### Winter 2021 Term

The purpose of the AARMS Advanced Course program is to give graduate students and upper year undergraduates from AARMS Member Universities an opportunity to take courses not offered at their home institution free of charge. In Winter 2021, the courses listed below will be offered in an exclusively online format.

Students may be able to claim credit for AARMS Advanced Courses from their home institution using mechanisms similar to the AARMS Summer School. Students hoping to receive academic credit for courses in this program are strongly encouraged to consult with their home institution about this process before the start of lectures.

Please send any questions about this program to Sanjeev Seahra.

### Important dates

- November 24, 2020: Registration opens
- December 7, 2020: Review of applications begins

### Analytic Function Theory

*Robert Milson (Dalhousie University)*

Topics include: review of analytic complex functions including topological properties of the plane, exponential, logarithmic, trigonometric and related functions, Cauchy-Riemann equations, integration and the Cauchy theorem. Cauchy’s integral formula, residues, the argument principle, applications to real integrals and zero counting, some results of conformal mapping, elliptic functions.

Minimal prerequisites include a course in vector calculus and an analysis course (real or complex).

**Scheduling:** MWF from 16:35-17:35 (Atlantic time), synchronous lectures

### Deep learning and deep reinforcement learning

*Alexander Bihlo (Memorial University)*

This course will provide a short overview of classical methods of machine learning before providing an introduction to the areas of deep learning, reinforcement learning, and deep reinforcement learning. An introduction to TensorFlow and Keras will be provided.

Undergraduate linear algebra, multivariate calculus as well as elementary probability. Elementary experience with Python.

**Scheduling:** to be announced

### Differential Equations and Dynamical Systems

*James Watmough (University of New Brunswick Fredericton)*

This course will provide an advanced introduction to differential equations and dynamical systems. Our approach to the subject will be largely geometrical and based on applications in ecology, epidemiology, and immunology. Topics to be covered include local stability of equilibria, Lyapunov functions, bifurcations of equilibria, limit cycles, bifurcations of limit cycles.

A good grounding in basic linear algebra and analysis is required, and some familiarity with Maple, Matlab, or R will be helpful. No previous exposure to differential equations is assumed.

**Scheduling:** to be announced

### General Relativity

*Alan Coley (Dalhousie University)*

*Viqar Husain (University of New Brunswick Fredericton)*

This course provides an introduction to the mathematical and physical ideas underlying the general theory of relativity. Topics covered include review of curved manifolds, followed by physics in curved spacetime, Einstein’s equations, spherical and cosmological solutions, and gravitational radiation.

Prerequisites include familiarity with partial differential equations, special relativity, and differential geometry; or permission of the instructors. Text: A first course in General Relativity (2nd. ed), B. Schutz.

**Scheduling:** to be announced

### Graph theory

*David Pike (Memorial University)*

The course will focus on concepts and proof techniques pertaining to Graph Theory. Three areas will be covered: matchings (including covers, the Konig-Egervary theorem, Hall’s theorem, Tutte’s 1-factor theorem), connectivity (including edge-connectivity, Menger’s theorem, Dirac’s fan lemma, the Chvatal-Erdos theorem), and network flows (including cuts, the Ford-Fulkerson algorithm, Menger’s theorem).

Previous exposure to basic concepts in Graph Theory, such as from anundergraduate course in the subject, is expected.

**Scheduling:** to be announced

### Harmonic Analysis

*Suresh Eswarathasan (Dalhousie University)*

Harmonic analysis is, roughly speaking, the quantitative study of functions on domains and similar objects such as measures or distributions. For simplicity, let us restrict to functions and consider the following two problems: P1) What is the most efficient way to decompose a function in a certain manner, or how does the size of it in one norm is related to the size in another? In fact, such problems are initially encountered in linear algebra. P2) Consider linear mappings that take a function as input and returns another as output, and understand how the size of the output (as quantified by various norms) relates to the size of the input. As in the first stated problem, one has already seen elementary forms of P2 when working with matrices. Our goal for the semester is understand these two infinite-dimensional linear algebra-type problems via the help of Fourier analysis, complex analysis, and abstract algebra. This area sits at the crossroads of a variety of fields in mathematics and our aim is to learn enough of the fundamentals to witness this in a single semester. Topics will include Fourier analysis on $\mathbb{R}^n$, Hausdorff/fractal dimension and measures, the Kakeya problem and its applications, and if time permits the connection between the Kakeya problem and local smoothing for wave equations.

