School of Mathematical Sciences 2017 / 2018 SUMMER VACATION RESEARCH PROJECTS Summer Vacation Period 20 November 2017 to 26 February 2018 Applications are open for the 2017 / 2018 AMSI and Monash summer vacation scholarships

Please contact the supervisor for more details, prerequisites and to obtain the required formal letter of support, before applying. How to apply .................................................................................................................................... 2 Applied and Computational Mathematics .................................................................................... 3 1.

Splitting integer linear programs with a Lagrangian axe ..................................................... 3

2.

Aggregation decomposition methods in a mining application ............................................. 3

3.

Simulating Lava Flows using Smoothed Particle Hydrodynamics ...................................... 3

4.

Numerical Optimization Methods for Big Data Analytics..................................................... 4

5.

Mathematics and Optimisation for Cryogenic Electron Microscopy .................................... 4

6.

Numerical Optimisation Applied to Monte-Carlo Algorithms for Finance ............................ 4

7.

MHD modelling of the evolution of proto-spicules in the solar chromosphere .................... 4

Pure Mathematics ........................................................................................................................ 5 8.

Functionals of higher-order derivatives of curvature for surfaces ....................................... 5

9.

Parallelogram polyominoes, partitions and polynomials ..................................................... 5

10.

How many ways are there to cover a sphere? .................................................................... 5

11.

Knots, polynomials and triangulations ................................................................................ 5

12.

Gravitational Waves: a mathematical analysis ................................................................... 6

13.

Exploring combinatorial geometry ....................................................................................... 6

14.

Fourier restriction estimates ................................................................................................ 7

15.

3D axisymmetric Navier-Stokes equation. .......................................................................... 7

16.

Some topics in nonlinear dispersive equations ................................................................... 7

Statistics and Stochastic Process ................................................................................................ 8 17.

The percolation on cellular automata .................................................................................. 8

18.

Inference of biological systems using approximate Bayesian computation ........................ 8

1

How to apply What does this AMSI/Monash scholarships offer me? the chance to work on a real research project for six weeks travel and accommodation to attend the AMSICONNECT student conference in Melbourne a six-week award of $450/week (total $2,700) Eligibility These scholarships are open to students who: are currently enrolled at an AMSI Member university are in their third year doing a major in the mathematical sciences (outstanding second year students with the support of their department may apply) have a strong academic record intend to go on to honours and/or postgraduate study in the mathematical sciences, this includes students doing joint degrees that include mathematics and statistics.

To apply visit: http:vrs.amsi.org.au/ Applications close: 17 September, 2017

IMPORTANT application information for students applying through AMSI website There are two steps to this application process, namely that you are required to submit two applications for a scholarship: 1. First application to AMSI at http:vrs.amsi.org.au/ . 2. Second application to Monash University at http://www.monash.edu.au/students/scholarships/research-projects/index.html (Monash applications can be made online from 11 September to 8 October, 2017) If you are unsuccessful with the AMSI application, then you will still be considered for the Monash University scholarship, which gives you a second chance. Scholarships offered, by either AMSI or Monash, are administered by Monash University, including your weekly payments. AMSI and Monash summer vacation scholarship offers Successful AMSI scholarship applicants will receive an offer directly from AMSI, and then a subsequent offer from Monash for the same project (not a second scholarship). You must accept both offers.

2

Applied and Computational Mathematics 1. Splitting integer linear programs with a Lagrangian axe Project title: Splitting integer linear programs with a Lagrangian axe Supervisor/s: Andreas Ernst and Dhananjay Thiruvady Prerequisites: Some programming skills required (will need to learn Julia or python, but a background in Matlab or any other programming language sufficient). MTH3170 or MTH3310 desirable but not essential. Additional details: must start before the Χmas break, extra funding available to extend the project for up to 10 weeks in total if the student is suited. Brief Description: Lagrangian methods are great for splitting large real-world optimisation problems into more manageable parts. However currently this is normally done manually. This project will look for ways to analyse the mathematical structure of a collection of such optimisation problems to try to automatically find good ways to split problems. The project provides an opportunity to learn about advanced optimisation methods and develop skills in computational mathematics.

