Courses: past

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In this course we develop elements of the theory of Gaussian and empirical processes that have proved useful for statistical inference in high-dimensional models, i.e. statistical models in which the number of parameters is much larger than the sample size. The course consists of three parts, with the first two parts laying the foundation for the third one: an introduction to modern techniques in Gaussian processes, a recap of empirical classical process theory emphasizing weak convergence on metric spaces, and lastly, a discussion of Gaussian approximation, high-dimensional CLTs, and the conditional multiplier bootstrap.

Course Contents:

• Part 1: Elements of Gaussian processes (concentration, comparison, anti-concentration, and super-concentration inequalities, Talagrand’s Generic chaining bounds).
• Part 2: Elements of empirical processes (convergence of laws on separable metric spaces, Glivenko-Cantelli and Donsker theorems under metric and bracketing entropy, applications to bootstrap)
• Part 3: A selection of theoretical problems in high-dimensional inference (including but not limited to Gaussian approximation, high-dimensional CLTs, and multiplier bootstrap when function classes are not Donsker).

Homework and Examinations

There will be regular homework assignments and an oral examination. The oral examination will work as follows: The lecture will be divided in roughly ten topics which will be shared with the students ahead of time. At the day of the examination the students will randomly draw two topics and give two 10-15 min presentations on their topics on the blackboard (no prepared notes allowed). Each presentation will conclude with ca. 5 minutes of follow-up questions. Textbooks for the first and second part:

• Dudley, R. M. (2014). “Uniform Central Limit Theorems”. CUP.
• Giné, E. and Nickl, R. (2016). “Mathematical Foundations of Infinite-Dimensional Statistical Models”. CUP.
• van der Vaart, A. and Wellner, J. (1996). “Weak Convergence and Empirical Processes”. Springer.

Typed lecture notes of all three parts will be provided.

Please note, the WDA registration deadline for this course at UBC will be Jan 6th, 2023.

Registration for this course is not currently available.

This course provides an extensive theoretical foundation for as well as hands-on introduction to several widely used methods for studying the properties of materials and structures, in particular at the nanoscale and mesoscale. The majority of the time is spent on quantum-mechanical methods: the first-principles approaches (starting from the Hartree-Fock theory and building up to Configuration Interaction and the Møller–Plesset Perturbation Theory) and, in particular, the Density Functional Theory, which are derived and discussed in detail. Semi-empirical methods such as Tight Binding and Molecular Dynamics are also covered, as well as strategies for modelling material properties (electronic, mechanical, optical, etc.). Practical activities include implementing some of the above theories in computer code, in addition to using established software (Gaussian, SIESTA, VASP, LAMMPS, etc.). Each student also works on a project of their choice using the methods discussed.

Introduction

• Modelling quantum systems and phenomena
• The many-body wave function and the Schrödinger equation
• The Born-Oppenheimer approximation
• Spin and the Pauli exclusion principle
• Representation of functions

Hartree-Fock theory

• Hartree products and Slater determinants
• The variation principle
• The expectation value of the Hamiltonian with a single Slater-determinant
• Lagrange’s method of undetermined multipliers
• Exchange interaction, the Fock operator, and the Hartree-Fock equations

Interpretation of Hartree-Fock orbitals

• Unitary transformations and the diagonalization of the Hartree-Fock equations
• The Koopmans theorem and the significance of canonical Hartree-Fock orbitals

Implementation of the Hartree-Fock equations

• Basis functions and basis sets
• The Roothaan equations
• Mulliken population analysis

Post-Hartree-Fock methods

• Many-electron excitations
• Basis set for many-electron wave functions
• Configuration interaction
• The Møller-Plesset perturbation theory

The density functional theory (DFT)

• Functional derivatives
• The theorems of Hohenberg and Kohn
• The Kohn-Sham method
• Total energy in DFT, and the significance of Kohn-Sham orbitals
• Correlation energy and exchange-correlation functionals
• The connection between DFT and the Thomas-Fermi-Dirac and Hartree-Fock theories
• Periodicity, the Bloch theorem, and band structure in DFT
• Finite-temperature DFT
• Time-dependent DFT

Semi-empirical approach to studying electronic structure

• Linear combination of atomic orbitals
• The Hückel method
• The Pariser-Parr-Pople method
• The tight-binding method

Semi-empirical approach to studying mechanical structure

• Molecular mechanics and molecular dynamics
• Force fields
• Time propagation
• Temperature, pressure, thermostats, and barostats

Please note, the WDA registration deadline for this course at UBC will be Jan 6th, 2023.

Registration for this course is not currently available.

