Courses: 2022-2023

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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

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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

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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.

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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.