2. Aggregation decomposition methods in a mining application Project title: Aggregation decomposition methods in a mining application Supervisor/s: Andreas Ernst and Davaatseren Baatar Prerequisites: Some programming skills required (will need to learn Julia or python, but a background in Matlab or any other programming language sufficient). MTH3170 or MTH3310 desirable but not essential. Additional details: must start before the Χmas break, extra funding available to extend the project for up to 10 weeks in total if the student is suited. Brief Description: When planning open cut mines, the area around the ore body is discretised by geologists. A high resolution discretisation allows greater accuracy but is computationally very challenging. This project will study the mathematics of a proposed variable resolution approach in which small blocks are adaptively aggregated. Computational experiments will be required to determine the efficacy of the approach.

3. Simulating Lava Flows using Smoothed Particle Hydrodynamics Project title: Simulating Lava Flows using Smoothed Particle Hydrodynamics Supervisors: Hans De Sterck, Binh Nguyen The goal of this project is to explore the use of simulation methods based on Smoothed Particle Hydrodynamics for modelling lava flow. A potential application is to assess hazards posed by lava flows to villages on the slopes of volcanos. Prerequisites: You should have taken an introductory numerical methods unit, and programming experience in Python or Matlab is required for this project. Major in computational/applied mathematics, computer science, engineering, physics, or a related discipline.

3

4. Numerical Optimization Methods for Big Data Analytics Project title: Numerical Optimization Methods for Big Data Analytics Supervisor: Hans De Sterck The data revolution is reshaping science, technology and business. Large-scale optimization is emerging as a key tool in extracting useful information from the deluge of data that arises in many areas of application. In this project you will explore optimization methods for big data that include the stochastic gradient descent (SGD) method, and use them to train models from machine learning such as logistic regression, with applications in classification and recommendation systems. You will learn about the algorithms and experiment with them, and we will investigate ways to improve important algorithmic properties such as the speed of convergence. Prerequisites: You should have taken an introductory numerical methods unit, and programming experience in Python or Matlab is required for this project. Major in computational/applied mathematics, computer science, engineering, or a related discipline.

5. Mathematics and Optimisation for Cryogenic Electron Microscopy Project title: Mathematics and Optimisation for Cryogenic Electron Microscopy Supervisors: Hans De Sterck, Tiangang Cui, Hans Elmlund Cryogenic electron microscopy enables the determination of the 3D structure of macromolecules from biology and nanophysics by observing many copies of the molecules at random orientations that are used for reconstructing the 3D shape like in tomography. However, it is difficult to design algorithms that do this reliably. The goal is to explore improving the reconstruction methods using mathematicsbased optimisation approaches. Prerequisites: Programming experience in Python or Matlab is required for this project. Experience in any of numerical methods, optimisation, signal processing, or tomography is an advantage. Major in computational/applied mathematics, computer science, engineering, physics, or a related discipline.

6. Numerical Optimisation Applied to Monte-Carlo Algorithms for Finance Project title: Numerical Optimisation Applied to Monte-Carlo Algorithms for Finance Supervisors: Hans De Sterck, Gregoire Loeper The Least-Squares Monte-Carlo Algorithm is broadly used for pricing Bermudean options. In this project we explore optimisation algorithms to extend this method to representations using nonlinear basis functions. One of the keys to the success of this approach is to solve efficiently a complex optimization problem, similar to those encountered in neural networks. Prerequisites: Knowledge of or interest in numerical methods, stochastic calculus, optimisation, and programming in Python or Matlab are required for this project. Major in computational/financial/applied mathematics, computer science, engineering, or a related discipline.

