From time to time, the AMSI member institutions offer short courses in the Advanced Collaborative Environment.

These courses are usually run outside general teaching weeks, in a short but intense period of a couple of weeks. The specialist subject matter might be of interest to students as well as academics. There is no assessment or examination.

- Be part of an AMSI Member Institution
- Have access to ACE facilities or apply for a Visimeet guest licence, to be able to attend via videoconference
- Register your participation with AMSI as per the instructions provided with the course information.

**Please note:** as these short courses are delivered as part of the Advanced Collaborative Environment, AMSI will not make provisions for travel or accommodation.

## Current short course listing

**WINTER 2017: PONCELET PORISM AND MATHEMATICAL BILLIARDS**

This course is open to postgraduate students, honours students and researchers. It is delivered via video conference through the ACE network. If you need a Visimeet licence contact Maaike Wienk at ace@amsi.org.au.

Please register by 1 July 2017 if you want to participate.

**Overview of course content**

Poncelet porism, dated in early XIXth century, represents one of the most beautiful and most important theorems in classical projective geometry. Suppose that two conics are given in the plane, together with a closed polygon inscribed in one of them and circumscribed about the other one. Then, the Poncelet porism states that there are infinitely many such polygons. Over the last few decades, the interest in the Poncelet theorem revived and exciting new advances are achieved, with implications several fields of mathematics. In this course, we will present historical review of classical results connected with Poncelet porism and some recent related developments in the mathematical billiards.

**Dates and times**

Monday 10 July – Friday 14 July, daily 11.00am-1.00pm AEST

**Venue **

AGR room, School of Mathematics and Statistics, University of Sydney

**Lecturer **

Dr Milena Radnovic (University of Sydney)

## Previous short courses

**SUMMER 2017: STOCHASTIC EQUATIONS AND PROCESSES IN PHYSICS AND BIOLOGY**

This course was delivered in February 2017.

**Overview of course content**

Fluctuations, noise and randomness determine basic properties of micro scale sized systems and are ubiquitous in nature. Our experience shows that if a classical system, no matter how large or small, is subjected to a process of a measurement, the outcome cannot be predicted with absolute certainty. Thus, the position of pollen grains, suspended in water, changes randomly in time as they jiggle due to Brownian motion; neural cells in any living organism generate a random series of electric pulses in response to the synaptic input from other cells; weather forecasts are unreliable because of the chaotic nature of global weather. In all these examples, the quantum effects can be neglected, and the origin of randomness is traced down to the complexity of the system or chaos. This course of lectures is designed to give an introduction into the phenomenological theory of stochastic equations and processes. The mathematical formalism of the stochastic calculus is explained using examples borrowed from physics and biology. The course is designed to be accessible for non-experts in the field.

**Lectures**

Lecture 1 (PDF) **updated**

Lecture 2 (PDF) **updated**

Lecture 3 (PDF) **updated**

Lecture 4 (PDF) **updated**

Lecture 5 (PDF) **updated**

Systems with interaction (PDF)

**Exercises**

Exercise 1 with solutions (PDF) **updated**

Exercise 2 with solutions (PDF) **updated**

Exercise 3 with solutions (PDF) **updated**

**Dates and times**

Monday 13 February – Friday 17 February, daily 11.00am-12.30pm

Monday 20 February -Tuesday 21 February, daily 11.00am-12.30pm

Friday 24 February, 11.00am-12.30pm

**Venue **

AMSI ACE room, Building 161, University of Melbourne

**Lecturer **

Dr Andrey Pototsky (Swinburne University)

##### Newcastle: Variational analysis and metric regularity theory

The University of Newcastle is offering a short course in the period 16 – 19 July 2013, entitled Variational analysis and metric regularity theory.

The course is open to postgraduate students, honours students and researchers who have access to Access Grid facilities.

If you are interested in participating, please register by sending an email to agr@amsi.org.au by 30 June with the following information: your full name, email address, University affiliation, and postal address.

**Overview of Course Content**

The classical regularity theory is centred around the implicit and Lyusternik-Graves theorems, on the one hand, and the Sard theorem and transversality theory, on the other. The theory (and a number of its applications to various problems of variational analysis) to be discussed in the course deals with similar problems for non-differentiable and set-valued mappings. This theory grew out of demands that came from needs of (mainly) optimization theory and subsequent understanding that some key ideas of the classical theory can be naturally expressed in purely metric terms without mention of any linear and/or differentiable structures.

