This course explores recent advances in geometry processing research with focus on computer graphics and vision applications. Topics span from shape reconstruction from images and video, through neural shape representations, garment modeling, fabrication, sketch-based modeling, geometry editing, and perceptual shape processing. Students will engage in depth with key readings and will gain hands-on experience through projects.

We will cover the following fundamental algorithms used for provably efficient convex optimization with a focus on rigorous convergence analysis: (1) Ellipsoid method; (2) Gradient Descent; (3) Mirror Descent; and (4) Interior Point Methods. At the end of the course, you should have an understanding of why these convex optimization algorithms work, and be able to effectively apply these tools to your own research.

This course is a modification on the course "Submodular Optimization" which was offered in 2023. You can see information on that course in my website       bshepherd.ca

Unfortunately, I am having issues with modifying the contents of that page so this gives some information on how things might change from past course. 

The course's first goal is to understand the behaviour of submodular set functions (as before). 

A set function f is defined on a finite ground set, call it V. It assigned a value f(A) to each subset of V. Such a function is submodular if it has the diminishing returns property. Namely, if $$A \subseteq B$$, then $f(A+i)-f(A) \geq f(B+i)-f(B)$. These functions have been studied in economics and operations research since the 1960s.

The course's second goal is to get practice seeing how submodular optimization problems arise in practice. This will be done either in lectures, or by working on assignment exercises. Sometimes these problems are cleverly disguised.

Of course, we will also discuss what algorithm paradigms work for submodular objectives. Classical approaches like greedy, local search, and the ellipsoid method need to be adapted. There are also more recent approaches which are sometimes needed such as multilinear extensions, continuous greedy and poison processes.

There have been several important breakthrough in the past 5 years, and these will be either part of the lecture sequence or can be selected for your final project. (Done in pairs).

I prefer people to work together in general, except on the short in class quizzes. The following is what I propose for the grading policy.

3 Assignments 45%

3 Quizzes (in class - usually about 20 minutes) 15%

Participation 10%

Projects 30%



Classes are Tuesday and Thursday (starting on Sept 10) in:

Civil and Mechanical Engineering Building (CEME) | Floor: 1 | Room: 1212

This is an introductory course in scientific computing considering both the underlying mathematics and implementation. The class is cross-listed for graduate credit as MATH 607E. The graduate version has an additional project requirement.

There is a coding component on assignments and tests that can be done with any platform (python or MATLAB recommended).

There are three major sections to the course material. First, we will look at basic numerical analysis ideas: root finding, interpolation, and quadrature (numerical integration). Second, we will consider numerical approximation of ODEs, both initial value problems (time stepping) and boundary value problems (finite difference and spectral approximation). Last, we will consider PDEs (parabolic, hyperbolic, and elliptic problems).

This is a continuation (with some review) of the material from MATH 405/607E in the first term. However, the course is open to all students with some background in numerical methods.

Three more advanced topics will be considered in the course. There will be assignments and a course project. Students will present proposals for their project during the term and their results in the last week.

The first will be the Finite Element Method (FEM) for elliptic problems. This method has very elegant mathematical theory which requires some Functional Analysis, which will be introduced in as painless a way as possible. As part of this topic, we will consider the conjugate gradient method for iterative solution of the corresponding linear systems. Some familiarity with freeware FEM codes (FreeFEM, FEniCS) will be gained in assignments.

The second topic will be hyperbolic conservation laws. Some of the phenomena and the corresponding mathematics will be introduced with a simple traffic flow model. Appropriate numerical methods will be developed. The computational framework, Clawpack, will be used for some application problems.

The final topic will be free and moving boundary value problems. Model problems (Stefan, Oxygen Depletion) will be considered and appropriate numerical methods developed.

Exact or numerical techniques are not the only way to solve problems or understand their solutions. This course describes the machinery of asymptotic analysis which can be applied to the solution of physical problems.
The syllabus:
I. Asymptotic series
II. Solution of algebraic systems
III. Integrals
IV. Differential equations
V. Matched asymptotics
VI. Multiple scales
VII. Improvement of series
Special emphasis will be given to applying the techniques to problems of physical relevance (e.g. analysis of wave dispersion relations, amplitude expansions for forming patterns, dynamics of nonlinear oscillators, fluid boundary layers).
Assessment will involve coursework (homework problems) and examination.

This is a course in modern techniques in applied mathematics, focusing on perturbation methods for partial differential equations. The material provides valuable skills and resources complementary to scientific computations, mathematical modelling in applications, analysis of PDEs and dynamical systems. The general concepts and methods are illustrated and developed for a wide variety of specific problems arising in mathematical biology, fluid mechanics, materials science, and wave propagation.

We develop tools and concepts from high-dimensional probability that underlie the mathematical theory of compressed sensing and related problems in data science. Major topics are:
1. Behaviour of sums of random variables, non-asymptotic deviation inequalities
2. Concentration of measure
3. Non-asymptotic random matrix theory and extrema of stochastic processes

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.

Tentative description (subject to change) --

Artificial Intelligence works far better than any existing theory predicts, and closing that gap has become one of the most active areas of modern applied mathematics. This course is a rigorous introduction to the mathematics of modern AI, organized around the three classical pillars: approximation (what neural networks can represent), optimization (what gradient-based training actually finds), and generalization (why interpolating models predict well), followed by two modern segments: the mathematics of large
language models and learned representations, and the mathematics of trustworthiness, where reliability, evaluation, and fairness are treated as formal problems in estimation, sampling, and geometry.

The course pairs theory with computation throughout. Nearly every result in this field was discovered empirically before it was proved, e.g. double descent, benign overfitting, neural collapse, in-context learning, and the course treats computational experiment as part of the mathematical methodology, not an afterthought. Each week combines lectures developing theorems and proof techniques (concentration of measure, convex and nonconvex optimization, harmonic analysis, random matrix theory, differential geometry) with a lab in which students probe trained and pretrained models the way researchers do: fitting linear probes, extracting and steering feature directions, measuring representation geometry, and reproducing landmark empirical findings from the research literature. The
emphasis is on research-style experimentation with existing models rather than reimplementing standard components from scratch.

Many particle systems
Second quantization, non-relativistic Fermi and Bose gases.
Action principle, Lagrangian density, canonical quantization, symmetries and Noether's theorem.
Space-time symmetries and the stress tensor
The Dirac equation and the Dirac quantum field theory
The photon as a quantized field, covariant quantization and quantum electrodynamics
Perturbation theory, Feynman diagrams
Functional methods, functional derivatives, functional integration, generating functionals for quantum field theory
Renormalization of quantum electrodynamics