7. MHD modelling of the evolution of proto-spicules in the solar chromosphere Project title: MHD modelling of the evolution of proto-spicules in the solar chromosphere Supervisors: Alina Donea About 100,000 spicules are active on the Sun’s surface at any given time. Spicules, are jets of dense gas ejected from the Sun’s chromosphere. Spicules occur at the edges of the chromospheric network,

4

where magnetic fields are stronger. The project takes advantage of the advanced numerical code which is used to perform extensive 1D/2D magneto- and hydrodynamic simulations. Our study aims at investigating the dynamics, morphology and stability of spicule-like features. The FLASH code solves the compressible Euler equations on a block-structured adaptive mesh, and its modular design permits the study of solar spicules, with different solvers. Prerequisites: Python for visualisation (basic), experience with Linux (desired), programming (basic), completed ODE,PDE maths units

Pure Mathematics 8. Functionals of higher-order derivatives of curvature for surfaces Project title: Functionals of higher-order derivatives of curvature for surfaces Supervisor: Yann Bernard The core aim of this project is to further the understanding of the Willmore energy, which arises in conformal geometry, elasticity mechanics, general relativity, and string theory. Using variational methods, the student will derive the Euler-Lagrange system of equations for critical points of energy functionals involving higher-order derivatives of curvature. With the help of Noether’s theorem, this system of higher-order equations will be reduced to a larger system of PDEs of lower order, paving the way for a rigorous analysis of the solutions of this system of equations. All which will be done is novel and could potentially appear in print at a later time. Prerequisites: contact supervisor for details.

9. Parallelogram polyominoes, partitions and polynomials Project title: Parallelogram polyominoes, partitions and polynomials Supervisor/s: Norm Do Prerequisites: The project is open to high-achieving students who have completed at least two years of their undergraduate degree and are interested in pursuing honours in Pure Mathematics. Some experience with computer programming would be desirable, but not essential.

10. How many ways are there to cover a sphere? Project title: How many ways are there to cover a sphere? Supervisor/s: Norm Do Prerequisites: The project is open to high-achieving students who have completed at least two years of their undergraduate degree and are interested in pursuing honours in Pure Mathematics. Some experience with computer programming would be desirable, but not essential.

11. Knots, polynomials and triangulations 5

Project title: Knots, polynomials and triangulations Supervisor/s: Norm Do and Josh Howie Prerequisites: The project is open to high-achieving students who have completed at least two years of their undergraduate degree and are interested in pursuing honours in Pure Mathematics. Some experience with computer programming would be desirable, but not essential.

12. Gravitational Waves: a mathematical analysis Project title: Gravitational Waves: a mathematical analysis Supervisor: Todd Oliynyk Project Abstract: Gravitational waves are variations in space-time that propagate at the speed of light. These waves are governed by the Einstein Field Equations of general relativity. Recently, gravitational waves have been directly observed by the LIGO detectors in the United States, which are extremely large (kilometres long!) and sensitive (able to detect spatial variations the size of an atom!) interferometers. The excitement surrounding the recent detection of gravitational waves is due to the fact that it opens up a whole new way of looking at the universe. This is because gravitational waves interact weakly with matter, and therefore events that create gravitational waves (for example, the merger of compact binary systems) can be seen from extremely large distances, and with exceptional clarity. In the coming years, this should allow us to look into large regions of the universe that are presently inaccessible to electromagnetic based detectors (i.e. light, microwaves, and x-ray astronomy). Project description: Due to the complexity of Einstein’s field equations, a general mathematical description of gravitational waves is presently out of reach. However, in regions where space-time is approximately flat, the mathematical description of gravitational waves is closely related to that of electromagnetic waves (i.e. radio waves, microwaves, visible light, x-rays, ect.). In fact, in the approximately flat regime, the mathematical description of gravitational waves reduces to the analysis of solutions to the flat space wave equation. a.

The goal of this project would be to understand how, after gauge fixing, the linearized vacuum Einstein field equations reduce to a constant coefficient wave equation for a symmetric two-tensor that describes perturbations of the space-time metric,

b.

to use Fourier analysis, and other partial differential equations techniques to analyse the class of solutions to the wave equations that describe gravitational waves, and

c.

to use this mathematical description to draw physical conclusions such as determining the amount of energy that is radiated from a dynamical system by gravitational waves, and to determine the dependence of the amplitude and profile of gravitational waves on the physical process that generated them.

