who want to effectively understand and apply convex optimization algorithms in their own research. The course will be different from and complementary to CPSC 536M taught by Michael Friedlander in Term 1.

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.

This course introduces fundamental principles and meth-
ods 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 condi-
tions, linear/quadratic/cone programs, duality and KKT conditions, first/second -order
methods, interior point methods, ADMM. General concepts will be illustrated through
applications in machine learning, statistics and signal processing.

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 1) 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 2) to develop the skills required to manipulate and use these principles to solve problems.

At the end of this course, students should be able to:

1. write down the `standard equations' of geophysical fluid dynamics (GFD), identify the different terms, evaluate their relative importance based on scaling arguments, and explain how different dynamical features depend on these terms. Examples include the geostrophic and quasi-geostrophic equations, boundary layer equations, and thermodynamic relationships;

2. define standard terms and concepts used in GFD (the ``language'' of GFD), and identify them when they arise in the context of dynamical interpretations. Examples include Eulerian, Lagrangian, hydrostatic, Boussinesq, the Coriolis acceleration/force, Ekman layers, vorticity, geostrophic, barotropic, and baroclinic;

3. use standard mathematical techniques to simplify complex equation sets relevant to GFD. Examples include linearization, scaling arguments, normal mode techniques, and complex exponentials in wave and instability problems;

4. use the appropriate approximations and mathematical techniques to simplify and solve particular ``canonical'' GFD problems. Examples include a description of Taylor columns, a description of Ekman layers, spin-down problems, Rossby adjustment problems, wave problems in non-rotating and rotating systems, and instability problems.

Combining physics-driven and data-driven methods for using geophysical and remote sensing data to solve problems in the geosciences. Case histories related to societal challenges due to climate change, environmental problems, and natural resources motivate the methods covered. This course emphasizes the practical application of quantitative methods in numerical modelling, inverse theory, and machine learning; a course project provides the opportunity for students to select a topic of interest to explore in more depth.

Writing and sharing scientific software is an essential part of any quantitative workflow. We will review fundamentals in the use of Python for scientific programming, and using version control including Git and GitHub for tracking changes and sharing work. Practices that facilitate the reproducibility of computational results are a theme throughout the course.

The goal of MATH 512 is to introduce mathematical methods of quantum theory. No prerequisite
of quantum physics is required.
The course will cover some aspects of functional analysis, operator theory and the calculus of
variations with short excursions into representation theory and operator algebras. The physical
axioms of quantum theory will be introduced and elementary results will be discussed.
We will introduce the Hilbert space formulation of quantum theory, discuss quantum dynamics
and its relation to spectral properties of linear operators. We will also study the role of symmetries
in quantum physics, in particular the rotation group thereby introducing the intrinsically quantum
notion of spin. The course will conclude with a short study of strongly interacting systems and
the role of locality in their analysis.

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

Evolution is the unifying theme in biology. Evolutionary processes are responsible for the emergence of the rich variety of species across the planet. Cooperation represents one of the key organizing principles in evolution, and the history of life and of societies could not have unfolded without the repeated cooperative integration of lower level units into higher level entities. Evolutionary theories have attracted increasing attention from other behavioural disciplines including sociology and economics.

This course provides an introduction into mathematical models of evolution and the theory of games. Modelling techniques include stochastic dynamics of invasion and fixation of mutants in finite & structured populations, evolutionary game theory, adaptive dynamics, as well as coding and simulations using the EvoLudo framework. Each student develops his/her own research project in consultation with the instructor. At the end of the term, all students hand in a written report, present their project to the class and participate in a peer review process assessing the projects of their fellow students.

For more information and interactive tutorials check out https://www.evoludo.org and https://wiki.evoludo.org

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, applied probability, monotone operator 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 block-coordinate, 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.

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.

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.