This is a first course in analytic number theory. In this course we will focus on the theory of the Riemann zeta function and of prime numbers. The goal of this course will include proving explicit bounds for the number $\pi(x)$ of primes which are less than a given number $x$. Building analytical tools to prove the prime number theorem (PNT) will be at the core of this course. We will explore and compare explicit formulas, whether they are using smooth weights or a truncated Perron formula, to relate averages over primes and $\pi(x)$ to sums over the zeros of zeta. Another originality of this course will be to explore each topic explicitly (essentially by computing all the hidden terms implied in the asymptotic estimates). With this respect, students will get an introduction to relevant programming languages and computational software. This will be closely connected to Analytic Number Theory 2 by Greg Martin (UBC), as the sequences of topics are coordinated between us; the intention is for students at all PIMS institutions to be able to take the second analytic number theory course as a continuation of the first one with maximum benefit. In addition, these two courses will provide excellent training for students who plan to attend the “Inclusive Paths in Explicit Number Theory” CRG summer school in 2023. All these events are part of the PIMS CRG “L-functions in Analytic Number Theory” (2022-2025).

Registration for this course is not currently available.

This is a graduate course on the structure and representation theory of real Lie groups. The course will have four parts: an introduction to topological and compact groups, the basics of Lie groups and differential geometry, the structure and representation theory of compact Lie groups, and (as time allows) the structure and representation theory of semisimple Lie groups.

https://personal.math.ubc.ca/~lior/teaching/2223/535_W23/

Other Information

Registration for this course is not currently available.

The course covers foundational mathematical tools that are useful in analyzing high-dimensional single-cell datasets, and modelling developmental stochastic processes. We cover basic probability theory, statistical inference, convex optimization, Markov stochastic processes, and advanced topics in optimal transport.

See the course website for the syllabus and other details.

https://sites.google.com/view/math612d/home

Other Information

Registration for this course is not currently available.

The course provides learning opportunities on statistical software, R, with some focus on data management and wrangling, reproducibility, and visualization. On top of that, there are basic introductions to Machine Learning such as k-NN, Naive Bayes, regression methods, etc. The focus is on hands-on skills with R and applications to real data.

Registration for this course is not currently available.

This course is a second graduate course in number theory, designed to follow Analytic Number Theory I taught by Prof. Habiba Kadiri (University of Lethbridge) in Fall 2022. We will learn about Dirichlet characters and sums involving them, Dirichlet L-functions and their zeros, and the prime number theorem in arithmetic progressions. With the explicit formula for the error term in this theorem, we will continue into limiting distributions of error terms and comparative prime number theory (“prime number races”). This course also precedes the summer school “Inclusive Paths in Explicit Number Theory” in Summer 2023 and is designed to give students the ideal preparation for that summer school program. All three of these events are part of the current PIMS CRG “L-functions in Analytic Number Theory” (2022-2025).

https://personal.math.ubc.ca/~gerg/index.shtml?613-Winter2023

Other Information

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This course discusses practical techniques for the analysis of multivariate data. Topics covered include estimation and hypothesis testing for multivariate means and variances; partial, multiple and canonical correlations; principal components analysis and factor analysis for data reduction; multivariate analysis of variance; discriminant analysis; and cluster analysis.

Registration for this course is not currently available.

This is a one semester three credit hour course. It is about the theory and applications of stochastic differential equations driven by Brownian motion. A stochastic differential equation (SDE) is a differential equation in which the rate of change is determined by the state of the system itself, some external known forces and some unknown external forces as well. The noise in the system is given by random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. Random differential equations are conjugate to stochastic differential equations. This course will concentrate on stochastic differential equations driven by Brownian motions. The stochastic differential equations are used to model various phenomena such as unstable stock prices or physical systems subject to thermal fluctuations. They have found applications in finance, signal processing, population dynamics and many other fields. It is the basis of some other applied probability areas such as filtering theory, stochastic control and stochastic differential games. To balance the theoretical and applied aspects and to include as much audience as possible, we shall focus on the stochastic differential equations driven only by Brownian motion (white noise). We will focus on the theory and not get into specic applied area (finance, signal processing, filtering, control and so on). We shall first briefly introduce some basic concepts and results on stochastic processes, in particular, the Brownian motions. Then we will discuss stochastic integrals, Ito formula, the existence and uniqueness of stochastic differential equations, some fundamental properties of the solution. We will concern with the Markov property, Kolmogorov backward and forward equations, Feynman-Kac formula, Girsanov formula. We will also concern with the ergodic theory and other stability problems. We may also mention some results on numerical simulations, Malliavin calculus and so on.