Pre-Requisites of the Project: Multivariable calculus and Linear Algebra. Any experience with differential geometry, general relativity, partial differential equations, or Fourier analysis would be beneficial, but certainly not necessary. Commencement Date of the Project: Commencement subject to negotiation with the Supervisor project on offer between the dates 15 Jan 2017 – 24 Feb 2017

13. Exploring combinatorial geometry Project title: Exploring combinatorial geometry. Supervisor: Michael Payne

6

Combinatorial geometry is the study of the combinatorial properties of arrangements of geometric objects, and is a rich source of simply stated but difficult open problems. It combines elements of combinatorics, especially graph theory, with ideas from linear algebra, convexity theory, topology and algebraic geometry. Together we will select an open problem, study its history and existing partial solutions in the literature, and look for new ways to attack it. Here is just one example: Given any drawing of the complete graph in the plane with straight edges, how many colours are needed to colour the edges so that no two edges of the same colour cross or share an endpoint? Start by considering the case when the vertices lie on a circle. Is this the hardest case, or do some drawings require more colours?

14. Fourier restriction estimates Project title: Fourier restriction estimates Supervisor: A/Prof Zihua Guo Fourier restriction estimate is one of the core topics in harmonic analysis. In 1960s, E. M. Stein observed for the first time the restriction phenomenon of the Fourier transform, and proposed the Fourier restriction conjecture. This conjecture in 3 and higher dimensions is still open and has been studied extensively. Besides revealing a fundamental property of the Fourier transform, Fourier restriction estimate has found tremendous applications in other fields, e.g. PDEs. This project will be devoted to study the classical Fourier restriction results and its applications in PDEs, providing a friendly introduction to the fascinating world of analysis and PDEs. Prerequisites: Please contact supervisor for prerequisites.

15. 3D axisymmetric Navier-Stokes equation. Project title: 3D axisymmetric Navier-Stokes equation Supervisor: A/Prof Zihua Guo The global existence of the large smooth solution for the 3D Navier-Stokes equation is a longstanding open problem. It is one of the millennium-problems. This project will focus on the axisymmetric case. This case was known to have some more structures so that there are rich results. Especially recently the criticality for this special case was revealed by Lei-Zhang basing on the work of Chen-Fang-Zhang. Their results showed it is only logarithmically far to finally solve the problem for this case. This project will be devoted to study the classical results, the rich structure as well as the recent mentioned works. Prerequisites: Please contact supervisor for prerequisites.

16. Some topics in nonlinear dispersive equations Project title: Some topics in nonlinear dispersive equations Supervisor: A/Prof Zihua Guo In the last 30 years, there were enormous developments in the fields of nonlinear dispersive equations, e.g. KdV, Schrodinger and wave equations. Many new tools were developed and new ideas from other fields have played important roles. This project will focus on one of the following 7

topics: low regularity well-posedness problems, long-time behavior, blowup behavior, and probabilistic well-posedness. Prerequisites: Please contact supervisor for prerequisites.

Statistics and Stochastic Process 17. The percolation on cellular automata Project title: The percolation on cellular automata Supervisor/s: Dr Andrea Collevecchio & A/Prof Kais Hamza Abstract. Consider the following self-organized system on Z+Z+. The vertices are labelled using the familiar euclidean coordinates. To each vertex we assign a value in {-1,1} according to the following rule. The value at vertex (i,j) is the product of the values at (I, j - 1) and (i -1, j). The system is well defined once we provide the boundary conditions, which are the following. The vertices with coordinates (1, j), with j 1 have all the same values. The values of the vertices of the form (i, 1), with i 1 are generated at random according to a certain rule. This project investigates the main features of this system. Prerequisites: A strong background in probability and stochastic processes is necessary for this project. You must have completed at least the unit MTH3241 (or equivalent). Duration and period: 4 to 6 weeks in January and/or February.

18. Inference of biological systems using approximate Bayesian computation Project title: Inference of biological systems using approximate Bayesian computation Supervisor/s: Tianhai Tian Prerequisites: Programming experience in MATLAB or R Additional details: Student should be in campus between November 20 and December 15, and extra funding is available if student is suited.

