AMSI Summer School
Subject MAST90079 (2016)
Note: This is an archived Handbook entry from 2016.
Credit Points:  12.5  

Level:  9 (Graduate/Postgraduate)  
Dates & Locations:  This subject has the following teaching availabilities in 2016: Summer Term, Parkville  Taught on campus.
Timetable can be viewed here. For information about these dates, click here.  
Time Commitment:  Contact Hours: 28 hours over 4 weeks Total Time Commitment: 170 hours  
Prerequisites:  Subject coordinator approval, upon endorsement by your current supervisor and course coordinator, is required upon application.  
Corequisites:  None  
Recommended Background Knowledge:  None  
Non Allowed Subjects:  None  
Core Participation Requirements: 
For the purposes of considering request for Reasonable Adjustments under the Disability Standards for Education (Cwth 2005), and Student Support and Engagement Policy, academic requirements for this subject are articulated in the Subject Overview, Learning Outcomes, Assessment and Generic Skills sections of this entry. It is University policy to take all reasonable steps to minimise the impact of disability upon academic study, and reasonable adjustments will be made to enhance a student's participation in the University's programs. Students who feel their disability may impact on meeting the requirements of this subject are encouraged to discuss this matter with a Faculty Student Adviser and Student Equity and Disability Support: http://services.unimelb.edu.au/disability 
Subject Overview: 
The AMSI Summer School is a fourweek program hosted at RMIT by the Australian Mathematical Sciences Institute. This subject will give students the opportunity to attend subjects taught by eminent lecturers from around Australia, creating an opportunity to study areas of mathematical sciences and cognate disciplines that may not be otherwise available at the University of Melbourne. AMSI Summer School is an exciting opportunity for mathematical sciences students from around Australia to come together over the summer break to develop their skills and networks. Students can choose from a selection of available modules which are individually detailed below. No course within the AMSI Summer School program that substantially covers material available in existing University of Melbourne postgraduate Mathematics and Statistics subjects will be available. The Schoolof Mathematics and Statistics determines the subset of allowed modules that students can choose from. This subject is only available to students enrolled in the Master of Science (Mathematics and Statistics) or the Graduate Diploma in Science (Advanced) in the Mathematics and Statistics stream.  Available Modules Calculus of variations: Theory and Practice In many physical problems, the solutions we seek minimise an energy. As a consequence, these solutions will also satisfy a partial differential equation. For example, minimisers of the Dirichlet energy satisfy the Laplace equation; in another example, surfaces that minimise area satisfy the minimal surface equation. These problems have beautiful geometric, analytic and practical aspects. In this course you will see how to find the partial differential equation associated with an energy; how to ensure that minimisers exist; how to deal with constraints; how to model these problems numerically; and the techniques of stability analysis. Topics covered will include:
 Conic Programming Conic programming is one of the core areas in modern optimisation. Conic models such as semidefinite programming have broad applications in real life, but at the same time present nontrivial computational challenges that result in rich underlying theory and interesting research problems. The aim of these lectures is to go over the fundamentals of conic optimisation and give an overview of the research field. This includes the basics of structured convex analysis, modern optimisation methods, modelling and relaxation techniques, complexity and illposedness issues, and a discourse on current research directions and open problems. Topics covered will include:
 Linear Control Theory and Structured Markov Chains This course covers the core elements of linear dynamical systems, control theory and Markov chains. The field of linear dynamical systems is all about deterministic mathematical models utilizing linear differential equations. Linear control theory considers the case where the dynamical systems are modified by means of a feedback mechanism. Here system behaviour is engineered to be governed by actuators and sensed by sensors. The general constructs are system models, feedback control, observers and optimal control laws under quadratic costs. The mathematics needed to master basic linear control theory is mostly centred around linear algebra and integral transforms. Moving onto inherently stochastic models arising in some engineered and biological systems, one considers Markov chains with discrete state spaces. Such Markov chains are naturally related to linear control theory since the state transition probabilities of Markov chains evolve as a linear dynamical system.Other relationships between these objects also exist. This merits studying linear dynamical systems, control theory and Markov chains simultaneously. In this course the students will explore these related fields together with applications, computation and theory. Topics covered will include:
 Modern Numerical Methods for Diffusion Equations on Generic Grids Porous media flows models, such as those involved in oil recovery or carbon storage, are too complex to be analytically solved. Numerical approximation is the only way to obtain quantitative and qualitative information on the solutions to these models. In field applications, several engineering constraints exist that must be taken into account in the mathematical study of the models and of their approximation: discontinuous permeability, unstructured grids with degenerate cells, etc. This course covers numerical methods — mostly finite volume schemes — developed in the past 15 years to tackle the numerical approximation of diffusion models under these engineering constraints. The construction of these schemes hinges on the analytical properties of the equations. We will therefore first detail these properties, before presenting the numerical schemes. Elements of the convergence analysis of the schemes, under constraints compatible with practical applications, will be given; we will also tackle questions pertaining to the implementation of the methods. For the sake of accessibility of the course, we will only consider linear steady diffusion; however, the material covered in this course also forms the core of the techniques required to deal with nonlinear and transient realworld models. Topics covered will include:
 Projective Geometry Cayley famously said that “projective geometry is all geometry” since it has the weakest formalisation of all the interesting geometries (i.e., those with large symmetry groups) and we can easily recover the Euclidean and affine geometries by adopting extra structure. There are essentially four school’s of practice in geometry: axiomatic, synthetic, analytic and transformational. In most of the modern day secondary and tertiary education in geometry, the analytic perspective holds sway. This course redresses this situation by emphasising the synthetic and transformational approach to projective geometry. The synthetic approach recaptures what geometry is all about, whilst we bring in the modern transformational perspective initiated by Hjelmslev, Hessenberg, Thomsen and Bachmann. This course will also be an introduction to reflection groups and permutation group theory, with continual interplay with the classical geometries that we have seen along the way. Topics covered will include:
 Further information on AMSI can be found at there website  http://ss16.amsi.org.au/program/ Students may gain credit for this subject only once. 

Learning Outcomes: 
After completing this subject students should:

Assessment: 
Calculus of variations: Theory and Practice
 Conic Programming
 Linear Control Theory and Structured Markov Chains
 Modern Numerical Methods for Diffusion Equations on Generic Grids
 Projective Geometry
 Assessment will be held during the 4 weeks of the summer school with exam components within 2 weeks of the end of the summer school. 
Prescribed Texts:  Details of prescibed text will be provided after enrolment. 
Breadth Options:  This subject is not available as a breadth subject. 
Fees Information:  Subject EFTSL, Level, Discipline & Census Date 
Generic Skills: 
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:

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