Maths Special Topics Class meeting Semester 2, 2023

Date: Monday, 24 July 2023 
Time: 10am Canberra
Venue: Seminar room - 1.33, HN building #145

Semester 2, 2023 - Topics classes

• Lie Algebras, Representation Theory, and Applications - Peter Bouwknegt.
  You cannot take this class if you took it in 2022, but it is open to students who took it in 2021.

  https://programsandcourses.anu.edu.au/2023/course/MATH3351#incompatibility

• Topics in Algebra - Algebraic Curves - Ian Le.
  There may be some overlap with Anand's Algebraic Geometry class, but it should be different enough that you may take this even if you took Anand's. Prerequisites are Algebra 1 and 2; Complex Analysis is encouraged as a co-enrolment for students who haven't previously taken it.)

• K-theory and Index Theory - Bryan Wang.
  You are encourage to contact him directly before the semester begins with questions or to express interest.  (Prerequisites: Algebra 1, An 2, DG)

• Geometric Analysis - Ben Andrews.
  It will be based loosely on the book "Riemannian Geometry and Geometric Analysis" (Jost).  Prerequisites include DG, An 2, and some background in PDE (e.g., co-enrolment in MATH4202).  Algebraic Topology is useful but not required.

• Algebraic Number Theory - Angus McAndrew.
  Consider a lattice in R^2, containing all the points (x,y) where x and y are integers. What possible distances can you make by drawing a line between two points? (Give it a go!) Classical questions like these motivated the development of algebraic number theory, a discipline where we use techniques of ring theory and algebra to study number theoretic questions. This course will discuss number fields, number rings, and ideal factorisation. Some of the major topics to be explored will be ramification, Dirichlet's unit theorem, class groups, and the Galois theory of number fields. Time permitting, we may explore further number theoretic applications. It'll be fun! (Prerequisites: Algebra 1 & 2)

• Operator Algebras - James Tener.
  This reading course will cover algebras of operators acting on Hilbert spaces (particularly C* algebras) and their application to spectral theory. This topic involves a blend of functional analysis, abstract algebra, and representation theory. We use Chapters 1, 2, and 4 of Arveson’s ‘A Short Course on Spectral Theory’ as a reference. Some major results we will cover include functional calculus for self-adjoint operators and the Gelfand-Naimark theorem. There will be weekly readings and exercises. The class will meet two or three times per week for discussions based on the exercises and readings. Assessment will consist of several assignments and a written project.  Students are required to have completed Algebra 1 and Analysis 2. Complex Analysis and/or Functional Analysis may be helpful but are not required.

• Topological Transforms for shape analysis - Katharine Turner.
  It will be largely an IBL style course where the students develop the existing theory about topological transforms such as the Persistent Homology Transform. The students are given a script with definitions and statements of lemmas and theorems, and exercises of examples. They will need to write up a a journal a completed version where they add in the proofs and details of examples. There will be an exam for the theory content. They will also have individual projects one exploring applications where they read a paper (or study a new database), write a summary and present it to the class.
  Prerequisites: Having already done algebraic topology would be a significant advantage. Any student who has not done algebraic topology will need to self learn the basics of complexes and homology such as from Hatcher’s book.

Course details

For more information about this course visit the Programs & courses website.