8

Please contact the supervisor for more details, prerequisites and to obtain the required formal letter of support, before applying. How to apply .................................................................................................................................... 2 Applied and Computational Mathematics .................................................................................... 3 1.

Splitting integer linear programs with a Lagrangian axe ..................................................... 3

2.

Aggregation decomposition methods in a mining application ............................................. 3

3.

Simulating Lava Flows using Smoothed Particle Hydrodynamics ...................................... 3

4.

Numerical Optimization Methods for Big Data Analytics..................................................... 4

5.

Mathematics and Optimisation for Cryogenic Electron Microscopy .................................... 4

6.

Numerical Optimisation Applied to Monte-Carlo Algorithms for Finance ............................ 4

7.

MHD modelling of the evolution of proto-spicules in the solar chromosphere .................... 4

Pure Mathematics ........................................................................................................................ 5 8.

Functionals of higher-order derivatives of curvature for surfaces ....................................... 5

9.

Parallelogram polyominoes, partitions and polynomials ..................................................... 5

10.

How many ways are there to cover a sphere? .................................................................... 5

11.

Knots, polynomials and triangulations ................................................................................ 5

12.

Gravitational Waves: a mathematical analysis ................................................................... 6

13.

Exploring combinatorial geometry ....................................................................................... 6

14.

Fourier restriction estimates ................................................................................................ 7

15.

3D axisymmetric Navier-Stokes equation. .......................................................................... 7

16.

Some topics in nonlinear dispersive equations ................................................................... 7

Statistics and Stochastic Process ................................................................................................ 8 17.

The percolation on cellular automata .................................................................................. 8

18.

Inference of biological systems using approximate Bayesian computation ........................ 8

1

How to apply What does this AMSI/Monash scholarships offer me? the chance to work on a real research project for six weeks travel and accommodation to attend the AMSICONNECT student conference in Melbourne a six-week award of $450/week (total $2,700) Eligibility These scholarships are open to students who: are currently enrolled at an AMSI Member university are in their third year doing a major in the mathematical sciences (outstanding second year students with the support of their department may apply) have a strong academic record intend to go on to honours and/or postgraduate study in the mathematical sciences, this includes students doing joint degrees that include mathematics and statistics.

To apply visit: http:vrs.amsi.org.au/ Applications close: 17 September, 2017

IMPORTANT application information for students applying through AMSI website There are two steps to this application process, namely that you are required to submit two applications for a scholarship: 1. First application to AMSI at http:vrs.amsi.org.au/ . 2. Second application to Monash University at http://www.monash.edu.au/students/scholarships/research-projects/index.html (Monash applications can be made online from 11 September to 8 October, 2017) If you are unsuccessful with the AMSI application, then you will still be considered for the Monash University scholarship, which gives you a second chance. Scholarships offered, by either AMSI or Monash, are administered by Monash University, including your weekly payments. AMSI and Monash summer vacation scholarship offers Successful AMSI scholarship applicants will receive an offer directly from AMSI, and then a subsequent offer from Monash for the same project (not a second scholarship). You must accept both offers.

2

Applied and Computational Mathematics 1. Splitting integer linear programs with a Lagrangian axe Project title: Splitting integer linear programs with a Lagrangian axe Supervisor/s: Andreas Ernst and Dhananjay Thiruvady Prerequisites: Some programming skills required (will need to learn Julia or python, but a background in Matlab or any other programming language sufficient). MTH3170 or MTH3310 desirable but not essential. Additional details: must start before the Χmas break, extra funding available to extend the project for up to 10 weeks in total if the student is suited. Brief Description: Lagrangian methods are great for splitting large real-world optimisation problems into more manageable parts. However currently this is normally done manually. This project will look for ways to analyse the mathematical structure of a collection of such optimisation problems to try to automatically find good ways to split problems. The project provides an opportunity to learn about advanced optimisation methods and develop skills in computational mathematics.