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This course offers a concise, but self-contained, introduction to the subject of mechanics, which combines geometrical view and physical insights. We will start with a formulation of classical mechanics in the framework of variational principles, translate from point to continuous systems, and analyze the effects of holonomic and nonholonomic constraints. The discussion of effects of friction and collision will naturally lead us to ergodic theory. A significant part of the course will be devoted to the geometric language of mechanics including analysis on manifolds, Lie groups, and differential topology. Among its applications, we will focus on symmetries, reduction, and geometric phase both in finite and infinite dimensions including fluid mechanics. Two key references which define the spirit of the course are “Lectures on Mechanics” by Jerrold Marsden and “Mathematical Methods of Classical Mechanics” by Vladimir Arnold.

Variational principles; celestial mechanics; holonomic and nonholonomic constraints; effects of friction and collision, ergodic theory and chaos; analysis on manifolds, Lie groups, differential topology; symmetries, reduction, and geometric phase; infinite-dimensional systems.

Registration for this course is not currently available.

The course is a first semester of algebraic topology. Broadly speaking, algebraic topology studies spaces and shapes by assigning algebraic invariants to them. Topics will include the fundamental group, covering spaces, CW complexes, homology (simplicial, singular, cellular), cohomology, and some applications.

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Much of our understanding of evolution, the process shaping the beautiful biological diversity in our world, is grounded in equally elegant mathematics. In this course we will cover the mathematical description of evolution. Involving a wide range of topics, from the analysis of non-linear dynamics to stochastic processes and partial differential equations, this course will challenge you to take mathematical principals and apply them to the natural world. Throughout this course we will focus particularly on addressing important contemporary existential questions with mathematical models, for example applications of evolution to conservation and public health.

Registration for this course is not currently available.

Cell biology provides many interesting challenges across many spatial scales. Mathematical and computational modeling are tools that can help gain a better understanding of cellular phenomena. At the small scales, there are puzzling examples of patterns formed by proteins inside cells, and dynamic rearrangement of cellular components that enable cells to actively move. At higher scales, cells sense chemical gradients, exhibit active motility, and interact with other cells to form functioning tissues and organs. Mathematical and computational models allow us to explore many of the leading questions at each of these levels. How do patterns form spontaneously? What are the limits of cell sensing? How do cells polarize and migrate in a directed manner? How does a collection of cells self-organize into a structured tissue? In this graduate course, we will explore such questions in the context of deterministic models (ordinary and partial differential equations) as well as stochastic simulations that emphasize multiscale approaches.

The course is designed to be equally suitable for mathematics graduate students looking to learn advanced modeling methods, interesting applications, and topics for further analysis, and biologists who want to understand and critically assess models and carry out advanced multiscale simulations. All participants will learn multiscale simulations (using the open source software Morpheus) to visualize behaviour that emerges from intracellular signaling systems, cell migration, and cell-cell interactions. An emphasis will be on communication across disciplines, matching students from distinct disciplines for joint presentations and projects. Learning goals, expectations, assignments, and grading will take into account the student background.

Registration for this course is not currently available.

In the second installment of OT+X series we take X=ML or machine learning. A number of problems equivalent or related to the Monge-Kantorovich Optimal Transport (OT) problem have appeared in recent years in machine learning, and data science at large. The fruitful connections between the two fields have led to several important advances impacting both. The Wasserstein metric defines a metric between probability measures, used to describe distributions over data or distributions over models, that improves upon existing metrics based on Hilbertian metrics and f-divergences, and that is now more easily amenable to efficient numerical computation.

The first part of the course will cover the mathematical basics of OT and introduce the geometry of Wasserstein spaces. The second part of the course will cover computational aspects of OT and describe the central role played by OT in convergence analysis of stochastic algorithms for deep learning, in distributionally robust statistical learning, and in combinatorial or geometrical problems arising in data science applications. The course is meant for a wide audience including graduate students and industry professionals. Prior knowledge of real analysis, probability, statistics, and machine learning will be particularly helpful. The course will be interspersed with numerical illustrations. Familiarity with coding in Python or R is a plus.

https://sites.math.washington.edu/~soumik/OTML.html

Other Information

Registration for this course is not currently available.

What can differential equations tell us about the solutions to systems of algebraic equations? Conversely, what are the special properties of differential equations, and their solutions, that “come from geometry”?

In this course, we will combine tools from both algebra and analysis in our concrete introduction to transcendental algebraic geometry. This includes the theory of differential forms and integration on families of algebraic curves, complex surfaces, and even Calabi-Yau threefolds. Along the way we will present the general theory of Fuchsian differential equations, their isomonodromic deformations, and associated completely integrable Pfaffian systems. Techniques of computation will be emphasized along with the theory.

What can differential equations tell us about the solutions to systems of algebraic equations? Conversely, what are the special properties of differential equations, and their solutions, that “come from geometry”?

In this course, we will combine tools from both algebra and analysis in our concrete introduction to transcendental algebraic geometry. This includes the theory of differential forms and integration on families of algebraic curves, complex surfaces, and even Calabi-Yau threefolds. Along the way we will present the general theory of Fuchsian differential equations, their isomonodromic deformations, and associated completely integrable Pfaffian systems. Techniques of computation will be emphasized along with the general theory.