**Topics to be covered**

The “theory” part of the course consists of five sections:

0. Classical theory;

1. Metric “phenomenological” theory;

2. Metric infinitesimal theory (with the concept of “slope” of DeGiorgi-Marino-Tosques at the

centre);

3. Banach theory (with subdifferentials and coderivatives as the main instrument of analysis);

4. Finite dimensional theory (mainly mappings with special structures, e.g. semi-algebraic, and non-smooth extensions of the Sard theorem and transversality theory).

In the second part of the course (some or all of) the following applications will be discussed:

5. Metric fixed point theory (with emphasis on two mappings models, e.g. F:X\to Y and G:Y\to X);

6. Subregularity, exact penalties and general approach to necessary optimality conditions (optimality

alternative);

7. Stability of solutions of systems of convex inequalities;

8. Curves of steepest descent for non-differentiable functions;

9. Von Neumann’s method of alternate projections for nonconvex sets;

10. Tame optimization and generically good behaviour;

11. Mathematical economics: extension of Debreu’s stability theorem for non-convex and non-smooth utilities.

Formally, for understanding of the course basic knowledge of functional analysis plus some acquaintance with convex analysis and nonlinear analysis in Banach spaces (e.g. Frechet and Gateaux derivatives, implicit function theorem) will be sufficient. Understanding of the interplay between analytic and geometric concepts would be very helpful.

**Dates and times**

Tuesday 16 July 2.00-4.00PM (AEST)

Wednesday 17 July 2.00-4.00Pm (AEST) ** NEW TIME ** Thursday 18 July 2:00-4:00pm (AEST)

Friday 19 July 2.00-4.00Pm (AEST)

Venue: University of Newcastle (CARMA)

Lecturer: Prof. Emeritus Alexander Ioffe, Technion, Israel

Contact person at Newcastle: Prof Jon Borwein

##### Wollongong: Geometric Flows of Curves

The University of Wollongong is offering a short course in the period 15 – 25 July 2013, entitled Geometric Flows of Curves. The course is open to postgraduate students, honours students and researchers who have access to Access Grid facilities.

If you are interested in participating, please register by sending an email to agr@amsi.org.au by 15 June 2013 with the following information: your full name, email address, University affiliation, and postal address.

**Overview of Course Content**

Geometric flows of curves are families of curves varying smoothly with time according to a given law. One example of this is the curve shortening flow, a geometric evolution equation for curves where each point moves with velocity given by the curvature vector at that point. This evolution equation is the main focus of our study in this course. Our main goal is to give in detail the original argument of Gage-Hamilton establishing that convex initial curves shrink to smooth round points. This argument has by now been replaced by much more efficient methods. The original argument contains many standard tools, and so it is hoped that a presentation of this in detail will introduce students to valuable techniques and methods with application far beyond the realm of curve flows.

A secondary goal is to improve Gage-Hamilton’s Theorem to Grayson’s Theorem, the statement that all embedded initial curves shrink to points. The original proof for this will be discussed and summarised. Time permitting, we will also detail how some modern methods due to Huisken and Andrews-Bryan have shortened and replaced many of the steps from the original proofs of these theorems.

Exercises are given throughout and form a crucial part of the learning process.

**Topics to be covered**

1. Elementary geometry of curves

2. Short-time existence for general curve flows

3. The proof of Gage-Hamilton that convex initial curves shrink to round points under curve

shortening flow

4. A discussion of Grayson’s original proof

5. A discussion of the alternative proofs of Grayson’s theorem due to Huisken and Gage-Hamilton

The course does not have any formal prerequisites apart from calculus and differential equations.Some knowledge of calculus of variations, the geometry of curves, and PDEs, would be helpful, but is not required.

The course is targeted at Honours students, since some mathematical maturity is required and a little bit of discipline to engage in the exercises given during lectures. That said, students from any year are welcome to participate.