2. Aggregation decomposition methods in a mining application Project title: Aggregation decomposition methods in a mining application Supervisor/s: Andreas Ernst and Davaatseren Baatar Prerequisites: Some programming skills required (will need to learn Julia or python, but a background in Matlab or any other programming language sufficient). MTH3170 or MTH3310 desirable but not essential. Additional details: must start before the Χmas break, extra funding available to extend the project for up to 10 weeks in total if the student is suited. Brief Description: When planning open cut mines, the area around the ore body is discretised by geologists. A high resolution discretisation allows greater accuracy but is computationally very challenging. This project will study the mathematics of a proposed variable resolution approach in which small blocks are adaptively aggregated. Computational experiments will be required to determine the efficacy of the approach.

3. Simulating Lava Flows using Smoothed Particle Hydrodynamics Project title: Simulating Lava Flows using Smoothed Particle Hydrodynamics Supervisors: Hans De Sterck, Binh Nguyen The goal of this project is to explore the use of simulation methods based on Smoothed Particle Hydrodynamics for modelling lava flow. A potential application is to assess hazards posed by lava flows to villages on the slopes of volcanos. Prerequisites: You should have taken an introductory numerical methods unit, and programming experience in Python or Matlab is required for this project. Major in computational/applied mathematics, computer science, engineering, physics, or a related discipline.

3

4. Numerical Optimization Methods for Big Data Analytics Project title: Numerical Optimization Methods for Big Data Analytics Supervisor: Hans De Sterck The data revolution is reshaping science, technology and business. Large-scale optimization is emerging as a key tool in extracting useful information from the deluge of data that arises in many areas of application. In this project you will explore optimization methods for big data that include the stochastic gradient descent (SGD) method, and use them to train models from machine learning such as logistic regression, with applications in classification and recommendation systems. You will learn about the algorithms and experiment with them, and we will investigate ways to improve important algorithmic properties such as the speed of convergence. Prerequisites: You should have taken an introductory numerical methods unit, and programming experience in Python or Matlab is required for this project. Major in computational/applied mathematics, computer science, engineering, or a related discipline.

5. Mathematics and Optimisation for Cryogenic Electron Microscopy Project title: Mathematics and Optimisation for Cryogenic Electron Microscopy Supervisors: Hans De Sterck, Tiangang Cui, Hans Elmlund Cryogenic electron microscopy enables the determination of the 3D structure of macromolecules from biology and nanophysics by observing many copies of the molecules at random orientations that are used for reconstructing the 3D shape like in tomography. However, it is difficult to design algorithms that do this reliably. The goal is to explore improving the reconstruction methods using mathematicsbased optimisation approaches. Prerequisites: Programming experience in Python or Matlab is required for this project. Experience in any of numerical methods, optimisation, signal processing, or tomography is an advantage. Major in computational/applied mathematics, computer science, engineering, physics, or a related discipline.

6. Numerical Optimisation Applied to Monte-Carlo Algorithms for Finance Project title: Numerical Optimisation Applied to Monte-Carlo Algorithms for Finance Supervisors: Hans De Sterck, Gregoire Loeper The Least-Squares Monte-Carlo Algorithm is broadly used for pricing Bermudean options. In this project we explore optimisation algorithms to extend this method to representations using nonlinear basis functions. One of the keys to the success of this approach is to solve efficiently a complex optimization problem, similar to those encountered in neural networks. Prerequisites: Knowledge of or interest in numerical methods, stochastic calculus, optimisation, and programming in Python or Matlab are required for this project. Major in computational/financial/applied mathematics, computer science, engineering, or a related discipline.