Dalhousie students taking this course have the recommended prerequisite of Math 4010, but are allowed to enroll if they have taken Math 3501, Math 3502, and Math 2135 (or equivalent courses). Students from outside Dalhousie are expected to have similar preparation. Note that the material can be modified depending on the background of students.

**Scheduling:** Lectures will be held asynchronously and recorded. Notes will also be uploaded.

### Homological Algebra

*Yorck Sommerhauser (Memorial University)*

The course will provide an introduction to homological algebra with focus on the cohomology of groups. The course will begin with the standard resolution used to define group cohomology, motivated by the consideration of group extensions. From the standard resolution, we will proceed to general projective resolutions and explain derived functors and the long exact homology sequence. The treatment of group cohomology will be based on the book “Group theory I” by Michio Suzuki; the exposition there will be complemented from other sources for the more general theory.

Students need to know the basic concepts of linear algebra, like the notion of an abstract vector space over a field, and the basic concepts of group theory. No advanced group theory will be required.

**Scheduling:** to be announced

### Module theory

Introduction to module theory: modules, submodules, quotient modules, module homomorphisms, generators for modules, direct sums, free modules, tensor products, exact sequences, projective modules, injective modules, and flat modules. Modules over principal ideal domains. Additional topics may include homological algebra, Ext and Tor Functors, symmetric and exterior algebras.

Students should have taken introductory course(s) similar to Dalhousie’s Math 3032. Specifically, students should be familiar with: rings, fields, integral domains, ideals, quotient rings, prime and maximal ideals, factorization of polynomials, Unique Factorization Domains, Euclidean Domains.

**Scheduling:** Tuesdays & Thursdays: 11:35 am-12:55 pm (Atlantic time)

### Multivariable Methods for Statistical Learning

*Jeff Picka (University of New Brunswick Fredericton)*

This course will cover basic multivariate statistical methods which are used in statistical learning. Topics to be covered include multivariate visualization, the multivariate Normal distribution, exploratory and confirmatory factor analysis, predictive versus model-based inference, various types of classification methods, and unsupervised learning.

An advanced introduction to statistical inference (e.g. STAT 3093 at UNB) and some familiarity with R and with matrices will be required.

**Scheduling:** to be announced

### Numerical Methods for Time-Dependent Differential Equations

*Scott MacLachlan (Memorial University)*

This course will cover the development of efficient and accurate tools for numerical simulation of time-dependent ordinary and partial differential equations. Standard numerical integration techniques for ODEs will be reviewed, followed by their extension to finite-difference and finite-volume discretizations of parabolic and hyperbolic PDEs. If time permits, advanced topics including finite-element discretization will be covered.

A basic background in numerical analysis and both ODEs and PDEs is required. Completion of the Memorial course Math 6210 is normally sufficient, as would be completion of upper-level undergraduate courses in numerical analysis and differential equations. A programming background is assumed, and the course will require homework assignments to be completed in python.

**Scheduling:** to be announced

### Topics in Logic and Computation

*Peter Selinger (Dalhousie University)*

This topic of this course will be Logic and Proof Assistants. Proof assistants are software in which one can write definitions, theorems, and proofs, and the software will check the correctness of the proofs. Students will learn about a particular kind of logic called dependent type theory. This also touches upon a number of other topics, such as predicate logic and lambda calculus. I do not plan to give an overview of all existing proof assistants (there are many), but rather, we will learn to use the particular proof assistant called Agda.

This course does not have specific prerequisites. Students must be comfortable with definitions, theorems, and proofs. Some exposure to formal logic may be useful, but familiarity with informal logic will do. Since we will actually be using Agda, access to a computer is required (and is hopefully a given, considering it is an online course). Prior programming experience is not required, but students should not be afraid of using the keyboard.

**Scheduling:** MWF 08:35-09:25 (Atlantic time); mostly synchronous with lectures being recorded for later review

### Topics in PDE's and dynamical systems

*Theodore Kolokolnikov (Dalhousie University)*

Maximum/comparison principles and applications; stability of PDE’s; pattern formation and applications; Green’s functions in 2D and log asymptotics; mean first passage time with small traps; swarming. See https://www.mathstat.dal.

Familiarity with basic differential equations (such as Dalhousie Math 3120 course) would be very helpful.

**Scheduling:** to be announced