# MATH 3349, MATH 4349, MATH 6209 - Special Topics in Mathematics

### Maths Topics Class meeting Semester 2, 2021

#### Time: 10am Canberra, Melbourne, Sydney

Zoom ink is avaliable on WATTLE MSI cominuty Forum page under general announcements.

*Enrolment in all topics classes is possible only with the permission of the lecturer.*

The process for obtaining permission codes will be discussed at the meeting on 26 July

### Semester 1, 2021 - Topics classes

#### Toric Varieties and Combinatorial Methods in Algebraic Geometry - M Helmer

Prerequisite: Algebra 1 (or similar first abstract algebra course).
Things to note: Some familiarity with algebraic geometry will be helpful for students but is not required.

#### Three-Manifolds (the best manifolds) - J Licata

Prerequisite: HD in Algebraic Topology

This class will introduce the basic tools of modern three-manifolds, including incompressible surfaces, the mapping class group, Heegaard splittings, open book decompositions and Dehn surgery.

#### Dirac Operators and the Atiyah-Singer Index Theorem - B Wang

Prerequisite: Differential Geometry and Lie Groups (MATH3342)

#### Symplectic Geometry - B Parker

Prerequisite: At a minimum, you should have taken either Differential Geometry or Algebraic Topology.

Classical mechanics takes place on a phase space that mixes position and momentum. This phase space has no natural metric, but instead has a natural 2-form, called a symplectic form, so it is a symplectic manifold. Symplectic manifolds arise frequently in modern mathematics, and symplectic geometry forms a beautiful, deep, and currently active field of research.

In this course, we'll study symplectic manifolds motivated by their origin in classical mechanics. Our starting text will be Arnold's Mathematical Methods of Classical Mechanics. The class time we spend together will mostly be you explaining theorems, definitions or solutions to problems to other students --- in small groups, if enrolment is over 5 students. Assessment will be based on these presentations, some in-class quizzes, and maybe a final exam, with weighting depending on student preference.

Although this course is mainly aimed at students who have not taken last year's introduction to symplectic geometry and gauge theory, there will be roles for a range of students including those who took that course.

#### Mathematics and Climate - N Kraitzman

Prerequisites: MATH2305 and MATH2306

### MATH3351/6211 Advanced Topics in Mathematical Physics

#### 2) Introduction to Integrable Models and Knot Theory  Lecturer: Murray Batchelor

Delivery Method:  Face-to-face, if possible. Recorded if needed.

Prerequisites: Permission code required.

#### 1) Mathematical Aspects of Gauge Theory

In this half of the course we will give a mathematical introduction to gauge theory, the underlying theory of high energy particle physics. We will introduce mathematical concepts from analysis, geometry and topology as we go along.  We will closely follow the book by Bleecker , but expand on the material where needed.

Other recommended textbooks are [2,3,4].

There will be some overlap with MATH3342: Advanced Differential Geometry, but we will try to keep this to a minimum.

 D. Bleecker, Gauge Theory and Variational Principles, Dover Publications, 2005.

 Y. Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick, Analysis, Manifolds and Physics, North Holland, 1982.

 M. Nakahara, Geometry, Topology and Physics, IOP Publishing, 2003.

 T. Frankel, The Geometry of Physics - An Introduction, Cambridge University Press, 1972.

#### 2) Introduction to Integrable Models and Knot Theory

This half of the course will begin with an introduction to the Yang-Baxter equation, the so-called master key to integrability, in the context of lattice models in statistical mechanics. This will be followed by the Yang-Baxter equation in other settings, such as factorised scattering in (1+1)-dimensional quantum field theory. A number of different solutions to the Yang-Baxter equation will be discussed. Aspects to be covered include related braid-monoid algebras and their pictorial representation via loop models. This leads to an emphasis on drawing diagrams using pictorial representations, which in turn leads to the connection with knot theory. This will include the connection between Yang-Baxter integrable models and knot invariants.

There are no set textbooks for this part of the course, however a number of key review articles will be provided.

A full list of topics will be available at the meeting on February 22, 2021