7. MHD modelling of the evolution of proto-spicules in the solar chromosphere Project title: MHD modelling of the evolution of proto-spicules in the solar chromosphere Supervisors: Alina Donea About 100,000 spicules are active on the Sun’s surface at any given time. Spicules, are jets of dense gas ejected from the Sun’s chromosphere. Spicules occur at the edges of the chromospheric network,

4

where magnetic fields are stronger. The project takes advantage of the advanced numerical code which is used to perform extensive 1D/2D magneto- and hydrodynamic simulations. Our study aims at investigating the dynamics, morphology and stability of spicule-like features. The FLASH code solves the compressible Euler equations on a block-structured adaptive mesh, and its modular design permits the study of solar spicules, with different solvers. Prerequisites: Python for visualisation (basic), experience with Linux (desired), programming (basic), completed ODE,PDE maths units

Pure Mathematics 8. Functionals of higher-order derivatives of curvature for surfaces Project title: Functionals of higher-order derivatives of curvature for surfaces Supervisor: Yann Bernard The core aim of this project is to further the understanding of the Willmore energy, which arises in conformal geometry, elasticity mechanics, general relativity, and string theory. Using variational methods, the student will derive the Euler-Lagrange system of equations for critical points of energy functionals involving higher-order derivatives of curvature. With the help of Noether’s theorem, this system of higher-order equations will be reduced to a larger system of PDEs of lower order, paving the way for a rigorous analysis of the solutions of this system of equations. All which will be done is novel and could potentially appear in print at a later time. Prerequisites: contact supervisor for details.

9. Parallelogram polyominoes, partitions and polynomials Project title: Parallelogram polyominoes, partitions and polynomials Supervisor/s: Norm Do Prerequisites: The project is open to high-achieving students who have completed at least two years of their undergraduate degree and are interested in pursuing honours in Pure Mathematics. Some experience with computer programming would be desirable, but not essential.

10. How many ways are there to cover a sphere? Project title: How many ways are there to cover a sphere? Supervisor/s: Norm Do Prerequisites: The project is open to high-achieving students who have completed at least two years of their undergraduate degree and are interested in pursuing honours in Pure Mathematics. Some experience with computer programming would be desirable, but not essential.

11. Knots, polynomials and triangulations 5

Project title: Knots, polynomials and triangulations Supervisor/s: Norm Do and Josh Howie Prerequisites: The project is open to high-achieving students who have completed at least two years of their undergraduate degree and are interested in pursuing honours in Pure Mathematics. Some experience with computer programming would be desirable, but not essential.

12. Gravitational Waves: a mathematical analysis Project title: Gravitational Waves: a mathematical analysis Supervisor: Todd Oliynyk Project Abstract: Gravitational waves are variations in space-time that propagate at the speed of light. These waves are governed by the Einstein Field Equations of general relativity. Recently, gravitational waves have been directly observed by the LIGO detectors in the United States, which are extremely large (kilometres long!) and sensitive (able to detect spatial variations the size of an atom!) interferometers. The excitement surrounding the recent detection of gravitational waves is due to the fact that it opens up a whole new way of looking at the universe. This is because gravitational waves interact weakly with matter, and therefore events that create gravitational waves (for example, the merger of compact binary systems) can be seen from extremely large distances, and with exceptional clarity. In the coming years, this should allow us to look into large regions of the universe that are presently inaccessible to electromagnetic based detectors (i.e. light, microwaves, and x-ray astronomy). Project description: Due to the complexity of Einstein’s field equations, a general mathematical description of gravitational waves is presently out of reach. However, in regions where space-time is approximately flat, the mathematical description of gravitational waves is closely related to that of electromagnetic waves (i.e. radio waves, microwaves, visible light, x-rays, ect.). In fact, in the approximately flat regime, the mathematical description of gravitational waves reduces to the analysis of solutions to the flat space wave equation. a.

The goal of this project would be to understand how, after gauge fixing, the linearized vacuum Einstein field equations reduce to a constant coefficient wave equation for a symmetric two-tensor that describes perturbations of the space-time metric,

b.

to use Fourier analysis, and other partial differential equations techniques to analyse the class of solutions to the wave equations that describe gravitational waves, and

c.

to use this mathematical description to draw physical conclusions such as determining the amount of energy that is radiated from a dynamical system by gravitational waves, and to determine the dependence of the amplitude and profile of gravitational waves on the physical process that generated them.

