MATH4349 - Special Topics in Mathematics

Topic: Maths Topics Class meeting S2 2020
Time: Jul 27, 2020 10 AM Canberra, Melbourne, Sydney

Join Zoom Meeting
https://anu.zoom.us/j/92594271447?pwd=N3FzZ3o5T1ZXWHB0NGkzRGkzb1k0QT09

Meeting ID: 925 9427 1447
Password: 623997

 

Semester 1

 

Introduction to Gauge Theory

B Wang

 

Gromov-Witten theory and mirror symmetry 

B Wang

 

Randomised Numerical Algorithms with Applications to Data Science

L Roberts

 

Mathematics & Climate

N Kraitzman

 

Fourier Analysis 

Yung

 

Vector Bundles & K-theory

V Angeltveit

 

Perverse sheaves & Deligne-Lusztig theory

Bapat & Onn

 

Fractal Tiling Theory

M Barnsley

 

Computational Algebraic Geometry

Helmer & Hegland

 

Semester 2 

 

Applied Topology

Robins & Turner

prerequisites: Algebraic Topology preferred, but speak to the lecturers

text: Ghrist's "Elementary Applied Topology"

 

 

Gauge theory and mirror symmetry (II) 

B Wang

Topics:  Categorification of the above Taubes’ theorem  (Atiyah-Floer conjecture),   Fukaya category To/From  mirror symmetry  and  any topics from the participants.   

Though as a continuation of two special topics courses of  semester one,  anyone familiar with  MATH3342/3320  and some mathematical maturity can learn on the go. 

 

 

Introduction to gauge theory and symplectic geometry

B Wang

Topics to be covered:  symplectic manifolds and Lagrangian submanifolds, Hamiltonian actions and moment maps, symplectic reduction; Principal bundles, connections and curvatures  (gauge fields and field strengths),  3 dimensional Chern-Simons theory and 2-dimensional Yang-Mills theory, Taubes’  theorem on gauge theoretic interpretation of Casson invariants.

Prerequisite:  MATH3342 (Advanced Differential Geometry) and MATH3320 (Advanced Analysis 2)

 

 

Ideals of Compact Operator

G Levitina

This will be an analysis course which, in a way, complements Analysis 3. For the first two weeks these two courses will cover pretty much the same material.  The only prerequisite is Analysis 2. 

 

 

Theory and numerical methods for time-dependent partial differential equations

K Duru

Pre-requisites: MATH2305/6, MATH3511 (optionally relevant MATH3501, MATH3015, ASTR3002)

The course will cover the fundamentals of the theory and numerical methods for time-dependent PDEs in bounded domains. It will begin with the well-posedness and stability theory of IBVPs at the PDE level.

It will cover classical numerical methods  such as FV, FE and FD methods. It will also introduce modern methods such as DG and spectral element methods. There will be special attention on the mathematical tools and techniques, like Fourier methods (including classical von Neumann analysis), energy methods, to analyze and prove numerical stability and convergence.   

Model problems will consist of advection and diffusion equations. Application problems will involve flow and wave propagation problems.

 

 

Foundations of Algebraic Geometry

Deopurkar

Prerequisites: Algebra 1, algebra 2, and knowledge of elementary topology.  Algebra 3 in the last two years is even better.

The plan (subject to change) is to read from Ravi Vakil's book-in-progress, supplemented by other sources. Luckily, Ravi is giving (and recording) lectures from this book, which are about a month ahead of our timeline. So I am planning to assign weekly reading and *viewing*. We will have (online) discussions twice a week for 1-1.5 hours. For the marks, there will be oral exam(s), and some written work, details to follow.