**Dates and Times**

Monday 15 July 1.30-3.30pm (AEST)

Thursday 18 July 1.30-3.30pm (AEST)

Tuesday 23 July 1.30-3.30pm (AEST)

Thursday 25 July 1.30-3.30pm (AEST)

Venue: University of Wollongong

Lecturer: Dr Glen Wheeler, University of Wollongong

##### Summer 2012: A beginner’s course in finite volume approximation of scalar conservation laws

Monash University is offering a short course in the period 3 – 12 December 2012, entitled A beginner’s course in finite volume approximation of scalar conservation laws. Preliminary course information can be found below. The course is open to postgraduate students, honours students and researchers who have access to Access Grid facilities. If you are interested in participating, please register by sending an email to agr@amsi.org.au by Thursday 1 November with the following information: your full name, email address, University affiliation, and postal address.

**Overview of Course Content**

We will present and study some methods to discretise scalar conservation laws in 1D space dimension. Considering first the (apparently) simple linear case, we will raise a few of the stability issues that can arise when discretising these partial differential equations. We will then introduce schemes for non-linear equation, based on the notion of monotone discrete fluxes. Classical examples (Lax-Friedrichs, Godounov) of such fluxes will be given, and we will try to understand the concept of numerical diffusion arising in those schemes. Theoretical study will be performed: stability, discrete entropy inequalities, convergence in the BV case, convergence in the bounded case. The numerical diffusion introduced by monotone fluxes is a key element in the stability of the schemes, but it also leads to poor approximation of the shocks appearing in the solution. We will conclude the course by looking at some higher order methods (MUSCL techniques) which preserve stability while giving better qualitative approximations of shocks.

**Dates and Times**

Monday 3 December 2.00-4.00pm (EST)

Wednesday 5 December 2.00-4.00pm (EST)

Thursday 6 December 12.30-2.30pm (EST)

Tuesday 11 December 1.00-3.00pm (EST)

Wednesday 12 December 2.00-4.00pm (EST)

Thursday 13 December, 2.00-4.00pm (EST)

Venue: Monash University, Melbourne

Lecturer: Dr Jerome Droniou, Monash University

Course information

##### Statistical Learning and Data Analysis

As part of the IDTC Student Mid-year conference organised by the Australian Technology Network of Universities, the Industry Doctoral Training Centre presents a short course on “Statistical Learning and Data Analysis”. Data are becoming easier to collect and store and many areas of application (e.g. biology, e-commerce, astronomy) are now extremely data rich. This course is a tour through some of the main foundations and methods of contemporary statistical learning and data analysis. Related areas of data mining and machine learning will be touched upon.

The course is built around several mathematical and computational exercises.

**Tentative List of Topics**

- Recap of statistical distribution theory.
- Graphical models.
- Elements of Bayesian inference and learning.
- Missing data models.
- Elements of statistical learning.
- Support vector machines.
- Semiparametric regression.
- Use of relevant R and BUGS software.

Course Outline

**Dates and Times (EST)**

Monday 23 July, 9.30-10.30 am, 11.00am-12.00pm, 2.00-3.00pm, 3.30-4.30pm

Tuesday 24 July, 9.00-11.00am, 2.00-3.00pm, 3.30-4.30pm

Wednesday 25 July, 9.00-11.00am, 1.45-2.45pm

Thursday 26 July, 9.00-11.00am, 11.30am-12.30pm, 3.00-5.00pm

Friday 27 July, 9.00-11.00am, 1.30-3.30pm

Venue: University of Technology, Sydney

Lecturer: Distinguished Professor Matt Wand, University of Technology, Sydney

The course is aimed at postgraduate students. Please note:

- For non-IDTC participants, there will be no assessment available for participating in this course;
- To participate in the computing exercises, it will be necessary to download some software,of which you will be notified if your participation is confirmed;
- To enable calculations and exercises it is advisable to bring a laptop to the classes.

Please contact agr@amsi.org.au for more information. Registrations for this course are now closed.

##### Winter 2012: Fredholm operators and index theory

The University of Wollongong is offering a short course entitled Fredholm operators and index theory (see course information).

**Dates and Times:**

Wednesday 13 June 13:30-15:30 EST

Tuesday 19 June and Wednesday 20 June 13:30-15:30 EST

Tuesday 26 June and Wednesday 27 June 13:30-15:30 EST

Location: UOW Access Grid Room (15.113)

Lecturer: Aidan Sims (asims@uow.edu.au)

The course is open to postgraduate students, honours students and researchers. This course will go ahead subject to sufficient interest.

If you are interested in participating, please register by sending an email to agr@amsi.org.au by Wednesday 30 May COB at the latest.

Course information