Pre-Requisites of the Project: Multivariable calculus and Linear Algebra. Any experience with differential geometry, general relativity, partial differential equations, or Fourier analysis would be beneficial, but certainly not necessary. Commencement Date of the Project: Commencement subject to negotiation with the Supervisor project on offer between the dates 15 Jan 2017 – 24 Feb 2017

13. Exploring combinatorial geometry Project title: Exploring combinatorial geometry. Supervisor: Michael Payne

6

Combinatorial geometry is the study of the combinatorial properties of arrangements of geometric objects, and is a rich source of simply stated but difficult open problems. It combines elements of combinatorics, especially graph theory, with ideas from linear algebra, convexity theory, topology and algebraic geometry. Together we will select an open problem, study its history and existing partial solutions in the literature, and look for new ways to attack it. Here is just one example: Given any drawing of the complete graph in the plane with straight edges, how many colours are needed to colour the edges so that no two edges of the same colour cross or share an endpoint? Start by considering the case when the vertices lie on a circle. Is this the hardest case, or do some drawings require more colours?

14. Fourier restriction estimates Project title: Fourier restriction estimates Supervisor: A/Prof Zihua Guo Fourier restriction estimate is one of the core topics in harmonic analysis. In 1960s, E. M. Stein observed for the first time the restriction phenomenon of the Fourier transform, and proposed the Fourier restriction conjecture. This conjecture in 3 and higher dimensions is still open and has been studied extensively. Besides revealing a fundamental property of the Fourier transform, Fourier restriction estimate has found tremendous applications in other fields, e.g. PDEs. This project will be devoted to study the classical Fourier restriction results and its applications in PDEs, providing a friendly introduction to the fascinating world of analysis and PDEs. Prerequisites: Please contact supervisor for prerequisites.

15. 3D axisymmetric Navier-Stokes equation. Project title: 3D axisymmetric Navier-Stokes equation Supervisor: A/Prof Zihua Guo The global existence of the large smooth solution for the 3D Navier-Stokes equation is a longstanding open problem. It is one of the millennium-problems. This project will focus on the axisymmetric case. This case was known to have some more structures so that there are rich results. Especially recently the criticality for this special case was revealed by Lei-Zhang basing on the work of Chen-Fang-Zhang. Their results showed it is only logarithmically far to finally solve the problem for this case. This project will be devoted to study the classical results, the rich structure as well as the recent mentioned works. Prerequisites: Please contact supervisor for prerequisites.

16. Some topics in nonlinear dispersive equations Project title: Some topics in nonlinear dispersive equations Supervisor: A/Prof Zihua Guo In the last 30 years, there were enormous developments in the fields of nonlinear dispersive equations, e.g. KdV, Schrodinger and wave equations. Many new tools were developed and new ideas from other fields have played important roles. This project will focus on one of the following 7

topics: low regularity well-posedness problems, long-time behavior, blowup behavior, and probabilistic well-posedness. Prerequisites: Please contact supervisor for prerequisites.

Statistics and Stochastic Process 17. The percolation on cellular automata Project title: The percolation on cellular automata Supervisor/s: Dr Andrea Collevecchio & A/Prof Kais Hamza Abstract. Consider the following self-organized system on Z+Z+. The vertices are labelled using the familiar euclidean coordinates. To each vertex we assign a value in {-1,1} according to the following rule. The value at vertex (i,j) is the product of the values at (I, j - 1) and (i -1, j). The system is well defined once we provide the boundary conditions, which are the following. The vertices with coordinates (1, j), with j 1 have all the same values. The values of the vertices of the form (i, 1), with i 1 are generated at random according to a certain rule. This project investigates the main features of this system. Prerequisites: A strong background in probability and stochastic processes is necessary for this project. You must have completed at least the unit MTH3241 (or equivalent). Duration and period: 4 to 6 weeks in January and/or February.

18. Inference of biological systems using approximate Bayesian computation Project title: Inference of biological systems using approximate Bayesian computation Supervisor/s: Tianhai Tian Prerequisites: Programming experience in MATLAB or R Additional details: Student should be in campus between November 20 and December 15, and extra funding is available if student is suited.

8