Textbooks

Course notes and excerpts from classic papers; For general differential equations content, the excellent new textbook “Linear Differential Equations in the Complex Domain: From Classical Theory to Forefront” by Yoshishige Haraoka (Springer Lecture Notes in Mathematics, Volume 2271).

Course grade

The course grade will be based on a research project/paper, tuned to each student’s background and interests, that will be completed during the term in consultation with the professor.

Registration for this course is not currently available.

Surveys topics in information security and privacy, with the purposes of cultivating an appropriate mindset for approaching security and privacy issues and developing basic familiarity with related technical controls.

This course may not be repeated for credit.

Learning Outcomes:

1. Recognize security and privacy threats, and enumerate possible defense mechanisms and their effectiveness in a distributed computer system
2. Identify mechanisms for controlling access to a computer system, and compare and contrast their effectiveness in practice.
3. Basic understanding of cryptographic tools and techniques and their applications in securing computer systems.
4. Outlining opinions and views about ethical and legal issues related to information security, their effect on digital and privacy rights, and research and development in this domain.
5. Identify network and software related attacks, and distinguish the role of different mechanisms in protecting the system.

Tentative Outline:

1. Introduction
2. Authentication
3. Access control
4. Malware
5. Introduction to cryptography
6. Modern cryptography - symmetric key
7. Modern cryptography - public-key
8. Web security
9. Introduction to blockchain
10. Network security

Registration for this course is not currently available.

We will be exploring topics in extremal combinatorics from problems for set systems to graph theory and hypergraphs. These include extremal results for chains and antichains, intersecting set systems, isoperimetric problems, extremal numbers for graphs, extremal properties of matchings, extremal numbers for small hypergraphs, graph eigenvalues, extremal problems for graph diameter, distance transitive graphs, and some extremal results from combinatorial matrix theory.

A more detailed list of topics is available in the preliminary syllabus.

Registration for this course is not currently available.

Asymptotic Geometric Analysis (AGA) lies at the border between geometry and analysis stemming from the study of geometric properties of finite dimensional normed spaces, especially the characteristic behavior that emerges when the dimension is suitably large or tends to infinity. Time permitting we plan to cover Banach-Mazur distance between convex bodies; John’s theorem; Dvoretsky’s theorem; properties of sections and projections of convex bodies; $MM^*$-estimate; M-ellipsoids, volumetric, entropic, and probabilistic methods for finite-dimensional convex bodies. We will also discuss methods of Fourier analysis in convex geometry. The idea of this approach is to express certain geometric quantities (such as sections or projections of convex bodies) in terms of the Fourier transform and then apply methods of harmonic analysis to solve geometric problems. In particular, we will talk about the following topics: the Fourier transform and sections of convex bodies; the Busemann-Petty problem; the Fourier transform and projections of convex bodies; Shephard’s problem; extremal sections of $l_p$-balls.

Banach-Mazur distance; John’s theorem and applications; Dvoretsky’s theorem; M-ellipsoids; the Fourier transform of distributions; the Busemann-Petty problem; Shephard’s problem; Additional topics at the discretion of the instructors.

Registration for this course is not currently available.

Introduction to the algebraic invariants that distinguish topological spaces. Focuses on the fundamental group and its applications, and homology. Introduction to the basics of homological algebra.

This course introduces the algebraic invariants that distinguish topological spaces. The course will focus on the fundamental group and its applications and homology. Students will also learn the basics of homological algebra. Over the last few decades, algebraic topology has developed many applications to data science, materials science, and robotics. Whenever possible, connections to these emerging research fields will be discussed.

Registration for this course is not currently available.

MATH 560 provides a broad overview of Mathematical Biology at an introductory level. The scope is obviously subject to the limitations of time and instructor knowledge and interests - this is a HUGE area of research.

It is intended for early stage math bio grad students, general applied math grad students interested in finding out more about biology applications, and grad students in other related departments interested in getting some mathematical and computational modelling experience.

The course is organized around a sample of topics in biology that have seen a significant amount of mathematical modelling over the years. Currently, I’m including content from ecology, evolution and evolutionary game theory, epidemiology, biochemistry and gene regulation, cell biology, electrophysiology, developmental biology. However, this list changes gradually from year to year, to reflect students’ and my own interests. The mathematical modelling methods and techniques covered are those that typically arise in the biological applications listed above. For example, I will cover models using ordinary and partial differential equations, stochastic processes, agent-based models and introduce techniques from bifurcation theory, asymptotics, dimensional analysis, numerical solution methods, and parameter estimation. An emphasis will be placed on reading and discussing classic and current papers.

A complete syllabus is available on the course website